Degenerations of the generic square matrix.
Polar map and determinantal structure
Rainelly Cunha1

Zaqueu Ramos2

Aron Simis3

arXiv:1610.07681v4 [] 18 Oct 2017

Contents
1 Preliminaries
1.1 Review of ideal invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Homaloidal polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4
4
5

2 Degeneration by cloning
2.1 Polar behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 The ideal of the submaximal minors . . . . . . . . . . . . . . . . . . . . . .
2.3 The dual variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6
7
12
16

3 Degeneration by zeros
3.1 Polar behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 The ideal of the submaximal minors . . . . . . . . . . . . . . . . . . . . . .
3.3 The dual variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21
22
30
36

Abstract
One studies certain degenerations of the generic square matrix over a field k along
with its main related structures, such as the determinant of the matrix, the ideal generated by its partial derivatives, the polar map defined by these derivatives, the Hessian
matrix and the ideal of the submaximal minors of the matrix. The main tool comes
from commutative algebra, with emphasis on ideal theory and syzygy theory. The
structure of the polar map is completely identified and the main properties of the ideal
of submaximal minors are determined. Cases where the degenerated determinant has
non-vanishing Hessian determinant show that the former is a factor of the latter with
the (Segre) expected multiplicity, a result treated by Landsberg-Manivel-Ressayre by
geometric means. Another byproduct is an affirmative answer to a question of F. Russo
concerning the codimension in the polar image of the dual variety to a hypersurface.

Introduction
As implicit in the title, we aim at the study of certain degenerations of the generic square
matrix over a field k along with its main related structures. The degenerations one has
in mind will be carried by quite simple homomorphisms of the ground polynomial ring
generated by the entries of the matrix over k, mapping any entry to another entry or to
0

AMS Mathematics Subject Classification (2010 Revision). Primary 13C40, 13D02, 14E05, 14M12;
Secondary 13H10,14M05.
1
Under a CAPES Doctoral scholarship. The paper contains parts of this author’s ongoing PhD thesis
2
Partially supported by a CNPq post-doctoral fellowship (151229/2014-7)
3
Partially supported by a CNPq grant (302298/2014-2) and by a CAPES-PVNS Fellowship
(5742201241/2016).

1

zero. Since the resulting degenerated matrices often “forget” their generic origins, the study
of the related structures becomes a hard step.
Concretely, let M denote an m × m matrix which is a degeneration of the m × m generic
matrix and let R denote the polynomial ring over k generated by its entries. The related
structures will mean primevally the determinant f ∈ R of M, the corresponding Jacobian
ideal J ⊂ R, the Hessian matrix H(f ) of f , the polar map of f defined by its partial
derivatives, and the ideal I ⊂ R of submaximal minors. The approach throughout takes
mainly the commutative algebra side of the structures, hence ideal theory and homological
aspects play a dominant role.
The usual geometric way is to look at J as defining the base scheme of the polar map
without further details about its scheme nature. Here, a major point is to first understand
the ideal theoretic features of J such as its codimension and some of its associated prime
ideals, as well as the impact of its linear syzygies. In a second step one focus on the
intertwining between J and the ideal I ⊂ R of the submaximal minors of the matrix. As
per default, we will have J ⊂ I and typically of the same codimension. Of great interest is
to find out when I is a prime ideal and to study its potential nature as the radical of J or
of its unmixed part. As far as we know, for the question of primality there are two basic
techniques. One is based on verifying beforehand the normality of R/I, the other through a
lucky application of the results on s-generic matrices, as developed in [5] and [6]. We draw
on both, depending on the available players. A beautiful question, still open in general to
our knowledge, is to determine when the singular locus of the determinantal variety defined
by I is set-theoretic defined by the immediately lower minors, just as happens in the generic
case.
Another major focus is the structure of the polar map and its image (freely called polar
image) in terms of its birational potentiality. Since the polar map is a rational map of
projective space to itself, the image can be defined in terms of the original coordinates
(variables of R) and we may at times make this abuse. Its homogeneous coordinate ring is
known as the special fiber (or special fiber cone) of the ideal J and plays a central role in
the theory of reductions of ideals. Its Krull dimension is also known as the analytic spread
of the ideal J. In this regard, we face two basic questions: first, to compute the analytic
spread of J (equivalently, in characteristic zero, the rank of the Hessian matrix H(f ) of f );
second, to decide when J is a minimal reduction of the ideal I of submaximal minors.
Complete answers to the above questions are largely dependent upon the sort of degenerations of the generic square matrix one is looking at. Throughout this work all matrices
will have as entries either variables in a polynomial ring over a field or zeros, viewed as
degenerations of the generic square matrix. The prevailing tone of this study is to understand the effect of such degenerations on the properties of the underlying ideal theoretic
structures. By and large, the typical degeneration one has in mind consists in replacing
some of the entries of the generic matrix by way of applying a homomorphism of the ambient polynomial ring to itself. There will be noted differences regarding the behavior of a
few properties and numerical invariants, such as codimension, primality, Gorensteiness and
Cohen–Macaulayness. In particular, degenerating variables to zero is a delicate matter and
may depend on strategically located zeros.
In this work we focus on two basic instances of these situations.
The first, called informally “entry cloning” is dealt with in Section 2. Here we show
that f is homaloidal. For this, we first prove that the Jacobian ideal J has maximal linear
2

rank and that the determinant of H(f ) does not vanish, both requiring quite some tour de
force. By using the birationality criterion in Theorem 1.1, it follows that the polar map is
a Cremona map – i.e., f is a homaloidal polynomial.
We then move on to the ideal I of the submaximal minors. It will be a Gorenstein ideal
of codimension 4, a fairly immediate consequence of specialization. Showing in addition
that it is a prime ideal required a result of Eisenbud drawn upon the 2-generic property of
the generic matrix – we believe that R/I is actually a normal ring. It turns out that I is the
minimal primary component of J and the latter defines a double structure on the variety
V (I) with a unique embedded component, the latter being a linear subspace of codimension
4m − 5. An additional result is that the rational map defined by the submaximal minors
is birational onto its image. We give the explicit form of the image, through its defining
equation, a determinantal expression of degree m − 1. From the purely algebraic side, this
reflects on showing that the ideal J is not a reduction of its minimal component I.
The last topic of the section is the structure of the dual variety V (f )∗ of V (f ). Here
we show that V (f )∗ is an arithmetically Cohen–Macaulay variety that has very nearly the
structure of a ladder determinantal ideal of 2-minors and has dimension 2m−2. We close the
section by proving that f divides its Hessian determinant and has the expected multiplicity
m(m − 2) − 1 thereof in the sense of B. Segre.
The second alternative is dealt with in Section 3. We replace generic entries by zeros
in a strategic position to be explained
 in the text. For any given 1 ≤ r ≤ m − 2, the
r+1
degenerated matrix will acquire 2 zeros. We prove that the ideal J still has maximal
linear rank. This time around, the Hessian determinant vanishes and the image of the polar
map is shown to have dimension m2 − r(r + 1) − 1. Moreover, its homogeneous coordinate
ring is a ladder determinantal Gorenstein ring.
Moving over again to the dual variety V (f )∗ we find that it is a ladder determinantal
variety of dimension 2m−2 defined by 2-minors. Thus, it has codimension (m−1)2 −r(r+1)
in the polar image. This result answers a question (oral communication) of F. Russo as
to whether there are natural examples where the codimension of the dual variety in the
polar image is larger than 1, assuming that the Hessian determinant of f vanishes – here
the gap is actually arbitrarily large and in addition it is representative of a well structured
class of determinantal hypersurfaces. We note that V (f )∗ is in particular an arithmetically
Cohen–Macaulay variety. It is arithmetically Gorenstein if r = m − 2.
In the sequel, as in the previous section, our drive is the nature of the ideal I of submaximal minors. Once again, we have geometry and algebra. The main geometric result is
that these minors define a birational map onto its image and the latter is a cone over the
polar variety of f with vertex cut by r+1
coordinate hyperplanes. The algebraic results
2
are deeper in the sense that one digs into other virtually hidden determinantal ideals coming from submatrices of the degenerate matrix. These ideals come naturally while trying
to uncover the nature of the relationship between the three ideals J, I, J : I. One of the
difficulties
is that I is not anymore prime for all values of r. We conjecture that the bound

r+1
≤ m − 3 is the exact obstruction for the primeness of I (one direction is proved here).
2
The second conjectured statement is that the ring R/J is Cohen–Macaulay if and only if
r = m − 2 (the “only if” part is proved here).
For a more precise discussion we refer to the statements of the various theorems. As
a guide, the main results are contained in Theorem 2.3, Theorem 2.4, Theorem 2.5, Theorem 2.6, Theorem 3.1, Theorem 3.7 and Theorem 3.16.
3

Unless otherwise stated, we assume throughout that the ground field has characteristic
zero.

1

Preliminaries

The aim of this section is to review some notions and tools from ideal theory and its role
in birational maps, including homaloidal ones.

1.1

Review of ideal invariants

Let (R, m) denote a Notherian local ring and its maximal ideal (respectively, a standard
graded ring over a field and its irrelevant ideal). For an ideal I ⊂ m (respectively, a
homogeneous ideal I ⊂ m), the special fiber of I is the ring R(I)/mR(I). Note that this is
an algebra over the residue field of R. The (Krull) dimension of this algebra is called the
analytic spread of I and is denoted `(I).
Quite generally, given ideals J ⊂ I in a ring R, J is said to be a reduction of I if there
exists an integer n ≥ 0 such that I n+1 = JI n . An ideal shares the same radical with all
its reductions. Therefore, they share the same set of minimal primes and have the same
codimension. A reduction J of I is called minimal if no ideal strictly contained in J is a
reduction of I. The reduction number of I with respect to a reduction J is the minimum
integer n such that JI n = I n+1 . It is denoted by redJ (I). The (absolute) reduction number
of I is defined as red(I) = min{redJ (I) | J ⊂ I is a minimal reduction of I}. If R/m is
infinite, then every minimal reduction of I is minimally generated by exactly `(I) elements.
In particular, in this case, every reduction of I contains a reduction generated by `(I)
elements.
The following invariants are related in the case of (R, m):
ht(I) ≤ `(I) ≤ min{µ(I), dim(R)},
where µ(I) stands for the minimal number of generators of I. If the rightmost inequality
turns out to be an equality, one says that I has maximal analytic spread. By and large,
the ideals considered in this work will have dim R ≤ µ(I), hence being of maximal analytic
spread means in this case that `(I) = dim R.
Suppose now that R is a standard graded over a field k and I is minimally generated
by n + 1 forms of same degree s. In this case, I is more precisely given by means of a free
graded presentation
R(−(s + 1))` ⊕

X

ϕ

R(−(s + j)) −→ R(−s)n+1 −→ I −→ 0

j≥2

for suitable shifts. Of much interest in this work is the value of `. The image of R(−(s+1))`
by ϕ is the linear part of ϕ – often denoted ϕ1 . It is easy to see that the rank of ϕ1 does
not depend on the particular minimal system of generators of I. Thus, we call it the linear
rank of I. One says that I has maximal liner rank provided its linear rank is n (=rank(ϕ)).
Clearly, the latter condition is trivially satisfied if ϕ = ϕ1 , in which case I is said to have
linear presentation (or is linearly presented).
4

Note that ϕ is a graded matrix whose columns generate the (first) syzygy module of I
(corresponding to the given choice of generators) and a syzyzy of I is an element of this
module – that is, a linear relation, with coefficients in R, on the chosen generators. In this
context, ϕ1 can be taken as the submatrix of ϕ whose entries are linear forms of the standard
graded ring R. Thus, the linear rank is the rank of the matrix of the linear syzygies.
Recall the notion of the initial ideal of a polynomial ideal over a field. For this one has
to introduce a monomial order in the polynomial ring. Given such a monomial order, if
f ∈ R we denote by in(f ) the initial term of f and by in(I) the ideal generated by the
initial terms of the elements of I – this ideal is called the initial ideal of I.
There are many excellent sources for the general theory of monomial ideals and Gröbner
bases; we refer to the recent book [8].

1.2

Homaloidal polynomials

Let k be an arbitrary field. For the purpose of the full geometric picture we may assume k
to be algebraically closed. We denote by Pn = Pnk the nth projective space, where n ≥ 1.
A rational map F : Pn 99K Pm is defined by m + 1 forms f = {f0 , . . . , fm } ⊂ R := k[x] =
k[x0 , . . . , xn ] of the same degree d ≥ 1, not all null. We often write F = (f0 : · · · : fm )
to underscore the projective setup. Any rational map can without lost of generality be
brought to satisfy the condition that gcd{f0 , . . . , fm } = 1 (in the geometric terminology,
F has no fixed part). The common degree d of the fj is the degree of F and the ideal
IF = (f0 , . . . , fm ) is called the base ideal of F.
The image of F is the projective subvariety W ⊂ Pm whose homogeneous coordinate ring
is the k-subalgebra k[f ] ⊂ R after degree renormalization. Write S := k[f ] ' k[y]/I(W ),
where I(W ) ⊂ k[y] = k[y0 , . . . , ym ] is the homogeneous defining ideal of the image in the
embedding W ⊂ Pm .
We say that F is birational onto its image if there is a rational map G : Pm 99K Pn ,
say, G = (g0 : · · · : gn ), with the residue classes of the gi ’s modulo I(W ) not all vanishing,
satisfying the relation
(g0 (f ) : · · · : gn (f )) = (x0 : · · · : xn ).
(1)
(See [4, Definition 2.10 and Corollary 2.12].) When m = n and F is a birational map of Pn ,
we say that F is a Cremona map. An important class of Cremona maps of Pn comes from
the so-called polar maps, that is, rational maps whose coordinates are the partial derivatives
of a homogeneous polynomial f in the ring R = k[x0 , . . . , xn ]. More precisely:
Let f ∈ k[x] = k[x0 , . . . , xn ] be a homogeneous polynomial of degree d ≥ 2. The ideal

J = Jf =

∂f
∂f
,...,
∂x0
∂xn


⊂ k[x]



∂f
∂f
is the Jacobian (or gradient) ideal of f . The rational map Pf := ∂x
:
·
·
·
:
is called
∂xn
0
the polar map defined by f . If Pf is birational one says that f is homaloidal.
We note that the image of this map is the projective subvariety on the target whose
homogeneous coordinate ring is given by the k-subalgebra k[∂f /∂x0 , . . . , ∂f /∂xn ] ⊂ k[x]
up to degree normalization. The image of Pf is called the polar variety of f .
The following birationality criterion will be largely used in this work:
5

Theorem 1.1. [[4], Theorem 3.2] Let F : Pn 99K Pm be a rational map, given by m + 1
forms f = {f0 , . . . , fm } of a fixed degree. If dim(k[f ]) = n + 1 and the linear rank of the
base ideal IF is m (maximal possible) then F is birational onto its image.
It is a classical result in characteristic zero that the Krull dimension of the k-algebra
k[f ] coincides with the rank of the Jacobian matrix of f = {f0 , . . . , fm }. Assuming that the
ground field has characteristic zero, the above criterion says that if the Hessian determinant
h(f ) does not vanish and the linear rank of the gradient ideal of f is maximal, then f is
homaloidal.
There are many sources for the basic material in these preliminaries; we refer to [4].

2

Degeneration by cloning

Quite generally, let (ai,j )1≤i≤j≤m denote an m × m matrix where ai,j is either a variable
on a ground polynomial ring R = k[x] over a field k or ai,j = 0. One of the simplest
specializations consists in going modulo a binomial of the shape ai,j −ai0 ,j 0 , where ai,j 6= ai0 ,j 0
and ai0 ,j 0 6= 0. The idea is to replace a certain nonzero entry ai0 ,j 0 (variable) by a different
entry ai,j , keeping ai,j as it was – somewhat like cloning a variable and keeping the mold.
It seems natural to expect that the new cloning place should matter as far as the finer
properties of the ideals are concerned.
The main object of this section is the behavior of the generic square matrix under this
sort of cloning degeneration. We will use the following notation for the generic square
matrix:


x1,1
x1,2
...
x1,m−1
x1,m
 x2,1
x2,2
...
x2,m−1
x2,m 


 ..
..  ,
.
.
.
..
(2)
G :=  .
. 
.
...


xm−1,1 xm−1,2 . . . xm−1,m−1 xm−1,m 
xm,1

xm,2

...

xm,m−1

xm,m

where the entries are independent variables over a field k.
Now, we distinguish essentially two sorts of cloning: the one that replaces an entry xi0 ,j 0
by another entry xi,j such that i 6= i0 and j 6= j 0 , and the one in which this replacement has
either i = i0 or j = j 0 .
In the situation of the second kind of cloning, by an obvious elementary operation and
renaming of variables (which is possible since the original matrix is generic), one can assume
that the matrix is the result of replacing a variable by zero on a generic matrix. Such a
procedure is recurrent, letting several entries being replaced by zeros. The resulting matrix
along with its main properties will be studied in Section 3.
Therefore, this section will deal exclusively with the first kind of cloning – which, for
emphasis, could be refereed to as diagonal cloning. Up to elementary row/column operations
and renaming of variables, we assume once for all that the diagonally cloned matrix has the
6

shape


x1,1
x2,1
..
.

x1,2
x2,2
..
.

...
...

x1,m−1
x2,m−1
..
.

x1,m
x2,m
..
.









GC := 
,
...


xm−1,1 xm−1,2 . . . xm−1,m−1 xm−1,m 
xm,1
xm,2 . . . xm,m−1 xm−1,m−1

(3)

where the entry xm−1,m−1 has been cloned as the (m, m)-entry of the m × m generic matrix.
The terminology may help us remind of the close interchange between properties associated
to one or the other copy of the same variable in its place as an entry of the matrix. The
question as to whether there is a similar theory for repeatedly many diagonal cloning steps
has not been taken up in this work, but it looks challenging.
Throughout Ir (M ) denotes the ideal generated by the r-minors of a matrix M .
The following notion has been largely dealt with in [6].
An m × n matrix M of linear forms (m ≤ n) over a ground field is said to be s-generic
for some integer 1 ≤ s ≤ m if even after arbitrary invertible row and column operations,
any s of its entries are linearly independent over the field. It was proved in [6] that the
m × n generic matrix over a field is m-generic; in particular, this matrix is s-generic for any
1 ≤ s ≤ m.
Most specializations of the generic matrix fail to be s-generic for s ≥ 2 due to their very
format. However many classical matrices are 1-generic.
One of the important consequences of s-genericity is the primeness of the ideal of rminors for certain values of r. With an appropriate adaptation of the original notation, the
part of the result we need reads as follows:
Proposition 2.1. ([6, Theorem 2.1]) One is given integers 1 ≤ w ≤ v. Let G denote the
w × v generic matrix over a ground field. Let M 0 denote a w × v matrix of linear forms
in the entries of G and let further M denote a w × v matrix of linear forms in the entries
of M 0 . Let there be given an integer k ≥ 1 such that M 0 is a (w − k)-generic matrix and
such that the vector space spanned by the entries of M has codimension at most k − 1 in
the vector space spanned by the entries of M 0 . Then the ideal Ik+1 (M ) is is prime.
The following result originally appeared in [7] in a different context. It has independently
been obtained in [12, Proposition 5.3.1] in the presently stated form.
Proposition 2.2. Let M denote a square matrix over R = k[x0 , . . . , xn ] such that every
entry is either 0 or xi for some i = 1, . . . , n. Then, for each i = 0, . . . , n, the partial
derivative of f = det M with respect to xi is the sum of the (signed) cofactors of the entry
xi , in all its slots as an entry of M .

2.1

Polar behavior

Throughout we set f := det(GC) and let J = Jf ∈ R denote the gradient ideal of f , i.e.,
the ideal generated by the partial derivatives of f with respect to the variables of R, the
polynomial ring in the entries of GC over the ground field k.
Theorem 2.3. Consider the diagonally cloned matrix as in (3). One has:
7

(i) f is irreducible.
(ii) The Hessian determinant h(f ) does not vanish.
(iii) The linear rank of the gradient ideal of f is m2 − 2 (maximum possible).
(iv) f is homaloidal.
Proof. (i) We induct on m, the initial step of the induction being subsumed in the general
step.
Expanding f according to Laplace rule along the first row yields
f = x1,1 ∆1,1 + g,
where ∆1,1 is the determinant of the (m − 1) × (m − 1) cloned generic matrix obtained from
GC by omitting the first row and the first column. Note that both ∆1,1 and g belong to the
subring k [x1,2 , . . . , . . . , xm,m−1 ]. Thus, in order to show that f is irreducible it suffices to
prove that it is a primitive polynomial (of degree 1) in k [x1,2 , . . . , xm,m−1 ] [x1,1 ].
Now, on one hand, ∆1,1 is the determinant of a cloned matrix of the same type, hence it is
irreducible by the inductive hypothesis. Therefore, it is enough to see that ∆1,1 is not factor
of g. For this, one verifies their initial terms in the revlex monomial order, noting that they
are slightly modified from the generic case: in(∆1,1 ) = (x2,m−1 x3,m−2 · · · xm−1,2 )xm−1,m−1
and in(g) = in(f ) = (x1,m−1 x2,m−2 · · · xm−1,1 )xm−1,m−1 .
An alternative more sophisticated argument is to use that the ideal P of submaximal
minors has codimension 4, as shown independently in Theorem 2.4 (i) below. Since P =
(J, ∆m,m ), as pointed out in the proof of the latter proposition, then J has codimension
at least 3. Therefore, the ring R/(f ) is locally regular in codimension one, so it must be
normal. But f is homogeneous, hence irreducible.
(ii) Set v := {x1,1 , x2,2 , x3,3 , . . . , xm−1,m−1 } for the set of variables along the main diagonal. We argue by a specialization procedure, namely, consider the ring endomorphism
ϕ of R by mapping any variable in v to itself and by mapping any variable off v to zero.
Clearly, it suffices to show that by applying ϕ to the entries of the Hessian matrix H(f )
the resulting matrix M has a nonzero determinant.
Note that the partial derivative of f with respect to any xi,i ∈ v coincides with the
signed cofactor of xi,i , for i ≤ m − 2, while for i = m − 1 it is the sum of the respective
signed cofactors of xi,i corresponding to its two appearances.
By expanding each such cofactor according to the Leibniz rule it is clear that it has a
unique (nonzero) term whose support lies in v and, moreover, the remaining terms have
degree at least 2 in the variables off v.
Now, for xi,j ∈
/ v, without exception, the corresponding partial derivative coincides with
the signed cofactor. By a similar token, the Leibniz expansion of this cofactor has no term
whose support lies in v and has exactly one nonzero term of degree 1 in the variables off v.
By the preceding observation, applying ϕ to any second partial derivative of f will return
zero or a monomial supported on the variables in v. Thus, the entries of M are zeros or
monomials supported on the variables in v.
To see that the determinant of the specialized matrix M is nonzero, consider the Jacobian matrix of the set of partial derivatives {fv | v ∈ v} with respect to the variables in v.
8

Let M0 denote the specialization of this Jacobian matrix by ϕ, considered as a corresponding
submatrix of M. Up to permutation of rows and columns of M, we may write


M0 N
M=
,
P M1
where M1 has exactly one nonzero entry on each row and each column. Now, by the
way the second partial derivatives of f specialize via ϕ, as explained above, one must have
N = P = 0. Therefore, det(M) = det(M0 ) det(M1 ), so it remains to prove the nonvanishing
of these two subdeterminants.
Now the first block M0 is the Hessian matrix of the form
!
m−2
Y
g :=
xi,i x2m−1,m−1 .
i=1

This is the product of the generators of the k-subalgebra
k[x1,1 , . . . , xm−2,m−2 , x2m−1,m−1 ] ⊂ k[x1,1 , . . . , xm−2,m−2 , xm−1,m−1 ].
Clearly these generators are algebraically independent over k, hence the subalgebra is isomorphic to a polynomial ring itself. Then g becomes the product of the variables of a
polynomial ring over k. This is a classical homaloidal polynomial, hence we are done for
the first matrix block.
As for M1 , since it has exactly one nonzero entry on each row and each column, its
determinant does not vanish.
(iii) Let fi,j denote the xi,j -derivative of f and let ∆j,i stand for the (signed) cofactor
of the (i, j)th entry of the matrix GC.
The classical Cauchy cofactor formula
GC · adj(GC) = adj(GC) · GC = det(GC) Im

(4)

yields by expansion a set of linear relations involving the (signed) cofactors of GC:
m
X

xi,j ∆j,k = 0, for 1 ≤ i ≤ m − 1 and 1 ≤ k ≤ m − 2 (k 6= i)

(5)

j=1
m−1
X

xm,j ∆j,k + xm−1,m−1 ∆m,k = 0, for 1 ≤ k ≤ m − 2

(6)

j=1
m
X
j=1
m
X

xi,j ∆j,i =

m
X

xi+1,j ∆j,i+1 , for 1 ≤ i ≤ m − 3

(7)

j=1

xi,k ∆j,i = 0, for 1 ≤ j ≤ m − 3 and j < k ≤ j + 2.

(8)

i=1
m
X

xi,m−1 ∆m−2,i = 0.

i=1

9

(9)

m−1
X

xi,m ∆m−2,i + xm−1,m−1 ∆m−2,m = 0.

(10)

i=1

Since fi,j = ∆j,i for every (i, j) 6= (m − 1, m − 1) and the above relations do not involve
∆m−1,m−1 or ∆m,m then they give linear syzygies of the partial derivatives of f .
In addition, (4) yields the following linear relations:
m−1
X

xm−1,j ∆j,m + xm−1,m ∆m,m = 0

(11)

xi,m ∆m−1,i + xm−1,m ∆m−1,m−1 + xm−1,m−1 ∆m−1,m = 0

(12)

j=1
m−2
X
i=1
m−1
X

xi,m−1 ∆m,i + xm,m−1 ∆m,m = 0

(13)

xm,j ∆j,m−1 + xm,m−1 ∆m−1,m−1 + xm−1,m−1 ∆m,m−1 = 0

(14)

i=1
m−2
X
j=1
m
X

xm−1,j ∆j,m−1 + xm−1,m−1 ∆m−1,m−1 =

xm−2,j ∆j,m−2

(15)

j=1

j=1,j6=m−1
m−1
X

m
X

xm,j ∆j,m + xm−1,m−1 ∆m,m =

j=1

m
X

xm−2,j ∆j,m−2 .

(16)

j=1

As fm−1,m−1 = ∆m−1,m−1 + ∆m,m , adding (11) to (12), (13) to (14) and (15) to (16),
respectively, outputs three new linear syzygies of the partial derivatives of f . Thus one has
a total of (m − 2)(m − 1) + (m − 3) + 2(m − 2) + 3 = m2 − 2 linear syzygies of J.
It remains to show that these are independent.
For this we order the set of partial derivatives fi,j in accordance with the following
ordered list of the entries xi,j :
x1,1 , x1,2 , . . . , x1,m

x2,1 , x2,2 , . . . , x2,m

xm−1,1 , xm,1

xm−1,2 , xm,2

...
...

xm−2,1 , xm−2,2 . . . , xm−2,m ,
xm−1,m−1 , xm,m−1

xm−1,m

Here we traverse the entries along the matrix rows, left to right, starting with the first row
and stopping prior to the row having xm−1,m−1 as an entry; then start traversing the last
two rows along its columns top to bottom, until exhausting all variables.
We now claim that, ordering the set of partial derivatives fi,j in this way, the above sets
of linear relations can be grouped into the following block matrix of linear syzygies:
10




























ϕ1
0
..
.
0

ϕ2
..
.
0

...
..
.
...

0m−1
2
0m−1
2

0m
2
0m
2

...
...

..
.
0m−1
2
0m−1
2

..
.
0m
2
0m
2

0m−1
1
01m−1
01m−1

0m
1
0m
1
0m
1

..
.
ϕm−2

...
...
...

0m
2
0m
2
..
.
0m
2
0m
2

ϕ32
022
..
.
022
022

ϕ43
..
.
022
022

.
...
...

ϕm−1
m−2
022

ϕm
m−1

...
...
...

0m
1
0m
1
0m
1

021
021
021

021
021
021

...
...
...

021
021
021

021
021
021

..

xm−1,m
2xm−1,m−1
0

xm,m−1
0
2xm−1,m−1

xm−1,m−1
xm,m−1
xm−1,m












.












Let us explain the blocks of the above matrix:
• ϕ1 is the matrix obtained from the transpose GC t of GC by omitting the first column;
• ϕ2 , . . . , ϕm−2 are each a copy of GC t (up to column permutation);
• ϕr+1
=
r



xm−1,r
xm,r



xm−1,r+1
xm−1,m−1
, r = 2, . . . , m − 2; ϕm
=
m−1
xm,r+1)
xm,m−1


xm−1,m
;
xm−1,m−1

• Each 0 under ϕ1 is an m × (m − 1) block of zeros and each 0 under ϕi is an m × m
block of zeros, for i = 2, . . . , m − 3 ;
• 0cr denotes an r × c block of zeros, for r = 1, 2 and c = 2, m − 1, m.
Next we justify why these blocks make up (linear) syzygies.
First, as already observed, the relations (5) through (16) yield linear syzygies of the
partial derivatives of f . Setting k = 1 in the relations (5) and (6) the resulting expressions
can be written, respectively, as
Pm−1
Pm
j=1 xm,j f1,j + xm−1,m−1 f1,m = 0.
j=1 xi,j f1,j = 0 for all i = 2, . . . , m − 1 and
Ordering the set of partial derivatives fi,j as explained before, the coefficients of these
relations form the first matrix above


x2,1
x3,1
...
xm−1,1
xm,1


..
..
..
..


.
.
...
.
.
ϕ1 := 

 x2,m−1 x3,m−1 . . . xm−1,m−1 xm,m−1 
x2,m
x3,m . . . xm−1,m xm−1,m−1
Note that ϕ1 coincides indeed with the submatrix of GC t obtained by omitting its first
column.
Getting ϕk , for k = 2, . . . ., m − 2, is similar, namely, use again relations (5) and (6)
retrieving a submatrix of GC t excluding the kth column and replacing it with an extra
column that comes from relation (7) taking i = k − 1.
Continuing, for each r = 2, . . . , m−2 the block ϕr+1
comes from the relation (8) (setting
r
m
j = r − 1) and ϕm−1 comes from the relations (9) and (10). Finally, the lower right corner
11

3 × 3 block of the matrix of linear syzygies comes from the three last relations obtained by
adding (11) to (12), (13) to (14) and (15) to (16).
This proves the claim about the large matrix above. Counting through the sizes of
the various blocks, one sees that this matrix is (m2 − 1) × (m2 − 2). Omitting its first row
obtains a block-diagonal submatrix of size (m2 −2)×(m2 −2), where each block has nonzero
determinant. Thus, the linear rank of J attains the maximum.
(iv) By (ii) the polar map of f is dominant. Since the linear rank is maximum by (iii),
one can apply Theorem 1.1 to conclude that f is homaloidal.

2.2

The ideal of the submaximal minors

In this part we study the nature of the ideal of submaximal minors (cofactors) of GC. As
previously, J denotes the gradient ideal of f = det(GC)
Theorem 2.4. Consider the matrix GC as in (3), with m ≥ 3. Let P := Im−1 (GC) denote
the ideal of (m − 1)-minors of GC. Then
(i) P is a Gorenstein prime ideal of codimension 4.
(ii) J has codimension 4 and P is the minimal primary component of J in R.
(iii) J defines a double structure on the variety defined by P , admitting one single embedded
component, the latter being a linear space of codimension 4m − 5.
(iv) Letting Di,j denote the cofactor of the (i, j)-entry of the generic matrix (yi,j )1≤i,j≤m ,
2
2
the (m − 1)-minors ∆ = {∆i,j } of GC define a birational map Pm −2 99K Pm −1 onto
a hypersurface of degree m − 1 with defining equation Dm,m − Dm−1,m−1 and inverse
e := {Di,j | (i, j) 6= (m, m)} modulo
map defined by the linear system spanned by D
Dm,m − Dm−1,m−1 .
(v) J is not a reduction of P .
Proof. (i) Let P denote the ideal of submaximal minors of the fully generic matrix (2).
The linear form xm,m − xm−1,m−1 is regular on the corresponding polynomial ambient and
also modulo P as the latter is prime and generated in degree m − 1 ≥ 2. Since P is a
Gorenstein ideal of codimenson 4 by a well-known result (“Scandinavian complex”), then
so is P .
In order to prove primality, we first consider the case m = 3 which seems to require a
direct intervention. We will show more, namely, that R/P is normal – and, hence a domain
as P is a homogeneous ideal. Since R/P is a Gorenstein ring, it suffices to show that R/P
is locally regular in codimension one. For this consider the Jacobian matrix of P :
12

x2,2 −x2,1
x2,3
0

 0
x2,3

x3,2 −x3,1

x2,2
0

 0
x
2,2

 0
0

 0
0
0
0


0
−x1,2 x1,1
0
−x2,1 −x1,3
0
x1,1
−x2,2
0
−x1,3 x1,2
0
0
0
0
−x3,1
0
x1,1
0
−x3,2
0
x1,2
0
0
x3,2 −x3,1
0
0
x2,2
x2,1 −x3,1
0
0
2x2,2 −x3,2


0
0
0
0 

0
0 

−x1,2 x1,1 

−x1,3
0 
.
0
−x1,3 

−x2,2 x2,1 

−x2,3
0 
0
−x2,3

Direct inspection yields that the following pure powers are (up to sign) 4-minors of this
matrix: x41,3 , x42,1 , x42,2 , x42,3 , x43,1 and x43,2 . Therefore, the ideal of 4-minors of the Jacobian
matrix has codimension at least 6 = 4 + 2, thus ensuring that R/P satisfies (R1 ).
For m ≥ 4 we apply Proposition 2.1 with M 0 = G standing for an m × m generic matrix
and M = GC the cloned generic matrix as in the statement. In addition, we take k = m − 2,
so k + 1 = m − 1 is the size of the submaximal minors. Since m ≥ 4 and the vector space
codimension in the theorem is now 1, one has 1 ≤ m − 3 = k − 1 as required. Finally, the
m × m generic matrix is m-generic as explained in [6, Examples, p. 548]; in particular, it
is 2 = m − (m − 2)-generic. The theorem applies to give that the ideal P = Im−1 (GC) is
prime.
(ii) By item (i), P is a prime ideal of codimension 4. We first show that cod(J : P ) > 4,
which ensures that the radical of the unmixed part of J has no primes of codimension < 4
and coincides with P – in particular, J will turn out to have codimension 4 as stated.
For this note that P = (J, ∆m,m ), where ∆m,m denotes the cofactor of the (m, m)th
entry. From the cofactor identity we read the following relations:
m
X

xk,j ∆j,m = 0, for k = 1, . . . , m − 1;

j=1
m−1
X

xm,j ∆m,j + xm−1,m−1 ∆m,m =

j=1

m
X

x1,j ∆j,1 ;

j=1
m
X

xi,k ∆m,i = 0, for k = 1, . . . , m − 1;

i=1

Since the partial derivative fi,j of f with respect to the variable xi,j is the (signed)
cofactor ∆j,i , with the single exception of the partial derivative with respect to the variable
xm−1,m−1 , we have that the entries of the mth column and of the mth row all belong to
the ideal (J : ∆m,m ) = (J : P ). In particular, the codimension of (J : P ) is at least 5, as
needed.
In addition, since P has codimension 4 then J : P 6⊂ P . Picking an element a ∈ J : P \P
shows that PP ⊂ JP . Therefore P is the unmixed part of J.
To prove that P is actually the entire minimal primary component of J we argue as
follows. In addition, also note that P = (J, ∆m−1,m−1 ), where ∆m−1,m−1 denotes the
cofactor of the (m − 1, m − 1)th entry. From the cofactor identity we read the following
relations:
13

m
X

xk,j ∆j,m−1 + xk,m−1 ∆m−1,m−1 = 0, for k = 1, . . . , m, (k 6= m − 1);

j=1,j6=m−1
m
X

xm−1,j ∆j,m−1 + xm−1,m−1 ∆m−1,m−1 =

m
X

x1,j ∆j,1 ;

j=1

j=1,j6=m−1
m
X

xi,k ∆m−1,i + xm−1,k ∆m−1,m−1 = 0, for k = 1, . . . , m (k 6= m − 1);

i=1,i6=m−1

Then as above we have that the entries of the (m − 1)th column and the (m − 1)th row
belong to the ideal (J : ∆m−1,m−1 ) = (J : P ).
From this, the variables of the last two rows and columns of GC multiply P into J. As
is clear that P is contained in the ideal generated by these variables it follows that P 2 ⊂ J
(of course, this much could eventually be verified by inspection). Therefore, the radical of
J – i.e., the radical of the minimal primary part of J – is P .
(iii) By (ii), P is the minimal component of a primary decomposition of J. We claim
that J : P is generated by the 4m − 5 entries of GC off the upper left submatrix of size
(m − 2) × (m − 2). Let I denote the ideal generated by these entries.
As seen in the previous item, I ⊂ J : P . We now prove the reverse inclusion by writing
I = I 0 + I 00 as sum of two prime ideals, where I 0 (respectively, I 00 ) is the ideal generated
by the variables on the (m − 1)th row and on the (m − 1)th column of GC (respectively, by
the variables on the mth row and on the mth column of GC). Observe that the cofactors
∆i,j ∈ I 00 for all (i, j) 6= (m, m) and ∆i,j ∈ I 0 for all (i, j) 6= (m − 1, m − 1). Clearly, then
∆m,m ∈
/ I 00 and ∆m−1,m−1 ∈
/ I 0.
Let b ∈ J : P = J : ∆m,m , say,
X

b ∆m,m =

ai,j fi,j + afm−1,m−1

(i,j)6=(m−1,m−1)

X

=

ai,j ∆j,i + a(∆m−1,m−1 + ∆m,m )

(17)

(i,j)6=(m−1,m−1)

for certain ai,j , a ∈ R. Then
X

(b − a)∆m,m =

ai,j ∆j,i + a∆m−1,m−1 ∈ I 00 .

(i,j)6=(m−1,m−1)

Since I 00 is a prime ideal and ∆m,m ∈
/ I 00 , we have c := b − a ∈ I 00 . Substituting for a = b − c
in (17) gives
(−b + c)∆m−1,m−1 =

X

ai,j ∆j,i − c∆m,m ∈ I 0 .

(i,j)6=(m−1,m−1)

By a similar token, since ∆m−1,m−1 ∈
/ I 0 , then −b + c ∈ I 0 . Therefore
b = c − (−b + c) ∈ I 00 + I 0 = I,
14

as required.
In particular, J : P is a prime ideal which is necessarily an associated prime of prime
of R/J. As pointed out, P ⊂ J : P , hence J : P is an embedded prime of R/J. Moreover,
this also gives P 2 ⊂ J, hence J defines a double structure on the irreducible variety defined
by P .
Let Q denotes the embedded component of J with radical J : P and let Q0 denote the
intersection of the remaining embedded components of J. From J = P ∩ Q ∩ Q0 we get
J : P = (Q : P ) ∩ (Q0 : P ),
√
in particular, passing to radicals, J : P ⊂ Q0 . This shows that Q is the unique embedded component of codimension ≤ 4m − 5 and the corresponding geometric component is
supported on a linear subspace.
(iv) By Theorem 2.3 (ii), the polar map is dominant, i.e., the partial derivatives of f
generate a subalgebra of maximum dimension (= m2 − 1). Since J ⊂ P is an inclusion
in the same degree, the subalgebra generated by the submaximal minors has dimension
m2 − 1 as well. On the other hand, since P is a specialization from the generic case, it is
linearly presented. Therefore, by Theorem 1.1 the minors define a birational map onto a
hypersurface.
To get the inverse map and the defining equation of the image we proceed as follows.
Write ∆j,i for the cofactor of the (i, j)-entry of GC. For the image it suffices to show
that Dm,m − Dm−1,m−1 belongs to the kernel of the k-algebra map
ψ : k[yi,j | 1 ≤ i, j ≤ m] → k[∆] = k[∆i,j | 1 ≤ i, j ≤ m],
as it is clearly an irreducible polynomial.
Consider the following well-known matrix identity
adj(adj(GC)) = f m−2 · GC,

(18)

where adj(M ) denotes the transpose matrix of cofactors of a square matrix M . On the
right-hand side matrix we obviously see the same element as its (m − 1, m − 1)-entry as its
(m, m)-entry, namely, f m−2 xm−1,m−1 .
As to the entries of the matrix on the left-hand side, for any (k, l), the (k, l)-entry
is Dl,k (∆). Indeed, the (k, l)-entry of adj(adj(GC)) is the cofactor of the entry ∆l,k in
the matrix adj(GC). Clearly, this cofactor is the (l, k)-cofactor Dl,k of the generic matrix
(yi,j )1≤i,j≤m evaluated at ∆.
Therefore, we get (Dm,m − Dm−1,m−1 )(∆) = 0, as required.
Finally, by the same token, from (18) one deduces that the inverse map has coordinates
e
D := {Di,j | (i, j) 6= (m, m)} modulo Dm,m − Dm−1,m−1 .
(v) It follows from (iv) that the reduction number of a minimal reduction of P is m − 2.
Thus, to conclude, it suffices to prove that P m−1 6⊂ JP m−2 .
We will show that ∆m−1 ∈ P m−1 does not belong to JP m−2 .
Recall from previous passages that J is generated by the cofactors
∆l,h , with (l, h) 6= (m − 1, m − 1), (l, h) 6= (m, m)
15

and the additional form ∆m,m + ∆m−1,m−1 .
If ∆m−1 ∈ JP m−2 , we can write
X

∆m−1
m,m =

∆l,h Ql,h (∆) + (∆m,m + ∆m−1,m−1 )Q(∆)

(19)

(l,h)6=(m−1,m−1)
(l,h)6=(m,m)

where Ql,h (∆) and Q(∆) are homogeneous polynomial expressions of degree m − 2 in the
set
∆ = {∆i,j | 1 ≤ i ≤ j ≤ m}
of the cofactors (generators of P ).
Clearly, this gives a polynomial relation of degree m − 1 on the generators of P , so the
corresponding form of degree m − 1 in k[yi,j |1 ≤ i ≤ j ≤ m] is a scalar multiple of the
defining equation H := Dm,m − Dm−1,m−1 obtained in the previous item. Note that H
contains only squarefree terms. We now argue that such a relation is impossible.
Namely, observe that the sum
X
∆l,h Ql,h (∆)
(l,h)6=(m−1,m−1)
(l,h)6=(m,m)

m−2
does not contain any nonzero terms of the form α∆m−1
m,m or β∆m−1,m−1 ∆m,m . In addition,
if these two terms appear in (∆m,m + ∆m−1,m−1 )Q(∆) they must have the same scalar
coefficient, say, c ∈ k. Bring the first of these to the left-hand side of (19) to get a polynomial
m−1
relation of P having a nonzero term (1 − c)ym−1,m−1
. If c 6= 1, this is a contradiction due
to the squarefree nature of H.
On the other hand, if c = 1 then we still have a polynomial relation of P having a nonzero
m−2 . Now, if m > 3 this is again a contradiction vis-à-vis the nature of H
term ym−1,m−1 ym,m
as the nonzero terms of the latter are squarefree monomials of degree m − 1 > 3 − 1 = 2.
Finally, if m = 3 a direct checking shows that the monomial ym−1,m−1 ym,m cannot be the
support of a nonzero term in H. This concludes the statement.

2.3

The dual variety

An interesting question in general is whether f is a factor of its Hessian determinant h(f )
with multiplicity ≥ 1. If this is the case, then f is said in addition to have the expected
multiplicity (according to Segre) if its multiplicity as a factor of h(f ) is m2 −2−dim V (f )∗ −
1 = m2 − 3 − dim V (f )∗ = cod(V (f )∗ ) − 1, where V (f )∗ denotes the dual variety to the
hypersurface V (f ) (see [2]).
Theorem 2.5. Let f = det(GC). Then dim V (f )∗ = 2m − 2. In particular, the expected
multiplicity of f as a factor of h(f ) is m(m − 2) − 1.
Proof. We develop the argument in two parts:
1. dim V (f )∗ ≥ 2m − 2.
We draw on a result of Segre ([17]), as transcribed in [16, Lemma 7.2.7], to wit:
dim V (f )∗ = rank H(f )
16

(mod f ) − 2,

where H(f ) denotes the Hessian matrix of f . It will then suffice to show that H(f ) has a
submatrix of rank at least 2m modulo f . Consider the submatrix


x2,1
...
x2,m−1
x2,m


..
..
..
..


.
.
.
.
ϕ=

 xm−1,1 . . . xm−1,m−1 xm−1,m 
xm,1 . . . xm,m−1 xm−1,m−1
of GC obtained by omitting the first row. The maximal minors of this (m − 1) × m matrix
generate a codimension 2 ideal. On the other hand, ϕ has the property F1 for its Fitting
ideals. Indeed, the Fitting ideals of its generic predecessor are prime ideals, hence the
Fitting ideals of ϕ are specializations thereof, and as such each has the same codimension
as the respective predecessor. It follows from this that the ideal is of linear type ([9]). In
particular the m maximal minors of ϕ are algebraically independent over k, hence their
Jacobian matrix with respect to the entries of ϕ has rank m.
Let A0 denote an m × m submatrix thereof with det(A0 ) 6= 0. Now, f ∈ (x1,1 , . . . , x1,m )
while det(A0 ) ∈
/ (x1,1 , . . . , x1,m ). This means that det(A0 ) 6= 0 even modulo f .
Write the Hessian matrix H(f ) in the block form


0 A
,
H(f ) =
At B
where the first block row is the Jacobian matrix of the maximal minors of ϕ in the order
of the variables starting with {x1,1 , . . . , x1,m } and At denotes the transpose of A. Since A0
above is a submatrix of A of rank m modulo f , then h(f ) = det H(f ) has rank at least 2m
modulo f .
2. dim V (f )∗ ≤ 2m − 2.
Here we focus on the homogeneous coordinate ring of V (f )∗ , namely, the following
k-subalgebra of k[x]/(f )
k[∂f /∂x1,1 , . . . , ∂f /∂xm,m−1 ]/(f ) ' k[yi,j | 1 ≤ i ≤ j ≤ m, (i, j) 6= (m, m)]/P,

(20)

for a suitable prime ideal P , the homogeneous defining ideal of V (f )∗ . The isomorphism is
an isomorphism of graded k-algebras induced by the assignment yi,j 7→ ∂f /∂xi,j , (i, j) 6=
(m, m).
Claim 1. The homogeneous defining ideal P of V (f )∗ contains the ladder determinantal
ideal generated by the 2 × 2 minors of the following matrix


y1,1
y1,2
...
y1,m−2
y1,m
y1,m−1
 y2,1
y2,2
...
y2,m−2
y2,m
y2,m−1 




..
..
.
.
.
.
.
.


.
.
.
.
.
L=
.
 ym−2,1 ym−2,2 . . . ym−2,m−2 ym−2,m ym−2,m−1 



 ym−1,1 ym−1,2 . . . ym−1,m−2 ym−1,m
ym,1
ym,2 . . . ym,m−2
To see this, we first recall that, for (i, j) 6= (m − 1, m − 1), the partial derivative ∂f /∂xi,j
coincides with the cofactor of xi,j in GC. Since ym−1,m−1 and ym,m are not entries of L,
17

the polynomial relations of the partial derivatives possibly involving the variables which are
entries of L are exactly relations of the cofactors other than the cofactor of xm−1,m−1 .
Thus, we focus on these factors, considering the following relation afforded by the cofactor identity:
adj(GC) · GC ≡ 0 (mod f ).
(21)
Further, for each pair of integers i, j such that 1 ≤ i < j ≤ m let Fij denote the 2 × m
submatrix of adj(GC) consisting of the ith and jth rows. In addition, let C stand for the
m × (m − 1) submatrix of GC consisting of its m − 1 leftmost columns. Then (46) gives the
relations
Fij C ≡ 0 (mod f ),
for all 1 ≤ i < j ≤ m. From this, since the rank of C modulo (f ) is obviously still m − 1,
the one of every Fi,j is necessarily 1. This shows that every 2 × 2 minor of adj(GC) vanishes
modulo (f ). Therefore, every such minor that does not involve either one of the cofactors
∆m−1,m−1 and ∆m,m gives a 2 × 2 minor of L vanishing on the partial derivatives. Clearly,
by construction, we obtain this way all the 2 × 2 minors of L. This proves the claim.
Now, since I2 (L) is a ladder determinantal ideal on a suitable generic matrix it is a
Cohen-Macaulay prime ideal (see [15] for primeness and [10] for Cohen–Macaulayness).
Moreover, its codimension is m(m−2)−2 = m2 −3−(2m−1) as follows from an application
to this case of the general principle in terms of maximal chains as described in [10, Theorem
4.6 and Corollary 4.7].
Note that by I2 (L) we understand the ideal generated by the 2 × 2 minors of L in the
polynomial ring A := k[I1 (L)1 ] spanned by the entries of L. Clearly, its extension to the full
polynomial ring B := k[yi,j | 1 ≤ i ≤ j ≤ m, (i, j) 6= (m, m)], is still prime of codimension
m(m − 2) − 2. Thus, P contains a prime subideal of codimension m(m − 2) − 2.
In addition, direct checking shows that the following two quadrics
g := y1,1 ym,m−1 − y1,m−1 ym,1 , h := y1,1 ym−1,m−1 − ym−1,1 y1,m−1 − ym,1 y1,m
belong to P and, furthermore:
Claim 2. {g, h} is a regular sequence modulo I2 (L)B.
It suffices to prove the assertion locally at the powers of y1,1 since I2 (L)B is prime and
y1,1 ∈
/ I2 (L)B. Now, locally at y1,1 and at the level of the ambient rings, f, g are like the
variables ym,m−1 and ym−1,m−1 , hence one has
By1,1 /(g, h) ' Ay1,1 .
Setting I = I2 (L) for lighter reading, it follows that
(By1,1 /IBy1,1 )/(g, h)(By1,1 /IBy1,1 ) ' By1,1 /(g, h, I)
' (By1,1 /(g, h))/((g, h, I)/(g, h))
' Ay1,1 /IAy1,1 .
But clearly, By1,1 /IBy1,1 ' (Ay1,1 /IAy1,1 )[ym−1,m−1 , ym,m−1 ]y1,1 , hence
dim Ay1,1 /IAy1,1 = dim By1,1 /IBy1,1 − 2.
18

This shows that
dim(By1,1 /IBy1,1 )/(g, h)(By1,1 /IBy1,1 ) = dim By1,1 /IBy1,1 − 2
and since By1,1 /IBy1,1 is Cohen–Macaulay, this proves the claim.
Summing up we have shown that P has codimension at least m(m−2)−2+2 = m(m−2).
Therefore, dim V (f )∗ = m2 − 2 − cod V (f )∗ ≤ m2 − 2 − m(m − 2) = 2m − 2, as was to be
shown.
Theorem 2.6. Let f = det(GC). Then f is a factor of its Hessian determinant h(f ) and
has the expected multiplicity as such.
Proof. Fix the notation of Theorem 2.4 (iv) and its proof in the next subsection. As seen
there, the identity
adj(adj(GC)) = det(GC)m−2 · GC
e :=
shows that the inverse map to the birational map defined by the minors ∆ has D
{Di,j | (i, j) 6= (m, m)} as its set of coordinates. By (1) one has the equality
e ◦ ∆(x) = f m−2 · (x).
D

(22)

Applying the chain rule to (22) yields
e
Θ(D)(∆)
· Θ(∆) = f m−2 · I +

1
f m−3 · (x)t · Grad(f ),
m−2

(23)

where Θ(S) denotes the Jacobian matrix of a set S of polynomials, Grad(f ) stands for the
row vector of the partial derivatives of f and I is the identity matrix of order m − 1.
Write



B
e
Θ(D)(∆) = A U , Θ(∆) =
,
V
e with
where U designates the column vector of the partial derivatives of the elements of D
respect to ym,m further evaluated at ∆, while V stands for the row vector of the partial
derivatives of ∆m,m with respect to the x-variables.
Letting E denote the elementary matrix obtained from I by adding the mth row to the
(m − 1)th row, one further has
!

H
f
e
Θ(D)(∆)
· E −1 = A0 U , E · Θ(∆) =
V
where Hf stands for the Hessian matrix of f with respect to the x-variables.
Applying these values to (23) obtains
A0 · Hf = f m−2 · I + A

(24)

1
where A = m−2
f m−3 · (x)t · Grad(f ) − U · V .
On the other hand, note that (∂Dij /∂yr,s ) (∆) is the (m−2)-minor of adj(GC) ommiting
the ith and rth rows and the jth and sth columns. Therefore, by a classical identity

19

(∂Dij /∂yr,s ) (∆) = f m−3 (∆i,j ∆r,s − ∆i,s ∆r,j ).

(25)

As a consequence, U = f m−3 · U 0 where the entries of U 0 are certain 2-minors of GC. We
now get
A0 · Hf = f m−3 (f · I + A0 )
where A0 =

1
m−2

· (x)t · Grad(f ) − U 0 · V. Thus,
det(A0 ) · det(Hf ) = f (m−3)(m

2 −1)

det(f · I + A0 )

(26)

Set n := m2 −1, [n] := {1, . . . , n} and let ∆[n]\{i1 ,...,ik } denote the principal (n−k)-minor
of A0 with rows and columns [n] \ {i1 , . . . , ik }. Also note that A0 has rank at most 2 since
it is a sum of matrices of rank 1. Using a classical formula for the determinant of a sum
where one of the summands is a diagonal matrix (see, e.g., [18, Lemma 2.3]), one has
det (f · I + A0 ) = det(A0 ) + f

∆[n]−{i}





!
X

X

+ . . . + f n−1 

∆[n]\{i1 ,...,in−1 }  + f n

1≤i1 <...<in−1 ≤n

i




X

= f n−2 

∆[n]\{i1 ,...,in−2 }  + f n−1 · trace(A0 ) + f n .

1≤i1 <...<in−2 ≤n

Setting G :=

X

∆[n]\{i1 ,...,in−2 } + f · trace(A0 ) + f 2 and substituting in (26)

1≤i1 <...<in−2 ≤n

gives
det(A0 ) · det(Hf ) = f (m−3)(m

2 −1)+m2 −3

· G = f (m

2 −1)(m−2)−2

·G

(27)

Suppose for a moment that G 6= 0. In this case, clearly det(A0 ) 6= 0 and a degree
argument shows that some positive power of f divides h(f ). Indeed, by construction,
2 −1)(m−2)−2

deg(det(A0 )) = (m2 − 1)(m(m − 3) + 2) < m((m2 − 1)(m − 2) − 2) = deg(f (m

).

Therefore, we are left with proving that G does not vanish.
Note that the vanishing of G would imply in particular that f is an integral element
over the k-subalgebra generated by the entries of A0 . This would possibly be forbidden if
the latter could be proved to be integrally closed. Due to the difficulty of this verification
we resort to a direct inspection. For this, note that
trace(A0 ) =
=

1
trace((x)t · Grad(f )) − trace(U 0 V )
m−2
X
∂∆m,m
m
f−
(aij am,m − ai,m amj )
m−2
∂xij
ij
i,j6=m


=

=

m

f − amm
m−2


X

aij

ij

∂∆m,m
−
∂xij

X
ij
i,j6=m

aim amj

∂∆m,m 

∂xij

X
∂∆m,m
m
f − (m − 1)am,m ∆m,m +
aim amj
.
m−2
∂xij
ij
i,j6=m

20

Setting qij := aij amm − aim amj , one has

X

∆[n]\{i1 ,...,in−2 }

=

1≤i1 <...<in−2 ≤n

X

det

(ij),(rs)

=

X


det

(ij),(rs)

=

X


det

(ij),(rs)

∂f
mm
xij ∂x
− qij ∂∆
∂xij
ij

∂f
mm
xij ∂x
− qij ∂∆
∂xrs
rs

∂f
mm
xrs ∂x
− qrs ∂∆
∂xij
ij

∂f
mm
xrs ∂x
− qrs ∂∆
∂xrs
rs
!
∂f
∂f



xij
xrs

−qij
−qrs

xij
xrs

aim amj
arm ams

· det

∂xij
∂∆mm
∂xij


· det

Thus, we can write G = G1 + G2 , where




X
X
∂∆m,m 
xij aim amj

G1 = f 
aim amj
det
· det
+
xrs arm ams
∂xij
ij
(ij),(rs)

i,j6=0

and


G2 = f

m
· f − (m − 1)am,m ∆m,m
m−2



!

∂xrs
∂∆mm
∂xrs

∂f
∂xij
∂∆mm
∂xij

∂f
∂xij
∂∆mm
∂xij

∂f
∂xrs
∂∆mm
∂xrs

!

∂f
∂xrs
∂∆mm
∂xrs

!

+ f 2.

Inspecting the summands of G1 and G2 one sees that the degree of xm−1,m−1 in G1 is at
most 3, while that of xm−1,m−1 in G2 is 4. This shows that G 6= 0.
Remark 2.7. (1) The factor h(f )/f m(m−2)−1 coincides with the determinant of the 2 × 2
submatrix with rows m − 1, m and columns m − 1, m.
(2) There is a mistaken assertion in the proof of [13, Proposition 3.2 (a)] to the effect that
the dual variety to the generic determinant f is the variety of submaximal minors. This is of
course nonsense since the dual variety is the variety of the 2×2 minors. This wrong assertion
in loc. cit. actually serves no purpose in the proof since (a) follows simply from the cofactor
relation. At the other end, this nonsense reflects on the second assertion of item (c) of the
same proposition which is therefore also flawed. The right conclusion is that the multiplicity
of f as a factor of its Hessian has indeed the expected multiplicity, since the variety of the
2 × 2 minors has codimension (m − 1)2 and hence m2 − 1 − (m − 1)2 = m(m − 2) as desired.
Likewise, [13, Conjecture 3.4 (a)] should be read as affirmative without exception.
(3) We have been kindly informed by J. Landsberg that either Theorem 2.5 or Theorem 2.6, or perhaps both – which were stated as a conjecture in the first version of this
prepint posted on the arXiv – have been obtained in [11] by geometric means. For our misfortune, we were not able to trace in the mentioned work the precise statements expressing
the above contents.

3

Degeneration by zeros

Recall from the previous section the cloning degeneration where an entry is cloned along the
same row or column of the original generic matrix. As mentioned before, up to elementary
operations of rows and/or columns the resulting matrix has a zero entry. A glimpse of this
first status has been tackled in [13, Proposition 4.9 (a)].
21

This procedure can be repeated to add more zeros. Aiming at a uniform treatment of
all these cases, we will fix integers m, r with 1 ≤ r ≤ m − 2 and consider the following
degeneration of the m × m generic matrix:
x1,1
..


.

 xm−r,1

 xm−r+1,1

 xm−r+2,1

..


.

 xm−1,1

xm,1


...

x1,m−r
..
.

x1,m−r+1
..
.

x1,m−r+2
..
.

...
...

xm−r,m−r
xm−r+1,m−r
xm−r+2,m−r
..
.
xm−1,m−r

xm−r,m−r+1
xm−r+1,m−r+1
xm−r+2,m−r+1
..
.
xm−1,m−r+1

...

xm,m−r

0

...
...
...
...

...

x1,m
..
.

xm−r,m−r+2
xm−r+1,m−r+2
xm−r+2,m−r+2
..
.
0

...
...
...
...
.
..
...

x1,m−1
..
.
xm−r,m−1
xm−r+1,m−1
0
..
.
0

xm−r,m
0
0
..
.
0

0

...

0

0

















(28)

Assuming m is fixed in the context, let us denote the above matrix by DG(r).

3.1

Polar behavior

Theorem 3.1. Let R = k[x] denote the polynomial ring in the nonzero entries of DG(r),
with 1 ≤ r ≤ m − 2, let f := det DG(r) and let J ⊂ R denote the gradient ideal of f . Then:
(a) f is irreducible.
(b) J has maximal linear rank.
r+1
2
(c) The homogeneous coordinate ring of the polar variety of f in Pm −( 2 )−1 is a Gorenstein ladder determinantal ring of dimension m2 − r(r + 1); in particular, the analytic
spread of J is m2 − r(r + 1).

Proof. (a) Expanding the determinant by Laplace along the first row, we can write f =
x1,1 ∆1,1 + g, where ∆1,1 is the cofactor of x1,1 . Clearly, both ∆1,1 and g belong to the
polynomial subring omitting the variable x1,1 . Thus, in order to show that f is irreducible
it suffices to prove that it is a primitive polynomial (of degree 1) in k[x1,2 , . . . , xm,m−r ][x1,1 ].
In other words, we need to check that no irreducible factor of ∆1,1 is a factor of g.
We induct on m ≥ r + 2. If m = r + 2 then ∆1,1 = x2,m x3,m−1 · · · xm−1,3 xm,2 , while the
initial term of g in the revlex monomial order is
in(g) = in(f ) = x1,m x2,m−1 · · · xm,1 .
Thus, assume that m > r + 2. By the inductive step, ∆1,1 is irreducible being the determinant of an (m − 1) × (m − 1) matrix of the same kind (same r). But deg(∆1,1 ) = deg(g) − 1.
Therefore, it suffices to show that ∆1,1 is not a factor of g. Supposing it were, we would
get that f is multiple of ∆1,1 by a linear factor – this is clearly impossible.
Once more, an alternative argument is to use that the ideal J has codimension 4, as will
be shown independently in Theorem 3.7 (b). Therefore, the ring R/(f ) is locally regular in
codimension at least one, so it must be normal. But f is homogeneous, hence irreducible.
(b) The proof is similar to the one of Theorem 2.3 (iii), but there is a numerical diversion
and, besides, the cases where r > m − r − 1 and r ≤ m − r − 1 keep slight differences.
22

Let fi,j denote the xi,j -derivative of f and let ∆j,i stand for the (signed) cofactor of xi,j
on DG(r). We first assume that r > m − r − 1. The Cauchy cofactor formula
DG(r) · adj(DG(r)) = adj(DG(r)) · DG(r) = det(DG(r)) Im
yields by expansion the following three blocks of linear relations involving the (signed)
cofactors of DG(r):
 Pm
for 1 ≤ i ≤ m − r, 1 ≤ k ≤ m − r (k 6= i)

 Pj=1 xi,j ∆j,k = 0
m−l
for 1 ≤ l ≤ r, 1 ≤ k ≤ m − r
j=1 xm−r+l,j ∆j,k = 0

 Pm x ∆ − Pm x
j=1 i+1,j ∆j,i+1 = 0 for 1 ≤ i ≤ m − r − 1
j=1 i,j j,i

(29)

with m2 − rm − 1 such relations;
( Pm

for 1 ≤ j ≤ m − r, 1 ≤ k ≤ m − r (k 6= j)
i=1 xi,j ∆k,i = 0
Pm−l
i=1 xi,m−r+l ∆k,i = 0 for 1 ≤ l ≤ 2r − m + 1, 1 ≤ k ≤ m − r

(30)

with (m − r)(m − r − 1) + (m − r)(2r − m + 1) = r(m − r) such relations; and

( Pm−l

Pm

i=1 xi,m−r+l ∆m−r+l,i −
Pm−k
i=1 xi,m−r+k ∆m−r+l,i =

j=1 xi,1 ∆1,i

= 0 for 1 ≤ l ≤ r − 1
for 1 ≤ l ≤ r − 2, l + 1 ≤ k ≤ r − 1

0

(31)

with r(r − 1)/2 such relations.
Similarly, when r ≤ m−r−1, the classical Cauchy cofactor formula outputs by expansion
three blocks of linear relations involving the (signed) cofactors of DG. Here, the first and
third blocks are, respectively, exactly as the above ones, while the second one requires a
modification due to the inequality reversal; namely, we get
( Pm

i=1
Pm

xi,j ∆k,i = 0 for 1 ≤ j ≤ r + 1, 1 ≤ k ≤ r (k 6= j)

i=1 xi,j ∆k,i

= 0 for 1 ≤ j ≤ r, r + 1 ≤ k ≤ m − r,

(32)

with r(m − r) such relations (as before).
Since fi,j coincides with the signed cofactor ∆j,i , any of the above relations gives a linear
syzygy of the partial derivatives
of f . Thus one has a total of m2 − rm − 1 + r(m − r) +

r+1
2
r(r − 1)/2 = m − 2 − 1 linear syzygies of J.
It remains to show that these are independent.
For this, we adopt the same strategy as in the proof of Theorem 2.3 (iii), whereby we
list the partial derivatives according to the following ordering of the nonzero entries: we
traverse the first row from left to right, then the second row in the same way, and so on
until we reach the last row with no zero entry; thereafter we start from the first row having
a zero and travel along the columns, from left to right, on each column from top to bottom,
till we all nonzero entries are counted.
23

Thus, the desired ordering is depicted in the following scheme, where we once more used
arrows for easy reading:
x1,1 , x1,2 , . . . , x1,m

x2,1 , x2,2 , . . . , x2,m

xm−r+1,1 , . . . , xm,1

xm−r+1,2 , . . . , xm,2

...

xm−r,1 , xm−r,2 . . . , xm−r,m

...

xm−r+1,m−r , xm−r+2,m−r , . . . , xm,m−r

xm−r+1,m−r+1 , . . . , xm−1,m−r+1

xm−r+1,m−r+2 , . . . , xm−2,m−r+2

...

xm−r+1,m−1

xm−r+1,m−2 , xm−r+2,m−2

With this ordering the above linear relations translate into linear syzygies collected in the
following block matrix












M=












ϕ1
0
..
.

...
ϕ2
..
.

0
0m−1
r
0m−1
r
..
.

0
0m
r
0m
r
..
.

0m−1
r
0m−1
r−1
0m−1
r−2
..
.

0m
r
0m
r−1
0m
r−2
..
.

0m−1
1

0m
1


...
..
.
...
...
...

ϕm−r
0m
r
0m
r
..
...
.
...
0m
r
. . . 0m
r−1
. . . 0m
r−2
..
..
.
.
...












,












..
.

0m
1

ϕ1r
0rr
..
.

ϕ2r
..
.

..

0rr
0rr−1
0rr−2
..
.

0rr
0rr−1
0rr−2
..
.

...
...
...
..
.

ϕr
0rr−1
0rr−2
..
.

Φ1
r−1
0r−2
..
.

Φ2
..
.

0r1

0r1

...

0r1

01r−1

0r−2
1

.
(m−r)

..

.
...

Φr−1

where:
• ϕ1 is the matrix obtained from the transpose DG(r)t of DG(r) by omitting the first
column
• ϕ2 , . . . , ϕm−r are each a copy of DG(r)t (up to column permutation);
• When r > m − r − 1, setting d = 2r − m + 1, for i = 1, . . . , m − r one has that ϕir is
the r × r minor omitting the ith column of the following submatrix of DG(r):

xm−r+1,1
..


.

 xm−d,1

xm−d+1,1


..

.

 xm−1,1


xm,1


. . . xm−r+1,m−r xm−r+1,m−r+1 xm−r+1,m−r+2 . . . xm−r+1,r+1
..
..
..
..
..
..


.
.
.
.
.
.

. . . xm−d,m−r
xm−d,m−r+1
xm−d,m−r+2 . . . xm−d,r+1 

.
. . . xm−d+1,m−r xm−d+1,m−r+1 xm−d+1,m−r+2 . . .
0


..
..
..
..
..
..

.
.
.
.
.
.


. . . xm−1,m−r
xm−1,m−r+1
0
...
0
...

xm,m−r

0
24

0

...

0

When r ≤ m − r − 1, consider the following submatrix of DG(r):




xm−r+1,1 . . . xm−r+1,r xm−r+1,r+1


..
..
..
..

.
.
.
.
.
...
xm,r
xm,r+1
m,1
Then, for i = 1, . . . , r (respectively, for i = r + 1, . . . , m − r) ϕir denotes the r × r
submatrix obtained by omitting the ith column (respectively, the last column).
• Each 0 under ϕ1 is an m × (m − 1) block of zeros and each 0 under ϕi is an m × m
block of zeros for i = 2, . . . , m − r − 1 ;
• 0cl denotes an l × c block of zeros.
• Φi is the following (r − i) × (r − i) submatrix of DG(r):




xm−r+1,m−r+i xm−r+1,m−r+i+1 . . . xm−r+1,m−1
xm−r+2,m−r+i xm−r+2,m−r+i+1 . . .

0


Φi = 
.
..
..
..
..


.
.
.
.
xm−i,m−r+i
0
...
0
Next we justify why these blocks make up (linear) syzygies. As already explained, the
relations in (29), (30) and (31) yield linear syzygies of the partial derivatives
of f . Setting
Pm
k = 1 in the first two relations of (29), the latter can be written as j=1 xi,j f1,j = 0, for
P
i = 2, . . . , m − r, and m−l
j=1 xm−r+l,j f1,j = 0, for all l = 1, . . . , r. By ordering the set of
partial derivatives fi,j as explained before, the coefficients of these relations become the
entries of the submatrix ϕ1 of DG(r)t obtained by omitting its first column, as mentioned
above, namely:


x2,1
..
.




 x2,m−r

 x2,m−r+1

 x2,m−r+2


..

.

 x2,m−1
x2,m

...

xm−r,1
..
.

...
...
xm−r,m−r
. . . xm−r,m−r+1
. . . xm−r,m−r+2
..
...
.
...
...

xm−r,m−1
xm−r,m

xm−r+1,1
..
.

xm−r+2,1
..
.

xm−r+1,m−r
xm−r+1,m−r+1
xm−r+1,m−r+2
..
.

xm−r+2,m−r
xm−r+2,m−r+1
xm−r+2,m−r+2
..
.

xm−r+1,m−1
0

0
0

...

xm−1,1
..
.

...
...
xm−1,m−r
. . . xm−1,m−r+1
...
0
..
.
..
.
...
...

0
0

xm,1
..
.
xm,m−r
0
0
..
.
0
0
















Getting ϕk , for k = 2, . . . , m − r, is similar, namely, we use again the first two relations
in the block (29) retrieving the submatrix of DG(r)t excluding the kth column and replacing
it with an extra column that comes from the last relation in (29) by taking i = k − 1.
25

Continuing, for each i = 1, . . . , m − r the block ϕir comes from the relations in the
blocks (30), if r > m − r − 1, or (32), if r ≤ m − r − 1, by setting k = i. Finally, for each
i = 1, . . . , r − 1, the block Φi comes from the relations in (31) by setting l = i.
This proves the claim about the large matrix above.
 Counting
through the sizes of the
2 − r+1 −1). Omitting its first
various blocks, one sees that this matrix is (m2 − r+1
)×(m
2
2
row obtains a square block-diagonal submatrix where each block has nonzero determinant.
Thus, the linear rank of J attains the maximum.
r+1
2
(c) Note that the polar map can be thought as the map of Pm −( 2 )−1 to itself defined
by the partial derivatives of f . As such, the polar variety will be described in terms of
defining equations in the original x-variables.

Let L = L(m, r) denote the set of variables in DG(r) lying to the left and above the
stair-like polygonal in Figure 1 and let Im−r (L) stand for the ideal generated by the (m −
r) × (m − r) minors of DG(r) involving only the variables in L.

Figure 1: stair-like polygonal.
Since L can be extended to a fully generic matrix of size (m − 1) × (m − 1), the ring
K[L]/Im−r (L) is one of the so-called ladder determinantal rings.
Claim: The homogeneous defining ideal of the image of the polar map of f contains
the ideal Im−r (L).
Let xi,j denote a nonzero entry of DG(r). Since the nonzero entries of the matrix are
independent variables, it follows easily from the Laplace expansion along the ith row that
the xi,j -derivative fi,j of f coincides with the (signed) cofactor of xi,j , heretofore denoted
∆j,i .
Given integers 1 ≤ i1 < i2 < . . . < im−r ≤ m − 1, consider the following submatrix of
the transpose matrix of cofactors:
26



∆i1 ,1
∆i2 ,1
..
.

∆i1 ,2
∆i2 ,2
..
.

∆i1 ,3
∆i2 ,3
..
.

···
···



F =

···
∆im−r ,1 ∆im−r ,2 ∆im−r ,3 · · ·
Letting






C=




x

∆i1 ,m−im−r +(m−r−1)
∆i2 ,m−im−r +(m−r−1)
..
.

x1,im−r +2
..
.

xm−r,im−r +1
xm−r+1,im−r +1
..
.

xm−r,im−r +2
xm−r+1,im−r +2
..
.

xm−im−r +(m−r−1),im−r +1



.


∆im−r ,m−im−r +(m−r−1)

x1,im−r +1
..
.

m−im−r +(m−r−2),im−r +1



···
···
···
···

···
xm−im−r +(m−r−2),im−r +2 · · ·
0
···

x1,m−1
..
.
xm−r,m−1
xm−r+1,m−1
..
.
0
0


x1,m
.. 
. 

xm−r,m 

0 
,
.. 
. 

0 
0

the cofactor identity adj(DG(r)) · DG(r) = det(DG(r))Im yields the relation
F · C = 0.
Since the columns of C are linearly independent, it follows that the rank of F is at most
m − im−r + (m − r − 1) − (m − im−r ) = (m − r) − 1. In other words, the maximal minors
of the following matrix


xi1 ,1
xi1 ,2
xi1 ,3
···
xi1 ,m−im−r +(m−r−1)
 xi2 ,1
xi2 ,2
xi2 ,3
···
xi2 ,m−im−r +(m−r−1) 


.
 ..
..
..
..

 .
.
.
···
.
xim−r ,1 xim−r ,2 xim−r ,3 · · · xim−r ,m−im−r +(m−r−1)
all vanish on the partial derivatives of f , thus proving the claim.

Claim: The codimension of the ideal Im−r (L(m, r)) is at least r+1
2 .
Let us note that the codimension of this ladder ideal could be obtained by the general
principle described in [10] (see also [3]), as done in the proof of Theorem 2.5. However, in
this structured situation we prefer to give an independent argument.
For this we induct with the following inductive hypothesis: let 1 ≤ i ≤ r − 1; then for
any (m − i) × (m − i) matrix of the form DG(r − i), the ideal Im−i−(r−i) (L(m − i, r − i))

has codimension at least r−i+1
. Note that m − i − (r − i) = m − r, hence the size of the
2
inner minors does not change in the inductive step.
We descend with regard to i; thus, the induction step starts out at i = r − 1, hence
r − i = 1 and m − i = m − (r − 1) = m − r + 1 and since m − r ≥ 2 by assumption,
then 3 ≤ m − (r − 1) ≤ m − 1. Rewriting n := m − r + 1, we are in the situation of an
n × n (n ≥ 3) matrix of the form DG(1). Clearly, then the ladder ideal In−1 (L(n − 1, 1)) is
a principal ideal generated by the (n − 1) × (n − 1) minor of DG(1) of the first n − 1 rows
and columns. Therefore, its codimension is 1 as desired.
To construct a suitable inductive precedent, let Le denote the set of variables that are
e the ideal
to the left and above the stair-like polygonal in Figure 2 and denote Im−r (L)
27

e Note
generated by the (m − r) × (m − r) minors of DG(r) involving only the variables in L.
e
that L is of the form L(m − 1, r − 1) relative to a matrix of the form DG(r − 1)). Clearly,
e it too is a ladder determinantal ideal on a suitable (m − 2) × (m − 2) generic
Im−r (L)
matrix; in particular, it is a Cohen-Macaulay prime ideal (see [15] for primeness and [10]
e is at
for Cohen–Macaulayness).
By the inductive hypothesis, the codimension of Im−r (L)

r
least 2 .

Figure 2: Sub-stair-like inductive.
Note that Le is a subset of L, hence there is a natural ring surjection
S :=

e
k[L]
k[L]
e  k[L]
=
[L \ L]
e
e
Im−r (L)
Im−r (L) k[L]
Im−r (L)



Since 2r + r = r+1
2 , it suffices to exhibit r elements of Im−r (L) forming a regular
e
sequence on the ring S := k[L]/Im−r (L)k[L].
Consider the matrices


x1,1
...
x1,m−r−1
x1,m−i


..
..
..
..


.
.
.
.
(33)


 xm−r−1,1 . . . xm−r−1,m−r−1
xm−r−1,m−i 
xm−r−1+i,1 . . . xm−r−1+i,m−r−1 xm−r−1+i,m−i
for i = 1, . . . , r. Let ∆i ∈ Im−r (L) denote the determinant of the above matrix, for i =
1, . . . , r.
The claim is that ∆ := {∆1 , . . . , ∆r } is a regular sequence on S.
Let δ denote the (m − r − 1)-minor in the upper left corner of (33). Clearly, δ is a regular
element on S as its defining ideal is a prime ideal generated in degree m − r. Therefore, it
suffices to show that the localized sequence
∆δ := {(∆1 )δ , . . . , (∆r )δ }
is a regular on Sδ . On the other hand, since S is Cohen-Macaulay, it is suffices to show that
dim Sδ /∆δ Sδ = dim Sδ − r.
28

Write X0 := {xm−r,m−1 , xm−r+1,m−2 , . . . , xm−2,m−r+1 , xm−1,m−r }. Note that, for every
i = 1, . . . , r, one has (∆i )δ = xm−r−1+i,m−i + (1/δ)Γi , with xm−r−1+i,m−i ∈ X0 and Γi ∈
k[L\X0 ]. The association xm−r−1+i,m−i 7→ −(1/δ)Γi therefore defines a ring homomorphism
k[L]δ /(∆δ ) = (k[X 0 ][L \ X 0 ])δ /(∆δ ) ' k[L \ X 0 ])δ
This entails a ring isomorphism
Sδ
k[L \ X0 ]δ
'
.
e
∆δ Sδ
(Im−r (L))k[L
\ X0 ]δ
e δ = dim Sδ − r
Thus, dim Sδ /∆δ Sδ = dim k[L]δ − r − codim (Im−r (L))

e
Therefore, codim (Im−r (L)) is at least codim (Im−r (L)) + r = r+1
2 .
In order to show that Im−r (L) is the homogeneous defining ideal
 of the polar variety
r+1
it suffices to show that the latter has codimension at most 2 . Since the dimension
of the homogeneous coordinate ring of the polar variety coincides with the rank of the
Hessian matrix of f , it now suffices to show that the latter is at least dim R − r+1
=
2


r+1
2 − r(r + 1).
=
m
−
m2 − r+1
2
2
For this, we proceed along the same line of the proof of Theorem 2.3 (ii). Namely, set
X := {xi,j | i + j = r + 2, r + 3, . . . , 2m − r} and consider the set of partial derivatives of
f with respect to the variables in X. Let M denote the Jacobian matrix of these partial
derivatives with respect to the variables in X. Observe that M is a submatrix of size
(m2 − r(r + 1)) × (m2 − r(r + 1)) of the Hessian matrix. We will show that det(M ) 6= 0.
Set v := {x1,m , x2,m−1 , . . . , xm,1 } ⊂ X, the set of variables along the main anti-diagonal
of DG(r).
As already pointed out, the partial derivative of f with respect to any xi,j ∈ X coincides
with the signed cofactor of xi,j . By expanding the cofactor of an entry in the set v one sees
that there is a unique (nonzero) term whose support lies in v and the remaining terms have
degree ≥ 2 in the variables off v. Similarly, the cofactor of a variable outside v has no term
whose support lies in v and has exactly one (nonzero) term of degree 1 in the variables off
v. In fact, if i + j 6= m + 1, one finds
∆j,i = xm+1−j,m+1−i (x1,m · · · xi,m−i+1
\ · · · xm−j+1,j
\ · · · xm,1 )
+ terms of degree at least 2 off v,
where the term inside the parenthesis has support in v.
Consider the ring endomorphism ϕ of R that maps any variable in v to itself and any
variable off v to zero. By the preceding observation, applying ϕ to any second partial
derivative of f involving only the variables of X will return zero or a monomial supported
f denote the resulting specialized matrix of M . Thus, any of
on the variables in v. Let M
its entries is either zero or a monomial supported on the variables in v.
f) is nonzero. For this, consider the Jacobian matrix of the set
We will show that det(M
of partial derivatives {fv : v ∈ v} with respect to the variables in v. Let M0 denote the
f.
specialization of this Jacobian matrix by ϕ considered as a corresponding submatrix of M
f
Up to permutation of rows and columns of M , we may write
29


f=
M

M0 N0
N1 M1


,

for suitable M1 . Now, by the way the second partial derivatives of f specialize via ϕ as
f) = det(M0 ) det(M1 ), so
explained above, one must have N0 = N1 = 0. Therefore, det(M
it remains to prove the nonvanishing of these two subdeterminants. Now the first block
is the Hessian matrix of the form g being taken as the product of the entries in the main
anti-diagonal of the matrix DG(r). By a similar argument used in the proof of Theorem 2.3
(ii), one has that g is a well-known homaloidal polynomial, hence we are done for the first
matrix block. As for the second block, by construction it has exactly one nonzero entry on
each row and each column. Therefore, it has a nonzero determinant.
To conclude the assertion of this item it remains to argue that the ladder determinantal
ring in question is Gorenstein. For this we use the criterion in [3, Theorem, (b) p. 120].
By the latter, we only need to see that the inner corners of the ladder depicted in Figure 1
have indices (a, b) satisfying the equality a + b = m − 1 + (m − r) − 1 = 2m − r − 2, where
the ladder is a structure in an (m − 1) × (m − 1) matrix.
This completes the proof of this item. The supplementary assertion on the analytic
spread of J is clear since the dimension of the latter equals the dimension of the k-subalgebra
generated by the partial derivatives.
Remark 3.2. One notes that the codimension of the polar variety in its embedding coincides with the codimension of DG(r) in the fully generic matrix of the same size, viewed as
vector spaces of matrices over the ground field k.

3.2

The ideal of the submaximal minors

We will need a couple of lemmas, the first of which is a non-generic version of [1, Theorem
10.16 (b)]:
Lemma 3.3. Let M be a square matrix with entries either variables over a field k or zeros,
such that det(M ) 6= 0. Let R denote the polynomial ring over k on the nonzero entries of
M and let S ⊂ R denote the k-subalgebra generated by the submaximal minors. Then the
extension S ⊂ R is algebraic at the level of the respective fields of fractions.
The proof is the same as the one given in [1, Theorem 10.16 (b)].
The second lemma was communicated to us by Aldo Conca, as a particular case of a
more general setup:
Lemma 3.4. The submaximal minors of the generic square matrix are a Gröbner base in
the reverse lexicographic order and the initial ideal of any minor is the product of its entries
along the main anti-diagonal.
This result is the counterpart of the classical result in the case of the lexicographic order,
where the initial ideals are the products of the entries along the main diagonals. In both
versions, the chosen term order should respect the rows and columns of M .
The content of the third lemma does not seem to have been noted before:
30

Lemma 3.5. Let G denote a generic m × m matrix and let X denote the set of entries none
of which belongs to the main anti-diagonal of a submaximal minor. Then X is a regular
sequence modulo the ideal generated by the submaximal minors in the polynomial ring of the
entries of G over a field k.
Proof. As for easy visualization,

•
•
•

•
•
•


•
•
•


•
•
•


..
..
..

.
.
.


•
xm−2,2 xm−2,3

 x
 m−1,1 xm−1,2 xm−1,3
xm,1
xm,2
•

X is the set of bulleted entries below (for m ≥ 6):

...
•
•
•
x1,m−1 x1,m
...
•
•
x2,m−2 x2,m−1 x2,m 

...
•
x3,m−3 x3,m−2 x3,m−1
• 

...
x4,m−4
x4,m−3 x4,m−2
•
• 

..
..
..
..
.. 
.
.
.
.
. 

. . . xm−2,m−4
•
•
•
• 

...
•
•
•
•
• 

...
•
•
•
•
•

(A similar picture can be depicted for m ≤5).
Clearly, the cardinality of X is 2 m−1
= (m − 1)(m − 2). Fix an ordering of the
2
elements {a1 , . . . , a(m−1)(m−2) } of X . By Lemma 3.4 and the assumption that every ai
avoids the initial ideal of any submaximal minor, it follows that the initial ideal of the
ideal (a1 , . . . , ai , P) is (a1 , . . . , ai , in(P)). Clearly, ai+1 is not a zero divisor modulo the
latter ideal, and hence, by a well known procedure, it is neither a zero divisor modulo
(a1 , . . . , ai , P).
In the subsequent parts we will relate the gradient ideal J ⊂ R of the determinant of
the matrix DG(r) in (3.3) to the ideal Im−1 (GC) ⊂ R of its submaximal minors. As an
easy preliminary, we observe that, for any nonzero entry xi,j of DG(r), since the nonzero
entries of the matrix are independent variables, it follows easily from the Laplace expansion
along the ith row that the xi,j -derivative fi,j of f coincides with the (signed) cofactor of
xi,j . In particular, one has J ⊂ Im−1 (GC) throughout the entire subsequent discussion and
understanding the conductor J : Im−1 (GC) will be crucial.
Proposition 3.6. Let DG(r) as in (3.3) denote our basic degenerate matrix, with 1 ≤ r ≤
m − 2. For every 0 ≤ j ≤ m, consider the submatrices Mj and Nj of DG(r) consisting of
its last j columns and the its last j rows, respectively. Write I := Im−1 (GC) ⊂ R for the
ideal of (m − 1)-minors of DG(r) and J for the gradient ideal of f := det(DG(r)). Then
Ij (Nj ) · Ir−j (Mr−j ) ⊂ J : I for every 0 ≤ j ≤ r.
Proof. For a fixed 1 ≤ j ≤ r, we write the matrices DG(r) and its adjoint adj(DG(r)) in
the following block form:
!


ej
N
Θ1,j Θ2,j
DG(r) =
, adj(DG(r)) =
;
(34)
Θ3,j Θ4,j
Nj
where Θ1,j , Θ2,j , Θ3,j , Θ4,j stand for submatrices of sizes (j + m − r) × (m − j), (j + m −
r) × j, (r − j) × (m − j) and (r − j) × j, respectively. Thus, we have
!
ej + Θ2,j Nj
Θ1,j N
adj(DG(r)) · DG(r) =
= f · Im .
(35)
ej + Θ4,j Nj
Θ3,j N
31

ej +
with Im denoting the identity matrix of order m. Since f belongs to J, then I1 (Θ1,j N
Θ2,j Nj ) ⊂ J. On the other hand, the entries of Θ1,j are cofactors of the entries on the
upper left corner of DG(r), hence belong to J as well. Therefore I1 (Θ2,j Nj ) ⊂ J as well.
From this by an easy argument it follows that
I1 (Θ2,j )Ij (Nj ) ⊂ J

(36)

I1 (Θ2,j )Ij (Nj ) · Ir−j (Mr−j ) ⊂ J

(37)

and, for even more reason,

Similarly, writing
DG(r) =



fr−j
M

Mr−j



we have:
DG(r) · adj(DG(r)) =



fr−j Θ1,j + Mr−j Θ3,j
M

fr−j Θ2,j + Mr−j Θ4,j
M



= f · Idm .

An entirely analogous reasoning leads to the inclusion I1 (Θ3,j )Ir−j (Mr−j ) ⊂ J, and for
even more reason
I1 (Θ3,j )Ij (Nj ) · Ir−j (Mr−j ) ⊂ J
(38)
fr−j Θ2,j +Mr−j Θ4,j , again I1 (M
fr−j Θ2,j +Mr−j Θ4,j ) ⊂
Arguing now with the second block M
fr−j Θ2,j + δMr−j Θ4,j ) ⊂ J. But, by (36), the
J, and hence for each δ ∈ Ij (Nj ), also I1 (δ M
fr−j Θ2,j belong to J. Thus, the entries de δMr−j Θ4,j belong to J and conseentries of δ M
quently
I1 (Θ4,j )Ij (Nj ) · Ir−j (Mr−j ) ⊂ J .
(39)
It follows from (37), (38) and (39) that
(I1 (Θ2,j ), I1 (Θ3,j ), I1 (Θ4,j )) Ij (Nj ) · Ir−j (Mr−j )) ⊂ J.

(40)

Since also I1 (Θ1,j ) ⊂ J, we have
(I1 (Θ1,j ), I1 (Θ2,j ), I1 (Θ3,j ), I1 (Θ4,j )) Ij (Nj ) · Ir−j (Mr−j )) ⊂ J.

(41)

From this equality it obtains
Ij (Nj ) · Ir−j (Mr−j )I ⊂ J

(42)

because
I = Im−1 (DG(r)) = I1 (adj(DG(r))) = (I1 (Θ1,j ), I1 (Θ2,j ), I1 (Θ3,j ), I1 (Θ4,j )).
This establishes the assertion above – we note that it contains as a special case (with j = 0
and j = r) the separate inclusions Ir (Mr ), Ir (Nr ) ⊂ J : I.
Theorem 3.7. Consider the matrix DG(r) as in (3.3), with 1 ≤ r ≤ m − 2. Let I :=
Im−1 (DG(r)) ⊂ R denote its ideal of (m − 1)-minors and J the gradient ideal of f :=
det(DG(r)). Then
32

(i) I is a Gorenstein ideal of codimension 4 and maximal analytic spread.
r+1
2
2
(ii) The (m − 1)-minors of DG(r) define a birational map Pm −( 2 )−1 99K Pm −1 onto a
cone over the polar variety of f with vertex cut by r+1
coordinate hyperplanes.
2

(iii) The conductor J : I has codimension 2(m − r) ≥ 4; in particular, J has codimension
4.
(iv) If r ≤ m − 3 then I is contained in the unmixed part of J; in particular, if R/J is
Cohen–Macaulay then r = m − 2.
(v) Let r = m − 2. Then the set of minimal primes of J is exactly the set of associated
primes of I and of the ideals (Ij (Nj ), Im−1−j (Mm−1−j )), for 1 ≤ j ≤ m − 2.

(vi) If, moreover, r+1
≤ m − 3, then I is a prime ideal; in particular, in this case it
2
coincides with the unmixed part of J.
Proof. (i) The analytic spread follows from Lemma 3.3.
The remaining assertions of the item follow from Lemma 3.5, which shows that I is a
specialization of the ideal of generic submaximal minors, provided we argue that the set

{xm,m−r+1 ; xm,m−r+2 , xm−1,m−r+2 ; . . . ; xm,m , xm−1,m , . . . , xm−r+1,m }
of variables on the voided entry places of the generic m × m matrix


x1,1
..
.




 xm−r,1

 xm−r+1,1

 xm−r+2,1


..

.

 xm−1,1

xm,1

...

x1,m−r
..
.

...
...
xm−r,m−r
. . . xm−r+1,m−r
. . . xm−r+2,m−r
..
...
.
...

xm−1,m−r

...

xm,m−r

x1,m−r+1
..
.

x1,m−r+2
..
.

xm−r,m−r+1
xm−r+1,m−r+1
xm−r+2,m−r+1
..
.

xm−r,m−r+2
xm−r+1,m−r+2
xm−r+2,m−r+2
..
.

xm−1,m−r+1

...

x1,m−1
..
.

...
...
xm−r,m−1
. . . xm−r+1,m−1
...
..
.
..
.
...
...

x1,m
..
.
xm−r,m
..
.

















is a subset of X as in the lemma. But this is immediate because of the assumption r ≤ m−2.
(ii) By Lemma 3.5, the ideal is a specialization of the ideal of submaximal minors in the
generic case; in particular, it is linearly presented. On the other hand, its analytic spread
is maximal by Lemma 3.3. Therefore, by Theorem 1.1 the minors define a birational map
onto the image. It remains to argue that the image is a cone over the polar variety, the
latter as described in Theorem 3.1 (c).
To see this note the homogeneous inclusion T := k[Jm−1 ] ⊂ T 0 := k[Im−1 ] of k-algebras
which are domains, where I is minimally generated by the generators of J and by r+1
2

0 = T [f , . . . , f ]. On
additional generators, say, f1 , . . . , fs , where s = r+1
,
that
is,
T
1
s
2

the other hand, one has dim T = m2 − r(r + 1) and dim T 0 = m2 − r+1
. Therefore,
2

r+1
0
tr.degk(T ) k(T )(f1 , . . . , fs ) = dim T − dim T = 2 = s, where k(T ) denotes the field of
fractions of T . This means that f1 , . . . , fs are algebraically independent over k(T ) and, a
fortiori, over T . This shows that T 0 is a polynomial ring over T in r+1
indeterminates.
2
Geometrically, the image of the
map
defined
by
the
(m
−
1)-minors
is
a
cone
over the polar

r+1
image with vertex cut by 2 independent hyperplanes.
33

(iii) The fact that the stated value 2(m−r) is an upper bound follows from the following
fact: first, because the only cofactors which are not partial derivatives (up to sign) are
those corresponding to the zero entries, then J is contained in the ideal Q generated by the
variables of the last row and the last column of the matrix. Since Q is prime and I 6⊂ Q
then clearly J : I ⊂ Q.
To see that 2(m − r) is a lower bound as well, we will use Proposition 3.6.
For that we need some intermediate results.
Claim 1. For every 1 ≤ j ≤ r, both Ij (Mj ) and Ij (Nj ) have codimension m − r.
By the clear symmetry, it suffices to consider Ij (Mj ). Note that Mj has r − j + 1 null
rows, so its ideal of j-minors coincides with the ideal of j-minors of its (m − (r − j + 1)) × j
submatrix Mj0 with no null rows. Clearly, this ideal of (maximal) minors has codimension
at most (m − (r − j + 1)) − j + 1 = m − r. Now, the matrix Mj0 specializes to the wellknown diagonal specialization using only m − r variables – by definition, the latter is the
specialization of a suitable Hankel matrix via the ring homomorphism mapping to zero the
variables of the upper left and lower right corner except the last variables of first column
and the first variable of the last column. This ensures that Ij (Mj0 ) has codimension at least
m − r.
Claim 2. For every 1 ≤ j ≤ r − 1, the respective sets of nonzero entries of Nj and
Mr−j+1 are disjoint. In particular, the codimension of (Ij (Nj ), Ir−j+1 (Mr−j+1 )) is 2(m−r).
The disjointness assertion is clear by inspection and the codimension follows from the
previous claim.
To proceed, we envisage the following chains of inclusions
I1 (N1 ) ⊃ I2 (N2 ) ⊃ . . . ⊃ Ir−1 (Nr−1 ) ⊃ Ir (Nr )

(43)

I1 (M1 ) ⊃ I2 (M2 ) ⊃ . . . ⊃ Ir−1 (Mr−1 ) ⊃ Ir (Mr ).

(44)

and
Let P denote a prime ideal containing the conductor J : I. By Proposition 3.6 one has
the inclusion I1 (N ) · Ir−1 (Mr−1 ) ⊂ P. Thus,
(A1 ) either I1 (N1 ) ⊂ P , or else
(B1 ) I1 (N1 ) 6⊂ J : I but Ir−1 (Mr−1 ) ⊂ P .
If (A1 ) is the case, then (I1 (N1 ), Ir (Mr )) ⊂ P , because Ir (Mr ) ⊂ J : I again by
Proposition 3.6 (with j = 0). By Claim 2 above, we then see that the codimension of J : I
is at least 2(m − r).
If (B1 ) takes place then we consider the inclusion I2 (N2 ) · Ir−2 (Mr−2 ) ⊂ J : I ⊂ P by
Proposition 3.6. The latter in turn gives rise to two possibilities according to which
(A2 ) either I2 (N2 ) ⊂ P, or else
(B2 ) I2 (N2 ) 6⊂ P but Ir−2 (Mr−2 ) ⊂ P.
Again, if (A2 ) is the case then (I2 (N2 ), Ir−1 (Mr−1 )) ⊂ P since Ir−1 (Mr−1 ) ⊂ P by
hypothesis. Once more, by Claim 2, the codimension of P is at least 2(m − r).
If intead (B2 ) occurs then we step up to the inclusion I3 (N3 )·Ir−3 (Mr−3 ) ⊂ P and repeat
the argument. Proceeding in this way, we may eventually find an index 1 ≤ j ≤ r − 1 such
34

that the first alternative (Aj ) holds, in which case we are through always by Claim 2.
Otherwise, we must be facing the situation where Ij (Nj ) 6⊂ P for every 1 ≤ j ≤ r − 1.
In particular, Ir−1 (Nr−1 ) 6⊂ P and I1 (M1 ) ⊂ P. Thus, (Ir (Nr ), I1 (M1 )) ⊂ P, and once
more by Claim 2, P has codimension at least 2(m − r). This concludes the proof of the
codimension of J : I.
The assertion that J has codimension 4 is then ensured as J ⊂ I and I has codimension
4 by item (i).
(iv) By (iii), if r ≤ m − 3 then J : I has codimension at least 2(m − r) ≥ 6. This implies
that I ⊂ J un and the two coincide up to radical.
The first assertion on the Cohen–Macaulayness of R/J is clear since then J is already
unmixed, hence J = I which is impossible since r ≥ 1.
(v) First, one has an inclusion J ⊂ (Ij (Nj ), Im−1−j (Mm−1−j )). Indeed, recall once more
that the partial derivatives are (signed) cofactors. Expanding each cofactor by Laplace
along j-minors, one sees that it is expressed as products of generators of Ij (Nj ) and of
Im−1−j (Mm−1−j ). Next, the ideal (Ij (Nj ), Im−1−j (Mm−1−j )) is perfect of codimension 4
– the codimension is clear by Claim 2 in the proof of (iii), while perfectness comes from
the fact that the tensor product over k of Cohen–Macaulay k-algebras of finite type is
Cohen–Macaulay.
At the other end, by (i) the ideal I is certainly perfect. Therefore, any associated prime
of either (Ij (Nj ), Im−1−j (Mm−1−j )) or I is a minimal prime of J.
Conversely, let P denote a minimal prime of J not containing I. Then J : I ⊂ P ,
hence the same argument in the proof of (iii) says that P contains some ideal of the form
(Ij (Nj ), Im−1−j (Mm−1−j )).
(vi) We will apply Proposition 2.1 in the case where M 0 = G is an m × m generic matrix
and M = DG(r) is the degenerated generic matrix as in the statement. In addition, we take
k = m − 2, so k + 1 = m − 1 is the size of the submaximal minors. Observe that the vector
space spanned by the entries of M has codimension r+1
in the vector space spanned by
2
the entries of M 0 . Since the m×m generic matrix is 2 = m−(m−2)-generic
 (it is m-generic
≤ k−1 = m−3
as explained in [6, Examples, p. 548]), the theorem ensures that if r+1
2
then Im−1 (DG) is prime.
Since in particular r ≤ m − 3 then item (d) says that I ⊂ J un . But I is prime, hence
I = J un .
Remark 3.8. The statement of item (i) in Theorem 3.7 depends not only the number of the
entries forming a regular sequence on P but also their mutual position. Thus, for example,
if more than r of the entries belong to one same column or row it may happen that I has
codimension strictly less than 4.
We end by filing a few natural questions/conjectures. The notation is the same as in
the last theorem.
Question 3.9. Does I = J un hold for r ≤ m − 3? (By (iv) above it would suffice to prove
that I is a radical ideal.)

Question 3.10. Is r+1
≤ m − 3 the exact obstrution for the primality of the ideal I of
2
submaximal minors?

Conjecture 3.11. If r+1
≤ m − 3 then J has no embedded primes.
2
35

Conjecture 3.12. Let r = m − 2. Then for any 1 ≤ j ≤ m − 2 one has the following
primary decomposition:
Ij (Mj ) = (xj+1,m−j+1 , δj ) ∩ (xj,m−j+2 , δi−1 ) ∩ . . . ∩ (x3,m−1 , δ2 ) ∩ (x2,m , x1,m ),
where δt denotes the determinant of the t × t upper submatrix of Mt . A similar result holds
for Ij (Nj ) upon reverting the indices of the entries and replacing δt by the determinant γt
of the t × t leftmost submatrix of Nt . In particular, Ij (Mj ) and Ij (Nj ) are radical ideals.
Conjecture 3.13. In the notation of the previous conjecture, one has
J :I

=

r
\

!
∩

(xm,1 , xm,2 , xt+1,m−t+1 , δt )

t=1

... ∩

1
\

r−1
\

!

(xm−1,3 , γ2 , xt+1,m−t+1 , δt )

∩

t=1

!
(xm,1 , xm,2 , xt+1,m−t+1 , δt )

∩ (x3,m−1 , γr , x2,m , δ1 ) .

t=1

In particular, J : I is a radical ideal.
Conjecture 3.14. If r = m − 2 then both R/J and R/J : I are Cohen–Macaulay reduced
rings and, moreover, one has J = I ∩ (J : I).

3.3

The dual variety

We keep the previous notation with DG(r) denoting the m × m matrix in (3.3).
In this part we describe the structure of the dual variety V (f )∗ of V (f ) for f =
det DG(r). The result in particular answers affirmatively a question posed by F. Russo
as to whether the codimension of the dual variety of a homogeneous polynomial in its polar
variety can be arbitrarily large when its Hessian determinant vanishes. In addition it shows
that this can happen in the case of structured varieties.
As a preliminary, we file the following lemma which may have independent interest.
Lemma 3.15. Let Ga,b denote the a × b generic matrix, with a ≥ b. Letting 0 ≤ r ≤ b − 2,
consider the corresponding degeneration matrix Ψ := DG a,b (r):


x1,1
x2,1
..
.

x1,2
x2,2
..
.






 xa−r,1
xa−r,2

 xa−r+1,1 xa−r+1,2


..
..

.
.

 xa−1,1
xa−1,2
xa,1
xa,2

...
...

x1,b−r
x2,b−r
..
.

x1,b−r+1
x2,b−r+1
..
.

...
...

x1,b−1
x2,b−1
..
.

x1,b
x2,b
..
.

. . . xa−r,b−r
xa−r,b−r+1 . . . xa−r,b−1 xa−r,b
. . . xa−r+1,b−r xa−r+1,b−r+1 . . . xa−r+1,b−1
0
..
..
.
..
.
..
..
.
.
.
. . . xa−1,b−r
xa−1,b−r+1 . . .
0
0
...
xa,b−r
0
...
0
0

Then the ideal of maximal minors of Ψ has the expected codimension a − b + 1.
36








.







Proof. The argument is pretty much the same as in the proof of Lemma 3.5, except that
the right lower corner of Ga,b whose entries will form a regular sequence is now as depicted
in blue below


x1,1
x2,1
..
.

x1,2
x2,2
..
.

x1,3
x2,3
..
.






 xa−b+2,1 xa−b+2,2 xa−b+2,3
X=
 xa−b+3,1 xa−b+3,2 xa−b+3,3


..
..
..

.
.
.

 xa−1,1
xa−1,2
xa−1,3
xa,1
xa,2
xa,3

...
...

x1,b−1
x2,b−1
..
.

x1,b
x2,b
..
.

. . . xa−b+2,b−1 xa−b+2,b
. . . xa−b+3,b−1 xa−b+3,b
..
..
.
..
.
.
. . . xa−1,b−1
xa−1,b
...
xa,b−1
xa,b








.







(Note that the “worst” case is the b-minor with of the last b rows of Ga,b , hence the top
right blue entry above.)
To finish the proof of the lemma, one argues that the blue entries form a regular sequence
on the initial ideal of Ib (Ga,b ) since the latter is generated by the products of the entries
along the anti-diagonals of the maximal minors (in any monomial order).
Next is the main result of this part. We stress that, in contrast to the dual variety in the
case of the cloning degeneration, here the dual variety will in fact be a ladder determinantal
variety.
Back to the notation of (3.3), one has:
Theorem 3.16. Let 0 ≤ r ≤ m − 2 and f := det DG(r). Then:
∗
(a) The dual variety
 V (f ) of V (f ) is a ladder determinantal variety of codimension
r+1
2
(m−1) − 2 defined by 2-minors; in particular it is arithmetically Cohen–Macaulay
and its codimension in the polar variety of V (f ) is (m − 1)2 − r(r + 1).

(b) V (f )∗ is arithmetically Gorenstein if and only if r = m − 2.
Proof. (a) The proof will proceed in parallel to the proof of Theorem 2.5, but there will
be some major changes.
We first show that once more dim V (f )∗ = 2m − 2 and, for that, check the sideinequalities separately.
1. dim V (f )∗ ≥ 2m − 2.
At the outset we draw as before upon the equality of Segre ([17]):
dim V (f )∗ = rank H(f )

(mod f ) − 2,

where H(f ) denotes the Hessian matrix of f . It will then suffice to show that H(f ) has a
submatrix of rank at least 2m modulo f .
37

For this purpose, consider the submatrix ϕ of DG(r) obtained by omitting the first row:


x2,1
..
.




 xm−r,1

 xm−r+1,1

 xm−r+2,1


..

.

 xm−1,1

xm,1

...

x2,m−r
..
.

...
...
xm−r,m−r
. . . xm−r+1,m−r
. . . xm−r+2,m−r
..
...
.

x2,m−r+1
..
.

x2,m−r+2
..
.

xm−r,m−r+1
xm−r+1,m−r+1
xm−r+2,m−r+1
..
.

xm−r,m−r+2
xm−r+1,m−r+2
xm−r+2,m−r+2
..
.

...

x2,m−1
..
.

...
...
xm−r,m−1
. . . xm−r+1,m−1
...
0
..
.
..
.



x2,m
..
.
xm−r,m
0
0
..
.

...

xm−1,m−r

xm−1,m−r+1

0

...

0

0

...

xm,m−r

0

0

...

0

0
















Claim: The ideal It (ϕ) has codimension at leat m − t + 1 for every 1 ≤ t ≤ m − 1.
To see this, for every 1 ≤ t ≤ m − 1, consider the m − 1 × t submatrix Ψt of ϕ of the
first t columns.
By Lemma 3.15, the codimension of It (Ψt ) is at least m − t – note that the lemma is
applied with a = m − 1, b = t ≤ m − 1 and r updated to r0 := r − (m − t), so indeed
b − r0 = t − r + (m − t) = m − r ≥ 2 as required.
It remains to find some t-minor of ϕ which is a nonzerodivisor on It (Ψt ). One choice is
the t-minor D of the columns 2, 3 . . . , t − 1, m and rows 1, m − t + 1, m − t + 2, . . . , m − 1.
Indeed, a direct verification shows that the variables in the support of the initial in(D) of D
in the reverse lexicographic order do not appear in the supports of the initial terms of the
maximal minors of Ψt . Therefore, in(D) is a nonzerodivisor on the initial ideal of It (Ψt ).
A standard argument then shows that It (Ψt ) has codimension at least m − t + 1.
This completes the proof of the claim.
The statement means that the ideal of maximal minors of ϕ satisfies the so-called property (F1 ). Since it is a perfect ideal of codimension 2 with ϕ as its defining Hilbert–Burch
syzygy matrix, it follows as in the proof of Theorem 2.5 that it is an ideal of linear type.
([9]). In particular the m maximal minors of ϕ are algebraically independent over k, hence
their Jacobian matrix with respect to the entries of ϕ has rank m.
Form this point the argument proceeds exactly as in the proof of item 1. of Theorem 2.5.
2. dim V (f )∗ ≤ 2m − 2.
Let P ⊂ k[y] := k[yi,j | 2 ≤ i + j ≤ 2m − r] denote the homogeneous defining ideal of
the dual variety V (f )∗ in its natural embedding, i.e.,
k[∂f /∂xi,j | 2 ≤ i + j ≤ 2m − r]/(f ) ' k[y]/P.

(45)

The isomorphism is an isomorphism of graded k-algebras induced by the assignment yi,j 7→
∂f /∂xi,j .
Claim 2. The homogeneous defining ideal P of V (f )∗ contains the ideal generated by
the 2 × 2 minors of the following ladder matrix:
To see this, we first recall that by Proposition 2.2, ∂f /∂xi,j coincides with the cofactor
of xi,j in DG(r). Now, consider the following relation afforded by the cofactor identity:
adj(DG(r)) · DG(r) ≡ 0 (mod f ).

(46)

Further, for each pair of integers i, j such that 1 ≤ i < j ≤ m let Fij denote the 2 × m
submatrix of adj(DG(r)) consisting of the ith and jth rows. In addition, let C stand for
38

Figure 3: Ladder ideal of the dual variety
the m × (m − 1) submatrix of DG(r) consisting of its m − 1 leftmost columns. Then (46)
gives the relations
Fij C ≡ 0 (mod f ),
for all 1 ≤ i < j ≤ m. From this, since the rank of C modulo (f ) is obviously still m − 1, the
rank of every Fi,j is necessarily 1. This shows that every 2 × 2 minor of adj(GD(r)) vanishes
modulo (f ). Therefore, each such minor involving only cofactors that are partial derivatives
gives a 2 × 2 minor of L vanishing on the partial derivatives. Clearly, by construction, we
obtain this way all the 2 × 2 minors of L. This proves the claim.
Now, since I2 (L) is a ladder determinantal ideal on a suitable generic matrix it is a
Cohen-Macaulay prime ideal (see [15] for primeness and [10] for Cohen–Macaulayness).

Moreover, its codimension is m2 − r+1
− (2m − 1) = (m − 1)2 − r+1
as follows from an
2
2
application to this case of the general principle in terms of maximal chains as described in
[10, Theorem 4.6 and Corollary 4.7].
(b) By the previous item, the homogeneous defining ideal of the dual variety is generated
by the 2×2 minors of the ladder matrix in Figure 3. Observe that the smallest square matrix
containing all the entries of the latter is the m × m matrix DG(r). By [3, Theorem, (b) p.
120], the ladder ideal is Gorenstein if only if the inner corners of the ladder have indices
(i, j) satisfying the equality i+j = m+1. In the present case, the inner corners have indices
satisfying the equation
i + j = m − r + (m − 1) = m − r + 1 + (m − 2) = · · · = m − 1 + (m − r) = 2m − r − 1.
Clearly this common value equals m + 1 if and only if r = m − 2.
Remark 3.17. It may of some interest to note that degenerating the matrix DG(m − 2) all
the way to a Hankel matrix, recovers the so-called sub-Hankel matrix thoroughly studied in
[2] from the homaloidal point of view and in [13], from the ideal theoretic side. The situation
is ever more intriguing since in the sub-Hankel case the determinant is homaloidal.
39

References
[1] W. Bruns, U. Vetter, Determinantal Rings, Lecture Notes in Mathematics 1327,
Springer-Verlag, 1988.
[2] C. Ciliberto, F. Russo and A. Simis, Homaloidal hypersurfaces and hypersurfaces with
vanishing Hessian, Advances in Math., 218 (2008) 1759–1805.
[3] A. Conca, Ladder determinantal rings, J. Pure Appl. Algebra 98 (1995), 119–134.
[4] A. Doria, H. Hassanzadeh and A. Simis, A characteristic free criterion of birationality,
Advances in Math., 230 (2012), 390–413.
[5] D. Eisenbud, On the resiliency of determinantal ideals, Proceedings of the U.S.-Japan
Seminar, Kyoto 1985. In Advanced Studies in Pure Math. II, Commutative Algebra and
Combinatorics, ed. M. Nagata and H. Matsumura, North-Holland (1987) 29–38.
[6] D. Eisenbud, Linear sections of determinantal varieties, Amer. J. Mathematics, 110
(1988), 541–575.
[7] M. A. Golberg, The derivative of a determinant, The American Mathematical Monthly,
79 (1972), 1124–1126.
[8] J. Herzog and T. Hibi, Monomial ideals, Graduate Texts in Mathematics 260, SpringerVerlag, 2011.
[9] J. Herzog, A. Simis and W. Vasconcelos, Koszul homology and blowing-up rings, in
Commutative Algebra, Proceedings, Trento (S. Greco and G. Valla, Eds.). Lecture
Notes in Pure and Applied Mathematics 84, Marcel-Dekker, 1983, 79–169.
[10] J. Herzog and N. N. Trung, Gröbner bases and multiplicity of determinantal and Pfaffian ideals, Adv. in Math. 96 (1992), 1–37.
[11] J. M. Landsberg, L. Manivel and N. Ressayre, Hypersurfaces with degenerate duals and
the geometric complexity theory program, Comment. Math. Helv. 88 (2013), 469-484.
[12] M. Mostafazadehfard, Hankel and sub-Hankel determinants – a detailed
study of their polar ideals, PhD Thesis, Universidade Federal de Pernambuco
(Recife, Brazil), July 2014.
[13] M. Mostafazadehfard and A. Simis, Homaloidal determinants, J. Algebra 450 (2016),
59-101.
[14] M. Mostafazadehfard and A. Simis, Corrigendum to “Homaloidal determinants”, 450
(2016), 59-101.
[15] H. Narasimhan, The Irreducibility of Ladder Determinantal Varieties, J. Algebra 102
(1986), 162–185.
[16] F. Russo, On the Geometry of Some Special Projective Varieties, Lecture Notes of the
Unione Matematica Italiana, Springer 2015.
40

[17] B. Segre, Bertini forms and Hessian matrices, J. London Math. Soc. 26 (1951), 164–176.
[18] A. Simis, R. Villarreal, Combinatorics of monomial Cremona maps, Math. Comp., 81
(2012), 1857–1867.

41

