F-THRESHOLDS, INTEGRAL CLOSURE AND CONVEXITY
MATTEO VARBARO

arXiv:1607.01147v1 [] 5 Jul 2016

To Winfried Bruns on his 70th birthday

A BSTRACT. The purpose of this note is to revisit the results of [HV] from a slightly different perspective, outlining how, if the integral closures of a finite set of prime ideals
abide the expected convexity patterns, then the existence of a peculiar polynomial f allows to compute the F-jumping numbers of all the ideals formed by taking sums of products of the original ones. The note concludes with the suggestion of a possible source of
examples falling in such a framework.

1. P ROPERTIES A, A+

AND

B FOR

A FINITE SET OF PRIME IDEALS

Let S be a standard graded polynomial ring over a field k and let m be a positive integer.
Fix homogeneous prime ideals of S:
p1 , p2 , . . . , pm .
For any σ = (σ1 , . . . , σm ) ∈ Nm and k = 1, . . . , m, denote by
I σ := pσ1 1 · · · pσmm

and

ek (σ ) := max{ℓ : I σ ⊆ pk }.
(ℓ)

(e (σ ))

k
. Since Spk is a regular local ring with maximal
Obviously we have I σ ⊆ m
k=1 pk
ℓ
ideal (pk )pk , we have that (pk )pk is integrally closed in Spk for any ℓ ∈ N. Therefore

(ℓ)

T

pk = (pk )ℓpk ∩ S is integrally closed in S for any ℓ ∈ N. Eventually we conclude that
(ek (σ ))
k=1 pk

Tm

is integrally closed in S, so:
Iσ

(1)

⊆

m
\
(ek (σ ))

pk

.

k=1

Definition 1.1. We say that p1 , . . . , pm satisfy condition A if
Iσ

=

m
\
(ek (σ ))

pk

∀ σ ∈ Nm .

k=1

If Σ ⊆ Nm , denote by I(Σ) := ∑σ ∈Σ I σ and by Σ ⊆ Qm the convex hull of Σ ⊆ Qm .
Lemma 1.2. For any Σ ⊆ Nm , I(Σ) ⊇ ∑v∈Σ I ⌈v⌉ , where ⌈v⌉ := (⌈v1 ⌉, . . . , ⌈vm ⌉) for v =
(v1 , . . ., vm ) ∈ Qm .
Proof. Since S is Noetherian, we can assume that Σ = {σ 1 , . . ., σ N } is a finite set. Take
v ∈ Σ. Then there exist nonnegative rational numbers q1 , . . . , qN such that
N

N

v = ∑ qi σ i

and

∑ qi = 1.

i=1

i=1

1

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MATTEO VARBARO

Let d be the product of the denominators of the qi ’s and σ = d · v ∈ Nm . Clearly:
(I ⌈v⌉ )d = I d·⌈v⌉ ⊆ I σ .
N
i
Setting ai = dqi , notice that σ = ∑N
i=1 ai σ and ∑i=1 ai = d. Therefore

(I ⌈v⌉ )d ⊆ I σ ⊆ I(Σ)d .
This implies that I ⌈v⌉ is contained in the integral closure of I(Σ).



From the above lemma, so, I(Σ) ⊇ ∑v∈Σ I ⌈v⌉ . In particular:
(2)

p1 , . . . , pm satisfy condition A =⇒ I(Σ) ⊇

∑
v∈Σ

m
\
(ek (⌈v⌉))

pk

k=1

!

.

Definition 1.3. We say that p1 , . . . , pm satisfy condition A+ if
!

∑

I(Σ) =

v∈Σ

m
\
(ek (⌈v⌉))

pk

∀ Σ ⊆ Nm

k=1

Remark 1.4. If p1 , . . . , pm satisfy condition A+, then they satisfy A as well (for σ ∈ Nm ,
just consider the singleton Σ = {σ }).
Lemma 1.5. Let σ 1 , . . . , σ N be vectors in Nm , and a1 , . . . , aN ∈ N. Then
!
N

ek

∑ ai σ i

i=1

N

= ∑ ai ek (σ i ) ∀ k = 1, . . . , m.
i=1

i
Proof. Set σ = ∑N
i=1 ai σ , and notice that


N 
N
N 
i ai
(∑N a e (σ i ))
(e (σ i )) ai
(a e (σ i ))
⊆ ∏ pk k
⊆ ∏ pk i k
⊆ pk i=1 i k
,
Iσ = ∏ Iσ
i=1

i=1

i=1

i
so the inequality ek (σ ) ≥ ∑N
i=1 ai ek (σ ) follows directly from the definition.
i
For the other inequality, for each i = 1, . . ., N choose fi ∈ I σ such that its image in
e (σ i )+1
Spk is not in (pk )pkk
. Then the class fi is a nonzero element of degree ek (σ i ) in the
associated graded ring G of Spk . Being G a polynomial ring (in particular a domain), the
ai
N
i
element ∏N
i=1 f i is a nonzero element of degree ∑i=1 ai ek (σ ) in G. Therefore
N

i
(∑N
i=1 ai ek (σ )+1)

∏ fiai ∈ Ipσk \ pk

.

i=1

i
This means that ek (σ ) ≤ ∑N
i=1 ai ek (σ ).



Consider the function e : Nm → Nm defined by

σ 7→ e(σ ) := (e1 (σ ), . . ., em (σ )).
From the above lemma we can extend it to a Q-linear map e : Qm → Qm .
Given Σ ⊆ Nm , the above map sends Σ to the convex hull PΣ ⊆ Qm of the set {e(σ ) :
σ ∈ Σ} ⊆ Qm . In particular we have the following:

F-THRESHOLDS, INTEGRAL CLOSURE AND CONVEXITY

3

Proposition 1.6. The prime ideals p1 , . . ., pm satisfy condition A+ if and only if
!
m
\
(⌈vk ⌉)

∑

I(Σ) =

(v1 ,...,vm )∈PΣ

If Σ ⊆

Nm

and s ∈ N, define

Σs

∀ Σ ⊆ Nm

pk

k=1

:= {σ i1 + . . . + σ is

: σ ik ∈ Σ}. Then

I(Σ)s = I(Σs).
Furthermore Σs = s · Σ, i.e. PΣs = s · PΣ. So:
Proposition 1.7. If p1 , . . . , pm satisfy condition A+, then
!
I(Σ)s =

∑

m
\
(⌈svk ⌉)

pk

(v1 ,...,vm )∈PΣ

∀ Σ ⊆ Nm , s ∈ N.

k=1

We conclude this section by stating the following definition:
Definition 1.8. We say that p1 , . . . , pm satisfy condition B if there exists a polynomial
T
ht(pk )
such that in≺ ( f ) is a square-free monomial for some term order ≺ on S.
f∈ m
k=1 pk
2. G ENERALIZED

TEST IDEALS AND

F - THRESHOLDS

Let p > 0 be the characteristic of k, I be an ideal of S and m be the homogeneous
maximal ideal of S. For all e ∈ N, define

νe (I) := max{r ∈ N : I r 6⊆ m[q] := (gq : g ∈ m)}, q = pe .
The F-pure threshold of I is then

νe (I)
.
e→∞ pe

fpt(I) := lim
e

e

The pe -th root of I, denoted by I [1/p ] , is the smallest ideal J ⊆ S such that I ⊆ J [p ] . By
the flatness of the Frobenius over S the q-th root is well defined. If λ is a positive real
number, then it is easy to see that

[1/pe ] 
[1/pe+1 ]
⌈λ pe ⌉
⌈λ pe+1 ⌉
I
⊆ I
.
The generalized test ideal of I with coefficient λ is defined as:
[1/pe ]

⌈λ pe ⌉
.
τ (λ · I) := I
e≫0

Note that τ (λ · I) ⊇ τ (µ · I) whenever λ ≤ µ . By [BMS, Corollary 2.16], ∀ λ ∈ R>0 ,
∃ ε ∈ R>0 such that τ (λ · I) = τ (µ · I) ∀ µ ∈ [λ , λ + ε ). A λ ∈ R>0 is called an Fjumping number for I if τ ((λ − ε ) · I) ) τ (λ · I) ∀ ε ∈ R>0 .
[

τ = (1)

)[
λ1

τ 6= (1)

)[
λ2

...

)[
λn

...

◮ λ -axis

(1) ) τ (λ1 · I) ) τ (λ2 · I) ) . . . ) τ (λn · I) ) . . .

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MATTEO VARBARO

The λi above are the F-jumping numbers. Notice that λ1 = fpt(I).
Theorem 2.1. If p1 , . . ., pm satisfy conditions A and B, then ∀ λ ∈ R>0 we have
σ

τ (λ · I ) =

m
\
(⌊λ ek (σ )⌋+1−ht(pk ))

pk

∀ σ ∈ Nm .

k=1

If p1 , . . . , pm satisfy conditions A+ and B, then ∀ λ ∈ R>0 we have
!

τ (λ · I(Σ)) =

∑

(v1 ,...,vm )∈PΣ

m
\
(⌊λ vk ⌋+1−ht(pk ))

pk

∀ Σ ⊆ Nm .

k=1

Proof. The first part immediately follows from [HV, Theorem 3.14], for if p1 , . . . , pm
satisfy conditions A and B, then I σ obviously enjoys condition (⋄+) of [HV] ∀ σ ∈ Nm .
Concerning the second part, Proposition 1.7 implies that I(Σ) enjoys condition (∗) of
[HV] ∀ Σ ⊆ Nm whenever p1 , . . . , pm satisfy conditions A+ and B. Therefore the conclusion follows once again by [HV, Theorem 4.3].

3. W HERE

GO FISHING ?

Let k be of characteristic p > 0. So far we have seen that, if we have graded primes
p1 , . . . , pm of S enjoying A and B, then we can compute lots of generalized test ideals. If
they enjoy A+ and B, we get even more.
That looks nice, but how can we produce p1 , . . . , pm like these? Before trying to answer
this question, let us notice that, as explained in [HV], the ideals p1 , . . . , pm of the following
examples satisfy conditions A+ and B:
(i) S = k[x1 , . . . , xm ] and pk = (xk ) for all k = 1, . . ., m.
(ii) S = k[X ], where X is an m × n generic matrix (with m ≤ n) and pk = Ik (X ) is the
ideal generated by the k-minors of X for all k = 1, . . . , m.
(iii) S = k[Y ], where Y is an m × m generic symmetric matrix and pk = Ik (Y ) is the
ideal generated by the k-minors of Y for all k = 1, . . ., m.
(iv) S = k[Z], where Z is a (2m + 1) × (2m + 1) generic skew-symmetric matrix and
pk = P2k (Z) is the ideal generated by the 2k-Pfaffians of Z for all k = 1, . . . , m.
Even for a simple example like (i), Theorem 2.1 is interesting: it gives a description of
the generalized test ideals of any monomial ideal.
In my opinion, a class to look at to find new examples might be the following: fix
f ∈ S a homogeneous polynomial such that in≺ ( f ) is a square-free monomial for
some term order ≺ (better if lexicographical) on S, and let C f be the set of ideals of S
defined, recursively, like follows:
(a) ( f ) ∈ C f ;
(b) If I ∈ C f , then I : J ∈ C f for all J ⊆ S;
(c) If I, J ∈ C f , then both I + J and I ∩ J belong to C f .
If f is an irreducible polynomial, C f consists of only the principal ideal generated by
f , but otherwise things can get interesting. Let us give two guiding examples:
(i) If u := x1 · · · xm , then the associated primes of (u) are (x1 ), . . ., (xm ). Furthermore
all the ideals of S = k[x1 , . . . , xm ] generated by variables are sums of the principal

F-THRESHOLDS, INTEGRAL CLOSURE AND CONVEXITY

5

ideals above, and all square-free monomial ideals can be obtained by intersecting
ideals generated by variables. Therefore, any square-free monomial ideal belongs
to Cu , and one can check that indeed:
Cu = {square-free monomial ideals of S}.
(ii) Let X = (xi j ) be an m × n matrix of variables, with m ≤ n. For positive integers
a1 < . . . < ak ≤ m and b1 < . . . < bk ≤ n, recall the standard notation for the
corresponding k-minor:


xa1 b1 xa1 b2 · · · xa1 bk
..
..  .
..
[a1, . . . , ak |b1 , . . . , bk ] := det  ...
.
.
.
xak b1 xak b2 · · · xak bk
For i = 0, . . . , n − m, let δi := [1, . . ., m|i + 1, . . ., m + i]. Also, for j = 1, . . . , m − 1
set g j := [ j + 1, . . ., m|1, . . ., m − j] and h j := [1, . . ., m − j|n − m + j + 1, . . ., n].
Let ∆ be the product of the δi ’s, the g j ’s and the h j ’s:
∆ :=

n−m

m−1

i=0

j=1

∏ δi · ∏ g j h j .

By considering the lexicographical term order ≺ extending the linear order
x11 > x12 > . . . x1n > x21 > . . . > x2n > . . . > xm1 > . . . > xmn ,
we have that
in(∆) =

n−m

m−1

i=0

j=1

∏ in(δi) · ∏ in(g j ) in(h j ) = ∏

xi j

i∈{1,...,m}
j∈{1,...,n}

is a square-free monomial. Since each (δi ) belongs to C∆ , the height-(n − m + 1)
complete intersection
J := (δ0 , . . . , δn−m )
is an ideal of C∆ too. Notice that the ideal Im (X ) generated by all the maximal
minors of X is a height-(n − m + 1) prime ideal containing J. So Im (X ) is an
associated prime of J, and thus an ideal of C∆ by definition. With more effort, one
should be able to show that the ideals of minors Ik (X ) stay in C∆ for any size k.
The ideals of C f have quite strong properties. First of all, C f is a finite set by [Sc].
Then, all the ideals in C f are radical. Even more, Knutson proved in [Kn] that they have
a square-free initial ideal!
In order to produce graded prime ideals p1 , . . . , pm satisfying conditions A (or even A+)
and B, it seems natural to seek for them among the prime ideals in C f . This is because, at
least, f is a good candidate for the polynomial needed for condition B: if f = f1 · · · fr is
the factorization of f in irreducible polynomials, then for each A ⊆ {1, . . ., r} the ideal
JA := ( fi : i ∈ A) ⊆ S
is a complete intersection of height |A|. If p is an associated prime ideal of JA , then f
obviously belongs to p|A| ⊆ p(|A|) . So such a p satisfies B.

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MATTEO VARBARO

Question 3.1. Does the ideal p above satisfy condition A? Even more, is it true that for
prime ideals p as above ps = p(s) for all s ∈ N?
If the above question admitted a positive answer, Theorem 2.1 would provide the generalized test ideals of p. A typical example, is when JA = (δ0 , . . . , δn−m ) and p = Im (X )
(see (ii) above), in which case it is well-known that Im (X )s = Im (X )(s) for all s ∈ N (e.g.
see [BV, Corollary 9.18].
Remark 3.2. Unfortunately, it is not true that p satisfies B for all prime ideal p ∈ C f :
for example, consider f = ∆ in the case m = n = 2, that is ∆ = x21 (x11 x22 − x12 x21 )x21 .
Notice that (x21 , x11 x22 − x12 x21 ) = (x21 , x11 x22 ) = (x21 , x11 ) ∩ (x21 , x22 ), so
p = (x21 , x11 ) + (x21 , x22 ) = (x21 , x11 , x22 ) ∈ C∆ .
However ∆ ∈
/ p(3) .
Problem 3.3. Find a large class of prime ideals in C f (or even characterize them) satisfying condition B.
If p1 , . . . , pm are prime ideals satisfying A+, then (by definition)

∑ pi = ∑ pi

i∈A

∀ A ⊆ {1, . . . , m}.

i∈A

If p1 , . . ., pm are in C f , then the above equality holds true because ∑i∈A pi , belonging to
C f , is a radical ideal.
Problem 3.4. Let P f be the set of prime ideals in C f . Is it true that P f satisfies condition
A+? If not, find a large subset of P f satisfying condition A+.
R EFERENCES
[BMS] M. Blickle, M. Mustaţă, K.E. Smith, Discreteness and rationality of F-thresholds, Michigan Math.
J. 57 (2008), 43–61.
[BV] W. Bruns, U. Vetter, Determinantal rings, Lecture Notes in Mathematics 1327, Springer-Verlag,
Berlin, 1988.
[HV] I.B. Henriques, M. Varbaro, Test, multiplier and invariant ideals, Adv. Math. 287 (2016), 704–732.
[Kn] A. Knutson, Frobenius splitting, point-counting, and degeneration, available at
http://arxiv.org/abs/0911.4941 (2009).
[Sc] K. Schwede, F-adjunction, Algebra & Number Theory 3 (2009), 907–950.
D IPARTIMENTO DI M ATEMATICA , U NIVERSIT À
E-mail address: varbaro@dima.unige.it

DEGLI

S TUDI DI G ENOVA , I TALY

