arXiv:1210.8203v2 [] 30 Apr 2013

THE ASYMPTOTICS OF SYMBOLIC GENERIC INITIAL
SYSTEMS OF SIX POINTS IN P2
SARAH MAYES

Abstract. Consider the ideal I ⊆ K[x, y, z] corresponding to six points
of P2 . We study the limiting behaviour of the symbolic generic initial
system, {gin(I (m) }m of I obtained by taking the reverse lexicographic
generic initial ideals of the uniform fat point ideals I (m) . The main result
of this paper is a theorem describing the limiting shape of {gin(I (m) }m
for each of the eleven possible configuration types of six points.

1. Introduction
Given a set of six points of Pn−1 with ideal I ⊆ k[Pn−1 ], we may consider
the ideal I (m) generated by the polynomials that vanish to at least order m
at each of the points. Such ideals are called uniform fat point ideals and,
although they are easy to describe, they have proven difficult to understand.
There are still many open problems and unresolved conjectures related to
finding the Hilbert function of I (m) and even the degree α(I (m) ) of the
smallest degree element of I (m) (for example, see [CHT11], [GH07], [GVT04],
[GHM09] , and [Har02]).
In this paper we will study a limiting shape that describes the behaviour of
the Hilbert functions of the set of fat point ideals {I (m) }m as m approaches
infinity. Studying asymptotic behaviour has been an important research
trend of the past twenty years; while individual algebraic objects may be
complicated, the limit of a collection of such objects is often quite nice (see,
for example, [Hun92], [Siu01],[ELS01], and [ES09]). Research on fat point
ideals has shown that certain challenges in understanding these ideals can be
overcome by studying the entire collection {I (m) }m . For instance, more can
(m)

be said about the limit (I) = limm→∞ α(Irm ) than the invariants α(I (m) )
of each ideal (see [BH10] and [Har02]).
To describe the limiting behaviour of the Hilbert functions of fat point
ideals, we will study the symbolic generic initial system, {gin(I (m) )}m , obtained by taking the reverse lexicographic generic initial ideals of fat point
ideals. When I ⊆ K[x, y, z] is an ideal of points of P2 , knowing the Hilbert
function of I (m) is equivalent to knowing the generators of gin(I (m) ); thus,
describing the limiting behaviour of the symbolic generic initial system of
I is equivalent to describing that of the Hilbert functions of the fat point
ideals I (m) as m gets large.
We define the limiting shape P of the symbolic generic initial system
1
{gin(I (m) }m of the ideal I to be the limit limm→∞ m
Pgin(I (m) ) , where Pgin(I (m) )
1

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SARAH MAYES

denotes the Newton polytope of gin(I (m) ). When I ⊆ K[x, y, z] corresponds
to an arrangement of points in P2 , each of the ideals gin(I (m) ) is generated
in the variables x and y, so Pgin(I (m) ) , and thus P , can be thought of as a
subset of R2 .
The main result of this paper is the following theorem describing the
limiting shape of the symbolic generic initial system of an ideal corresponding to any collection of 6 points in P2 . The concept of configuration type
mentioned is intuitive; for example, {p1 , . . . , p6 } are of configuration type B
pictured in Figure 1 when there is one line through three of the points but
no lines through any other three points and no conics through all six points
(see Definition 2.3).
Theorem 1.1. Let I ⊆ K[x, y, z] be the ideal corresponding to a set of
six points in P2 . Then the limiting polytope P of the reverse lexicographic
symbolic generic initial system {gin(I (m) )}m is equal to the limiting shape
P shown in Figures 1 and 2 corresponding to the configuration type of the
six points.
This theorem will be proved in Section 4; Sections 2 and 3 contain background information necessary for the proof. In Section 5 we discuss how
characteristics of the arrangement of an arbitrary set of points in P2 are, or
may be, reflected in the limiting shape of the corresponding symbolic generic
initial system.

SYMBOLIC GENERIC INITIAL SYSTEMS OF SIX POINTS
3

Figure 1. The limiting shape P of the generic initial systems {gin(I (m) )}m when I is the ideal corresponding
to points {p1 , . . . , pr } in configuration types A through F pictured.

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SARAH MAYES

Figure 2. The limiting shape P of the generic initial systems {gin(I (m) )}m when I is the ideal corresponding
to points {p1 , . . . , pr } in configuration types G through K pictured.

SYMBOLIC GENERIC INITIAL SYSTEMS OF SIX POINTS

5

2. Background
In this section we will introduce notation, definitions, and results related
to fat points in P2 , generic initial ideals, and systems of ideals. Unless stated
otherwise, R = K[x, y, z] is the polynomial ring in three variables over a field
K of characteristic 0 with the standard grading and the reverse lexicographic
order > with x > y > z.
2.1. Fat Points in P2 .
Definition 2.1. Let p1 , . . . , pr be distinct points of P2 , Ij be the ideal
of K[P2 ] = R consisting of all forms vanishing at the point pj , and I =
I1 ∩ · · · ∩ Ir be the ideal of the points p1 , . . . , pr . A fat point subscheme
Z = m1 p1 + · · · + mr pr , where the mi are nonnegative integers, is the
subscheme of P2 defined by the ideal IZ = I1m1 ∩ · · · ∩ Irmr consisting of
forms that vanish at the points pi with multiplicity at least mi . When
mi = m for all i, we say that Z is uniform; in this case, IZ is equal to the
mth symbolic power of I, I (m) .
The following lemma relates the symbolic and ordinary powers of I in the
case we are interested in (see, for example, Lemma 1.3 of [AV03]).
Lemma 2.2. If I is the ideal of distinct points in P2 ,
(I m )sat = I (m) ,
where J sat =

S

k≥0 (J

: mk ) denotes the saturation of J.

The precise definition of a configuration type mentioned in the statement
of Theorem 1.1 is as follows.
Definition 2.3 ([GH07]). Two sets of points {p1 , . . . , pr } and {p01 , . . . , p0r } of
P2 have the same configuration type if for all sequences of positive integers
m1 , . . . , mr the ideals of the fat point subschemes Z = m1 p1 + · · · + mr pr
and Z 0 = m1 p01 + · · · + mr p0r have the same Hilbert function, possibly after
reordering.
Proposition 2.4 ([GH07]). The configuration types for six distinct points
in P2 are exactly the configurations A through K shown in Figures 1 and 2.
2.2. Generic Initial Ideals. An element g = (gij ) ∈ GLn (K) acts on
R = K[x1 , . . . , xn ] and sends any homogeneous element f (x1 , . . . , xn ) to the
homogeneous element
f (g(x1 ), . . . , g(xn ))
Pn
where g(xi ) = j=1 gij xj . If g(I) = I for every upper triangular matrix g
then we say that I is Borel-fixed. Borel-fixed ideals are strongly stable when
K is of characteristic 0; that is, for every monomial m in the ideal such that
x m
xi divides m, the monomials xj i are also in the ideal for all j < i. This
property makes such ideals particularly nice to work with.
To any homogeneous ideal I of R we can associate a Borel-fixed monomial
ideal gin> (I) which can be thought of as a coordinate-independent version

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SARAH MAYES

of the initial ideal. Its existence is guaranteed by Galligo’s theorem (also
see [Gre98, Theorem 1.27]).
Theorem 2.5 ([Gal74] and [BS87b]). For any multiplicative monomial order > on R and any homogeneous ideal I ⊂ R, there exists a Zariski open
subset U ⊂ GLn (K) such that In> (g(I)) is constant and Borel-fixed for all
g ∈ U.
Definition 2.6. The generic initial ideal of I, denoted gin> (I), is defined
to be In> (g(I)) where g ∈ U is as in Galligo’s theorem.
The reverse lexicographic order > is a total ordering on the monomials of
R defined by:
(1) if |I| = |J| then xI > xJ if there is a k such that im = jm for all
m > k and ik < jk ; and
(2) if |I| > |J| then xI > xJ .
For example, x21 > x1 x2 > x22 > x1 x3 > x2 x3 > x23 . From this point on,
gin(I) = gin> (I) will denote the generic initial ideal with respect to the
reverse lexicographic order.
Recall that the Hilbert function HI (t) of I is defined by HI (t) = dim(It ).
The following result is a consequence of the fact that Hilbert functions
are invariant under making changes of coordinates and taking initial ideals
([Gre98]).
Proposition 2.7. For any homogeneous ideal I in R, the Hilbert functions
of I and gin(I) are equal.
We now describe the structure of the ideals gin(I (m) ) where I is an ideal
corresponding to points in P2 . The proof of this result is contained in
[May12a] and follows from results of Bayer and Stillman ([BS87a]) and of
Herzog and Srinivasan ([HS98])
Proposition 2.8 (Corollary 12.9 of [May12a]). Suppose I ⊆ K[x, y, z] is
the ideal of distinct points in P2 . Then the minimal generators of gin(I (m) )
are
{xα(m) , xα(m)−1 y λα(m)−1 , . . . , xy λ1 (m) , y λ0 (m) }
for λ0 (m), . . . , λα(m)−1 such that λ0 (m) > λ1 (m) > · · · > λα(m)−1 (m) ≥ 1.
Since Borel-fixed ideals generated in two variables are determined by their
Hilbert functions (see, for example, Lemma 3.7 of [May12b]), we have the
following corollary of Propositions 2.7 and 2.8.
Corollary 2.9. If I and I 0 are ideals corresponding to two point arrangements of the same configuration type, gin(I (m) ) = gin(I 0(m) ) for all m.
Actually finding the Hilbert functions of fat point ideals is not easy and is
a significant area of research. (for example, see [CHT11], [GH07], [GVT04],
[GHM09] , and [Har02]) When I is the ideal of less than 9 points, however,
techniques exist for computing these Hilbert functions. In Section 3 we will

SYMBOLIC GENERIC INITIAL SYSTEMS OF SIX POINTS

7

outline the method used in this paper, following [GH07]. Other techniques,
such as those in [CHT11], can also be used for some of the point arrangements A through K.
2.3. Graded Systems. In this subsection we introduce the limiting shape
of a graded system of monomial ideals.
Definition 2.10 ([ELS01]). A graded system of ideals is a collection of
ideals J• = {Ji }∞
i=1 such that
Ji · Jj ⊆ Ji+j

for all i, j ≥ 1.

Definition 2.11. The generic initial system of a homogeneous ideal I
is the collection of ideals J• such that Ji = gin(I i ). The symbolic generic
initial system of a homogeneous ideal I is the collection J• such that
Ji = gin(I (i) ).
The following lemma justifies calling these collections ‘systems’; see Lemma
2.5 of [May12c] and Lemma 2.2 of [May12a] for proofs.
Lemma 2.12. Generic initial systems and symbolic generic initial systems
are graded system of ideals.
Let J be a monomial ideal of R = K[x1 , . . . , xn ]. We may associate to J
a subset Λ of Nn consisting of the points λ such that xλ ∈ J. The Newton
polytope PJ of J is the convex hull of Λ regarded as a subset of Rn . Scaling
the polytope PJ by a factor of r gives another polytope that we will denote
rPJ .
If a• is a graded system of monomial ideals in R, the polytopes of { 1q Paq }q
1
are nested: 1c Pac ⊂ c+1
Pac+1 for all c ≥ 1. The limiting shape P of a• is the
limit of the polytopes in this set:
[ 1
Pa .
P =
q q
∗
q∈N

When I is the ideal of points in P2 gin(I (m) ) is generated in the variables
x and y by Proposition 2.8, so we can think of each Pgin(I (m) ) , and thus P ,
as a subset of R2 .
3. Technique for computing the Hilbert function
Here we summarize the method that is used to compute HI (m) (t) in this
paper. It follows the work of Guardo and Harbourne in [GH07]; details can
be found there.
Suppose that π : X → P2 is the blow-up of distinct points p1 , . . . , pr of
2
P . Let Ei = π −1 (pi ) for i = 1, . . . , r and L be the total transform in X of
a line not passing through any of the points p1 , . . . , pr . The classes of these
divisors form a basis of Cl(X); for convenience, we will write ei in place of
[Ei ] and e0 in place of [L]. Further, the intersection product in Cl(X) is
defined by e2i = −1 for i = 1, . . . , r; e20 = 1; and ei · ej = 0 for all i 6= j.

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SARAH MAYES

Let Z = m(p1 + · · · + pr ) be a uniform fat point subscheme with sheaf of
ideals IZ ; set
Fd = dE0 − m(E1 + E2 + · · · + Er )
and Fd = OX (Fd ).
The following lemma relates divisors on X to the Hilbert function of I (m) .
Lemma 3.1. If Fd = dE0 − m(E1 + · · · + Er ) then h0 (X, Fd ) = HI (m) (d).
Proof. Since π∗ (Fd ) = IZ ⊗ OP2 (d),
HI (m) (d) = dim((IZ )d ) = h0 (P2 , IZ ⊗ OP2 (d)) = h0 (X, Fd )
for all d.



For convenience, we will sometimes write h0 (X, F ) = h0 (X, OX (F )). Recall that if [F ] not the class of an effective divisor then h0 (X, F ) = 0. On the
other hand, if F is effective, then we will see that we can compute h0 (X, F )
by finding h0 (X, H) for some numerically effective divisor H.
Definition 3.2. A divisor H is numerically effective if [F ] · [H] ≥ 0 for
every effective divisor F , where [F ]·[H] denotes the intersection multiplicity.
The cone of classes of numerically effective divisors in Cl(X) is denoted by
NEF(X).
Lemma 3.3. Suppose that X is the blow-up of P2 at r ≤ 8 points in general
position and that F ∈ NEF(X). Then F is effective and
h0 (X, F ) = ([F ]2 − [F ] · [KX ])/2 + 1
where KX = −3E0 + E1 + · · · + Er .
Proof. This is a consequence of Riemann-Roch and the fact that h1 (X, F ) =
0 for any numerically effective divisor F . See Lemma 2.1b of [GH07] for a
discussion.

The set of classes of effective, reduced, and irreducible curves of negative
intersection is
NEG(X) := {[C] ∈ Cl(X) : [C]2 < 0, C is effective, reduced, and irreducible}.
The set of classes in NEG(X) with self intersection less than −1 is
neg(X) := {[C] ∈ NEG(X) : [C]2 < −1}.
The following result of Guardo and Harbourne allows us to easily identify
divisor classes belonging to NEG(X). In the lemma, the curves defining
the configuration type are lines that pass through any three points or conics
that pass through any six points. For example, the divisors defining the
configuration type shown in Figure 3 are E0 − E1 − E2 − E3 and E0 − E1 −
E4 − E5 .

SYMBOLIC GENERIC INITIAL SYSTEMS OF SIX POINTS

9

Figure 3. Points p1 , . . . , p6 of configuration type H.
Lemma 3.4 (Lemma 2.1d of [GH07]). The elements of neg(X) are the
classes of divisors that correspond to the curves defining the configuration
types. Further,
NEG(X) = neg(X)∪{[C] ∈ B∪L∪Q : [C]2 = −1, [C]·[D] ≥ 0 for all D ∈ neg(X)}
where B = {ei : i > 0}, L = {e0 −ei1 −· · ·−eir : r ≥ 2, 0 < i1 < · · · < ir ≤ 6},
and Q = {2e0 − ei1 − · · · − eir : r ≥ 5, 0 < i1 < · · · < ir ≤ 6}.
The following result will be used in Procedure 3.6; see Section 2 of [GH07].
Lemma 3.5. Suppose that [C] ∈ NEG(X) is such that [F ] · [C] < 0. Then
h0 (X, F ) = h0 (X, F − C).
Knowing how to compute h0 (X, H) for a numerically effective divisor H
will allow us to compute h0 (X, F ) for any divisor F . In particular, given
a divisor F , there exists a divisor H such that h0 (X, F ) = h0 (X, H) and
either:
(a) H is numerically effective so
h0 (X, F ) = h0 (X, H) = (H 2 − H · KX )/2 + 1
by Lemma 3.3; or
(b) there is a numerically effective divisor G such that [G]·[H] < 0 so [H]
is not the class of an effective divisor and h0 (X, F ) = h0 (X, H) = 0.
The method for finding such an H is as follows.
Procedure 3.6 (Remark 2.4 of [GH07]). Given a divisor F we can find a
divisor H with h0 (X, F ) = h0 (X, H) satisfying either condition (a) or (b)
above as follows.
(1) Reduce to the case where [F ]·ei ≥ 0 for all i = 1, . . . , n: if [F ]·ei < 0
for some i, h0 (X, F ) = h0 (X, F − ([F ] · ei )Ei ), so we can replace F
with F − ([F ] · ei )Ei .
(2) Since L is numerically effective, if [F ] · e0 < 0 then [F ] is not the
class of an effective divisor and we can take H = F (case (b)).
(3) If [F ] · [C] ≥ 0 for every [C] ∈ NEG(X) then, by Lemma 3.4, F is
numerically effective, so we can take H = F (case (a)).
(4) If [F ]·[C] < 0 for some [C] ∈ NEG(X) then h0 (X, F ) = h0 (X, F −C)
by Lemma 3.5. Then replace F with F − C and repeat from Step 2.
There are only a finite number of elements in NEG(X) to check by Lemma
3.4 so it is possible to complete Step 3. Further, [F ] · e0 > [F − C] · e0 when

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SARAH MAYES

[C] ∈ NEG(X), so the condition in Step 2 will be satisfied after at most
[F ] · e0 + 1 repetitions. Thus, the process will terminate.
Taking these results together we can compute the Hilbert function of I (m)
as follows.
(1) Compute NEG(X) from neg(X) using Lemma 3.4.
(2) Find Ht corresponding to Ft using Procedure 3.6 for all t.
(3) Compute HI (m) (t) = h0 (X, Ft ) = h0 (X, Ht ) with Lemma 3.3.
4. Proof of the Main Theorem
In this section, we will outline the proof of Theorem 1.1. Recall that ideals
of points with the same configuration type have the same symbolic generic
initial system by Corollary 2.9 so the statement of the theorem makes sense.
Further, Proposition 2.4 ensures that the theorem includes all possible sets
of six points.
If I is the ideal of a set of six points having configuration type E, G, or
K, the theorem follows from the main result of [May12c]. Likewise, if I is
the ideal of six points of configuration type A, the theorem follows from the
main result of [May12a].
For the remaining cases we can find the limiting polytope of {gin(I (m) )}m
by following the five steps below. First, we record a lemma that will be used
in Step 2.
Lemma 4.1. Let J be a monomial ideal of K[x, y, z] generated in the variables x and y. Then the number of elements of J of degree t only involving
the variables x and y is equal to HJ (t) − HJ (t − 1). The number of minimal
generators of J in degree t is equal to HJ (t) − HJ (t − 2) − 1.
Proof. The first statement follows from the fact that there are exactly HJ (t−
1) monomials of J of degree t involving the variable z. The number of
generators in degree t is equal to the number of monomials of J in the
variables x and y of degree t minus the number of monomials of J that arise
from multiplying the elements of degree t − 1 in x and y by the variables x
and y. Using this, the last statement follows from the first.

Step 1: Find the Hilbert function of I (m) for infinitely many m by using the
method outlined in Section 3.
Step 2: Find the number of minimal generators of gin(I (m) ) of each degree t
for infinitely many m. We can use Lemma 4.1 for this computation
because gin(I (m) ) is an ideal generated in the variables x and y
(Proposition 2.8) and we know the Hilbert function of gin(I (m) ) from
Proposition 2.7 and Step 1.
Step 3: Write down the generators of gin(I (m) ) for infinitely many m. Note
that this follows from Step 2 since
gin(I (m) ) = (xα(m) , xα(m)−1 y λα(m)−1 , . . . , xy λ1 (m) , y λ0 (m) )
where λ0 (m) > · · · > λk−1 (m) ≥ 1 by Proposition 2.8.

SYMBOLIC GENERIC INITIAL SYSTEMS OF SIX POINTS

11

Step 4: Compute the Newton polytope Pgin(I (m) ) of each gin(I (m) ) for infinitely many m. Recall that the boundary of these polytopes is
determined by the convex hull of the points (i, λi (m)) and (α(m), 0).
Step 5: Find the limiting polytope of the symbolic generic initial system of
I. To do this it suffices to take the limit
[ 1
P =
Pa
m m
∗
m∈N

over an infinite subset of N∗ .
All of the remaining calculations are similar but long so, for the sake of
space, we will only record the proof here for configuration H.
4.1. Proof of main theorem for configuration H. Let I be the ideal of
points p1 , . . . , p6 of configuration type H, ordered as in Figure 3.
Step 1.
First we will follow the method outlined in Section 3 to find HI (m) for
infinitely many m. We will use the notation from Section 3 and will often
denote the divisor a0 E0 − (a1 E1 + a2 E2 + a3 E3 + a4 E4 + a5 E5 + a6 E6 ) by
(a0 ; a1 , a2 , a3 , a4 , a5 , a6 ). Also, if F1 and F2 are divisors, F1 · F2 denotes
[F1 ] · [F2 ], the intersection multiplicity of their classes.
First we need to determine NEG(X). Note that the configuration type H
is defined by a line through points 1, 2, and 3 and another line through points
1, 4, and 5. Thus, neg(X) consists of the classes of A1 := E0 − E1 − E2 − E3
and A2 := E0 − E1 − E4 − E5 . The other elements of NEG(X) are exactly
those [C] ∈ B ∪ L ∪ Q such that [C]2 = −1 and [C] · [D] ≥ 0 for all
[D] ∈ neg(X) by Lemma 3.4. Using this, one can check that NEG(X)
consists of the classes of the divisors
A1 := E0 − E1 − E2 − E3 , A2 := E0 − E1 − E4 − E5 ,
B := E0 − E1 − E6 ,
Ci := E0 − Ei − E6 for i = 2, 3, 4, 5,
Dij := E0 − Ei − Ej for i = 2, 3 and j = 4, 5,
Q := 2E0 − E2 − E3 − E4 − E5 − E6 .
Next, we will follow Procedure 3.6 for each Ft once we fix m divisible by
12. The procedure produces a divisor Ht that is either numerically effective
or is in the class of an effective divisor such that
HI (m) (t) = h0 (X, Ft ) = h0 (X, Ht ).
First, we will make some observations about which elements of NEG(X)
may be subtracted during the procedure.
Suppose that J is a divisor of the form J := (a; b, c, c, c, c, d). We will
show that if the procedure allows us to subtract one Ai (respectively, one
Ci or one Dij ) from J, we can subtract them all consecutively. This is
equivalent to showing that if the intersection multiplicity of J with A1 is

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SARAH MAYES

negative then the intersection multiplicity of J −A1 with A2 is also negative;
parallel statements hold for the Ci and Dij .
Ai :
J · A1 = a − b − 2c
(J − A1 ) · A2 = (a − 1; b − 1, c − 1, c − 1, c, c, d) · A2
= a − 1 − (b − 1) − 2c = a − b − 2c
Ci :
J · C2 = a − c − d
(J − C2 ) · C3 = (a − 1; b, c − 1, c, c, c, d − 1) · C3
= (a − 1) − c − (d − 1) = a − c − d
(J − C2 − C3 ) · C4 = (a − 2; b, c − 1, c − 1, c, c, d − 2) · C4
= (a − 2) − c − (d − 2) = a − c − d
(J − C2 − C3 − C4 ) · C5 = (a − 3; b; c − 1, c − 1, c − 1, c, d − 3) · C5
= (a − 3) − c − (d − 3) = a − c − d
Dij :
J · D24 = a − 2c
(J − D24 ) · D25 = (a − 1; b, c − 1, c, c − 1, c, d) · D25
= (a − 1) − (c − 1) − c = a − 2c
(J − D24 − D25 ) · D34 = (a − 2; b, c − 2, c, c − 1, c − 1, d) · D34
= (a − 2) − c − (c − 1) = a − 2c − 1
(J − D24 − D25 − D34 ) · D35 = (a − 3; b, c − 2, c − 1, c − 2, c − 1, d) · D35
= (a − 3) − 2(c − 1) = a − 2c − 1
Define
A := A1 + A2 , C := C2 + C3 + C4 + C5 , D := D24 + D25 + D34 + D35 .
The calculations above show that if J · A1 < 0 (if J · C2 < 0, J · D24 < 0,
respectively) then the procedure will allow us to subtract one entire copy of
A (C, D). If we begin with a divisor of the form J = (a; b, c, c, c, c, d) then
J − A, J − B, J − C, J − D, and J − Q have the same form. These facts
taken together mean that that Ht is obtained from Ft - a divisor with the
same form as J - by subtracting off copies of A, B, C, D, and Q.
In Procedure 3.6, the requirement for being able to subtract an element
of NEG(X) from J is that the intersection of that element with J is strictly
negative. Thus, it is of interest how the intersection multiplicities with
elements of NEG(X) change as other elements of NEG(X) are subtracted
from a divisor of the form (a; b, c, c, c, c, d).
If J = (a; b, c, c, c, c, d) as above, we have the following.

SYMBOLIC GENERIC INITIAL SYSTEMS OF SIX POINTS

13

value of G
Ai B Ci Dij Q
(J − A) · G − J · G 2 0 -1
0
0
(J − B) · G − J · G 0 1 0
-1 -1
(J − C) · G − J · G -2 0 1
-2 0
(J − D) · G − J · G 0 -4 -2
0
0
(J − Q) · G − J · G 0 -1 0
0
1
We now use this set-up to obtain Ht from Ft by successively subtracting
elements of NEG(X) that have negative intersection with the remaining
divisor. First note that
Ft · Ai = t − 3m < 0 ⇐⇒ t < 3m,
Ft · B = Ft · Ci = Ft · Dij = t − 2m < 0 ⇐⇒ t < 2m,
and

5m
.
2
Therefore, [Ft ] = [Ht ] (that is, Ft is numerically effective) if and only if
t ≥ 3m. In this case, h0 (X, Ft ) = 12 t2 − 3m2 + 32 t − 3m + 1 by Lemma 3.3.
We will assume from this point on that 12|m.
Now suppose that 3m > t ≥ 5m
2 . In this case, [Ai ]·[Ft ] < 0, but [C]·[Ft ] ≥
0 for all other [C] ∈ NEG(X); thus, Procedure 3.6 allows us to subtract Ai
- and thus A - but no other divisors initially. How many copies can we
subtract? From the table, we see that the intersection multiplicity of the
remaining divisor with Ai increases by 2 each time we subtract a copy of Ai .
We can keep subtracting copies of A as long as the intersection multiplicity
with Ai is strictly negative; thus, we can subtract exactly
' &
'
&
3m − t
F t · Ai
=
−
2
2
Ft · Q = 2t − 5m < 0 ⇐⇒ t <

copies of A. The only other intersection multiplicity that changes through
the process subtracting As is with the Ci , which decreases by one for each
copy of A subtracted. Thus,
&
' !
&
'
3m − t
3m − t
Ft −
A · Ci = t − 2m −
2
2
7m
and this is never negative when t ≥ 5m
2 (t must be at most 3 for this expres
sion to be negative). Thus, the intersection multiplicity of Ft − 3m−t
with
2
all [C] ∈ NEG(X) is nonnegative, so
&
'
&
'
&
'
!
3m − t
3m − t
3m − t
Ht = t − 2
;m − 2
,m −
,...,m .
2
2
2

When t is even,


t−m
Ht = 2t − 3m; t − 2m,
,...,m
2

14

SARAH MAYES

and h0 (X, Ft ) = t2 − 3tm − 32 m2 + 32 t − 3m + 1 while when t is odd


t−m−1
,...,m
Ht = 2t − 3m − 1; t − 2m − 1,
2
3 2
3
1
0
2
and h (X, Ft ) = t − 3tm − 2 m + 2 t − 3m + 2 .
7m
Now suppose that 5m
2 > t ≥ 3 . In this case, Procedure 3.6 allows us
to subtract copies of Q because Ft · Q < 0. From the table, for each copy
of Q subtracted, the intersection multiplicity increases by 1; since we can
keep subtracting copies of Q as long as the intersection multiplicity with the
remaining divisor is negative, we
l can msubtract exactly −Ft · Q = 5m − 2t
A by the same argument as in the
copies. We may also subtract 3m−t
2
previous case, since subtracting copies of A doesn’t change the intersection
multiplicity with Q and vice versa.
Through the process of subtracting As and Qs the intersection multiplicities with Ci and B have changed; in particular,
&
'
l 3m − t m
3m − t
A − (5m − 2t)Q) · Ci = t − 2m −
(Ft −
2
2
and
(Ft −

l 3m − t m

A − (5m − 2t)Q) · Ci = t − 2m − (5m − 2t) = 3t − 7m.
2
These are both nonnegative, as t ≥ 7m
3 , so the intersection multiplicity
of the remaining divisor with all elements of NEG(X) is nonnegative and
Procedure 3.6 terminates.1 Therefore, when t is even
5t − 11m
, . . . , 2t − 4m)
Ht = (6t − 13m; t − 2m,
2
and h0 (X, Ft ) = 3t2 − 13tm + 14m2 + 52 t − 11
2 m + 1, while when t is odd
5t − 11m − 1
, . . . , 2t − 4m)
2
1
and h0 (X, Ft ) = 3t2 − 13tm + 14m2 + 52 t − 11
2 m + 2.
7m
Now suppose
as above, we can

 that2mt = 3 − 1. By the same arguments
m
subtract 3m−t
=
copies
of
A
and
5m
−
2t
=
+
2
copies
of Q when
2
6
3
following Procedure 3.6. Then
m



2m
m
m
2m
F 7m −1 −
A−
+ 2 Q = m − 7; − 2, − 3, . . . ,
−2
3
6
3
3
3
3
has intersection multiplicity 1 with Ai and -2 with Ci . At this point, Procedure 3.6 allows us to do the following.
• Subtract one copy of C. Now the intersection multiplicity with Ai
is −1 and the intersection multiplicity with Ci is −1.
Ht = (6t − 13m − 1; t − 2m − 1,

1 7m is always an integer under the divisibility assumption so we don’t have to worry about
3

t being the smallest odd integer less than

7m+1
.
3

SYMBOLIC GENERIC INITIAL SYSTEMS OF SIX POINTS

15

• Subtract one copy of A. Now the intersection multiplicity with A is
1 and the intersection multiplicity with Ci is −2.
It is clear that we can repeat this process as many times as we wish when
we follow the procedure; eventually, we will end up with a divisor that has
a negative E0 coefficient. We have that HI (m) (t) = h0 (X, Ft ) = 0 when
7m
t = 7m
3 − 1 and thus HI (m) (t) = 0 for all t < 3 .
Step 2.
Assume that 12|m.
Now we will turn our attention to the generic initial ideals of I (m) . We
compute the number of generators of gin(I (m) ) in each degree using Lemma
4.1 and the Hilbert function values from Step 1. We have the following.
Value of t
Number of generators of degree t
t < 7m
0
3
1
7m
t= 3
3m + 1
2
t = 7m
+
1
3
3m + 3
5m
7m
>
t
≥
+
2,
t
even
6
2
3
5m
7m
4
2 > t ≥ 3 + 2, t odd
t = 5m
6
2
+
1
1
t = 5m
2
5m
3m > t ≥ 2 + 2, t even
2
3m > t ≥ 5m
+
2,
t
odd
0
2
t = 3m
2
t > 3m
0
Step 3.
Assume once again that 12|m.
Note that there are
m
5m
7m
−2
m
2 − 3 −2
= 6
=
−1
2
2
12
7m
even (or odd) integers t such that 5m
2 > t ≥ 3 + 2, and
3m −

5m
2

−2

m
−1
2
4
even (or odd) integers t such that 3m > t ≥ 5m
2 + 2.
Using the results of Step 2, we can find strictly decreasing λi such that
=

gin(I (m) ) = (xk , xk−1 y λk−1 , . . . , xy λ1 , y λ0 ).
7m
Since the smallest degree generator is of degree 7m
3 , k = 3 .
The values of λi that we obtain are shown in the following table.

16

SARAH MAYES

7m
7m
degree 7m
3
3 +1
3 +2
7m
7m
4m
4m
4m
i 3
−
1
·
·
·
2m
2m
−
1
2m
−
2
·
·
·
−
3
3
3
3 − 4 ···
3 −9
m
m
m
· · · m + 4 m + 6 · · · m + 11
λi 0
1
···
3
3 +2
3 +3
5m
degree 7m
+
3
·
·
·
3
2
4m
4m
m
m
m
i 4m
−
10
·
·
·
−
13
·
·
·
−
4
−
10(
··· m
3
3
3
12 − 1) = 2 + 6
2 +5
2 +1
m
λi m + 13 · · · m + 16 · · · m + 6 + 12( 12 − 1) = 2m − 6 2m − 5 · · · 2m − 1
5m
5m
degree 5m
···
3m
2 +1
2 +2
2 +4
m
m
m
m
m
m
m
i
−
1
−
2
−
3
−
4
·
·
·
−
1
−
2(
0
2
2
2
2
2
2
4 − 1) = 1
m
λi 2m + 1 2m + 3 2m + 4 2m + 7 2m + 8 · · · 2m + 3 + 4( 4 − 1) = 3m − 1 3m

Step 4.
Assume that 12|m.
The Newton polytope of gin(I (m) ) is the convex hull of the ideal when
thought of as a subset of R2 . In particular, its boundary is determined by
the points (i, λi ) recorded in the table from Step 3. Plotting these points,
one can see that the boundary of Pgin(I (m) ) is defined by the line segments
m
4m
through the points (0, 3m), ( m
2 −2, 2m+4), ( 2 +1, 2m−1), ( 3 −9, m+11),
m
7m
4m
( 3 − 3, m + 4), (2m, 3 ), and ( 3 , 0).
Step 5.
1
and taking the limit as m
Scaling Pgin(I (m) ) from the previous step by m
approaches infinity, the limiting shape of the symbolic generic initial system
is defined by the line segments through the following points.
 3m 
(0, 3) = lim 0,
m→∞
m

 m/2 − 2 2m + 4 
 m/2 + 1 2m − 1 
, 2 = lim
,
= lim
,
m→∞
m→∞
2
m
m
m
m

1

 4m/3 − 9 m + 11 
 4m/3 − 3 m + 4 

= lim
, 1 = lim
,
,
m→∞
m→∞
3
m
m
m
m
 1
 2m m/3 
2,
= lim
,
m→∞
3
m m
7 
 7m/3 
, 0 = lim
,0
m→∞
3
m
Note that (2, 31 ) lies on the line segment connecting ( 43 , 1) with ( 37 , 0) so it
is not a vertex of the boundary of the limiting shape.
4

5. Point Configurations and Limiting Shapes: Questions and
Observations
In this section we investigate how the arrangement of points in a point
configuration influences the limiting shape of the symbolic generic initial
system of the corresponding ideal. Throughout I will be the ideal of a point
configuration in P2 and P will denote the limiting shape of {gin(I (m) )}m .

SYMBOLIC GENERIC INITIAL SYSTEMS OF SIX POINTS

17

The following is Lemma 2.5 of [May12a] and is proven there; it describes
how the number of points in a configuration is reflected in the limiting shape.
Lemma 5.1. Let I be the ideal corresponding to an arrangement of r distinct
points in P2 . If Q is the complement of the limiting shape P of {gin(I (m) )}m
in R2≥0 , then the area of Q is equal to 2r .
While this lemma imposes strong restrictions on where the limiting shape
can lie, we would like a more precise description in terms of geometry.
Question 5.2. What is the meaning of the intercepts of the boundary of
P ? Can one see these intercepts in the point configuration?
We may partially answer this question. The x-intercept of P is equal to
limm→∞ α(m)
m where α(m) is the degree of the smallest degree element of
(m)
I . This limit has been studied for some special point configurations and
is sometimes equal to the Seshadri constant of I (see, for example, [BH10]
and [Har02]). Unfortunately, it does not seem as though there is a simple
connection to the point configuration.
(m)
The y-intercept of the boundary of P is equal to limm→∞ reg(Im ) (see
Lemma 3.1 of [May12a]). This limit is not as well-studied as the previous
one, but appears to have a nice geometric meaning in certain cases. For
example, when there is a line passing through at least three points and
algorithms similar to the one outlined in Section 3 may be used to find the
(m)
Hilbert function of I (m) , limm→∞ reg(Im ) is equal to the maximum number
of points lying on a single line.
Question 5.3. What features does a point configuration possess when the
boundary of P consists of a fixed number of line segments?
To describe one potential answer to Question 5.3, we will distinguish
between different ‘types’ of points within a configuration.
A curve of degree

d+2
d defines a point configuration if at least 2 points in the configuration
lie on the curve. If N points of the configuration lie on such a curve, we will
denote this curve by Cd,N . For example, in Configuration H shown in Figure
3, there are two curves defining the point configuration. Since they are both
lines containing three points, the set of curves defining the configuration is
denoted {C1,3 , C1,3 }.
Each point within a configuration may then be associated with the set of
curves defining the configuration that pass through that point. For example,
in Figure 3, points 2, 3, 4, and 5 correspond to the set {C1,3 }, point 1
corresponds to the set {C1,3 , C1,3 }, and point 6 corresponds to the empty
set. We will call such sets the incidence type of a point. Thus, Configuration
H has three distinct incidence types. Configuration F also has three distinct
incidence types: there are two points of incidence type {C1,3 }, three of
type {C1,4 }, and one of type {C1,3 , C1,4 }. Configuration I has two distinct
incidence types.

18

SARAH MAYES

Observation 5.4. Suppose that I is the ideal corresponding to one of the
following subsets of P2 : a point configuration of at most six points; a point
configuration arising from a complete intersection; a generic set of points;
points on an irreducible conic; a point configuration where all but one point
lies on a line; a point configuration where all but two points lies on a line
and no other line passes through three points; or a star point configuration.
Then the number of line segments forming the boundary of the limiting
shape of {gin(I (m) )}m is equal to the number of distinct incidence types of
the points in the corresponding point configuration ([May12a], [May13a],
[May13b], [May12c]).
While we do not have enough evidence to claim that the answer to Question 5.3 is always given by the number of incidence types in a configuration,
it is interesting to note that it holds for all of the cases that have been studied up to this point. This provides further evidence that our asymptotic
viewpoint reveals information that cannot be seen by looking at the Hilbert
functions of individual fat point ideals I (m) .
Question 5.5. Is there a geometric interpretation for the coordinates of
the ‘crux points’ lying on the intersection of the line segments defining the
boundary of P ?
The answer to this final question seems mysterious; it is likely that many
more configurations will need to be studied to formulate a reasonable conjecture to answer this question.
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A. Arsie and J.E. Vatne, A note on symbolic and ordinary powers of homogeneous ideals, Annali dell’Universita di Ferrara 49 (2003), no. 1, 19–30.
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Department of Mathematics, University of Michigan, 530 Church Street,
Ann Arbor MI 48109
E-mail address: mayess@umich.edu

