A semigroup defining the Gröbner degeneration of a toric ideal

We give an explicit set of generators for the semigroup of the Gröbner degeneration of a toric ideal. This set of generators is used to study algebraic properties of the semigroup it generates: approximation of semigroups, non-preservation of saturation, Betti elements, uniqueness of presentations, and Möbius functions.


Introduction
The Gröbner degeneration of an ideal is a flat family that deforms the ideal into a simpler one, for instance, a monomial ideal.Together with the theory of Gröbner bases, Gröbner degenerations are a powerful tool in the study of algebraic and geometric properties of an ideal or the corresponding algebraic variety.
In this paper we study Gröbner degenerations of toric ideals.The starting point for the whole discussion is the fact that the Gröbner degeneration of a toric ideal is also a toric ideal.Hence, it is defined by some semigroup.Several natural questions can be deduced concerning the algebraic or combinatorial properties of this semigroup.Yet, somehow surprisingly, they have not been systematically explored as far as we know.This paper aims at giving a first step towards that study.
Let S ⊂ Z d be a semigroup defining a toric ideal I. Let S w ⊂ Z d+1 be the semigroup of the Gröbner degeneration of I with respect to some weight vector w.Our first result provides an explicit set of generators of S w (Theorem 1.2).The resulting set of generators is our main tool to explore to what extent properties of the semigroup S are preserved for S w .
We first study approximations of semigroups.It is known that the semigroup S can be approximated by its saturation S.This means that there exists a ∈ S such that a+S ⊂ S. We prove that the corresponding result for the semigroup S w also involves the element a ∈ S (Theorem 2.3).We stress that approximation of semigroups has a great deal of applications, including the study of valuations on graded algebras, the computation of dimension and degree of projective varieties, and intersection theory [10].
Next, we show that saturation is not preserved under Gröbner degenerations.We exhibit an infinite family of saturated semigroups for which a degeneration is saturated only for finitely many of them (Section 3).This suggests that saturation is actually rarely preserved under degenerations.Even though this phenomenon was somehow predictable, there are no explicit examples illustrating this fact in the literature, as far as we are aware.
We then study Betti elements of semigroups, a concept that is closely related to syzygies of toric ideals [7].As before, the goal is to study the behaviour of Betti elements under Gröbner degenerations.We first show that each Betti element of the semigroup S gives rise to a Betti element of S w (Theorem 4.4).However, in general not all Betti elements of S w can be described like this (Example 4.7).
A particular case that we study is that of semigroups having a unique minimal generating set.These semigroups appear, for instance, in Algebraic Statistics [15].We show that this property is preserved in the degeneration for certain families (Sections 4.1 and 4.2).In our opinion, this gives enough evidence to conjecture that the uniqueness of a minimal generating set is always preserved under Gröbner degenerations (Conjecture 4.18).
The last section concerns Möbius functions of semigroups.This notion was introduced by G.-C. Rota and has a great deal of applications [13].For a long time, Möbius functions were investigated only for numerical semigroups, although more general results were obtained in recent years [1].In particular, an explicit formula for the Möbius function of semigroups with unique Betti element was given in [1,Theorem 4.1].As an application of this formula, we conclude this paper by showing that the Möbius function of S w can be computed using only data of S and w, whenever both S and S w have a unique Betti element (Theorem 5.4).

An explicit semigroup defining the Gröbner degeneration of a toric ideal
Let us start by recalling the construction of a Gröbner degeneration of an ideal with respect to a weight vector.Let J ⊂ K[x 1 , . . ., x n ] be an ideal and w = (w 1 , . . ., w n ) ∈ N n .For The ideal J(t, w) := f t |f ∈ J is called the Gröbner degeneration of J.It is a classical fact that J(t, w) gives rise to a flat family deforming the affine algebraic variety defined by J to the variety of the initial ideal of J with respect to w [3, Theorem 15.17].
The following remark, which will be constantly used in this paper, provides a method to compute generators for the ideal J(t, w).
Our first goal is to give an explicit generating set of the semigroup defining the Gröbner degeneration of a toric ideal.This generating set is described in terms of generators of the original semigroup and the weight vector.
The following notation will be constantly used throughout this paper.
Let A = {a 1 , . . ., a n } ⊂ Z d be a finite subset.Consider the following map of semigroups, The image of π A is denoted as NA.The previous map induces a map of K-algebras, Denote I A := ker πA .It is well known that this ideal is prime and generated by binomials.More precisely [14,Chapter 4], The ideal I A is called the toric ideal defined by A.
The following theorem describes an explicit subset of Z d+1 that determines the Gröbner degeneration of I A .
Then I A (t, w) = I Aw .
Proof.Let u = (u 1 , . . ., u n ) ∈ N n and u n+1 ∈ N. We have the following relation: Let us first show that I A (t, w) ⊂ I Aw .Let G ⊂ I A be a Gröbner basis consisting of binomials with respect to a refined monomial order > w and such that G(t, w) = {g t |g ∈ G} generates I A (t, w) (see Remark 1.1).We show that G(t, w) Thus, g t ∈ I Aw and so I A (t, w) ⊂ I Aw .Now we show In particular, π A (u) = π A (v) and so and we conclude that The previous theorem is our main tool to explore several combinatorial and algebraic properties of the Gröbner degeneration of toric ideals.
Remark 1.3.García-Puente, Sottile and Zhu introduced a set similar to A w to define regular subdivisions of A [4,16].
We conclude this section by defining the class of semigroups considered in this paper.Definition 1.4.An affine semigroup S is a finitely generated submonoid of Z d such that the group generated by S is Z d .Remark 1.5.Given an affine semigroup generated by A = {a 1 , . . ., a n } ⊂ Z d and w ∈ N n , we have that NA w ⊂ Z d+1 is a finitely generated semigroup.In addition, since NA generates Z d as a group and (0, . . ., 0, 1) ∈ A w , we obtain that NA w generates Z d+1 as a group.Hence, NA w is an affine semigroup.

Approximations of semigroups
Our first application of Theorem 1.2 concerns approximations of affine semigroups.
A well-known result in the theory of affine semigroups states that such semigroups can be approximated by their saturation.Recall that the saturation of an affine semigroup S ⊂ Z d is defined as S := R ≥0 S ∩ Z d .Equivalently, S = {a ∈ Z d |ka ∈ S, for some k ∈ N}.
Theorem 2.1.[10, Theorem 1.4] Let S ⊂ Z d be an affine semigroup.There exists a ∈ S such that a + S ⊂ S.
Our next goal is to study this theorem for the affine semigroup of the Gröbner degeneration of Assume that there exists an element a ∈ NA such that a + NA ⊂ NA.Let w ∈ N n .Consider the following sets: i=1 β i a i for some β i ∈ N. Now consider the following cases.

Saturation
In this section we show that saturation of affine semigroups is not preserved by Gröbner degeneration.Recall that an affine semigroup S is saturated if Let m ∈ N. Consider the set It is well known that NA(m) is a saturated semigroup.Let w = (1, 1, 1).We show that NA(m) w is saturated if and only if m ≤ 2. We use this notation throughout this section.
We want to show that (a, b, c) ∈ NA(1) w .There exist α i ∈ N such that For the other inclusion it is enough to show that A 1 ∩ Z 3 ⊂ NA(2) w , where By the third equation it follows that z ≤ 4.
Now we are ready to study the remaining values of z.
Proposition 3.3.The semigroup NA(m) w is not saturated for all m ≥ 3.

Betti elements
As in previous sections, let S ⊂ Z d be an affine semigroup generated by A = {a 1 , . . ., a n }.Given w ∈ N n , denote S w := NA w .In this section we study the behaviour of Betti elements of affine semigroups under Gröbner degenerations.We assume that S is pointed, that is, S ∩ (−S) = {0}.We first recall the basic definitions we need.Let ∼ A denote the kernel congruence of π A , i.e., α for S is a system of generators of ∼ A .A minimal presentation for S is a minimal system of generators of ∼ A .Notice that this is equivalent to ask for a minimal set of binomial generators of the corresponding toric ideal.Our first result relates the Betti elements of S with those of S w .First we prove a simple lemma.Lemma 4.3.Let w ∈ N n .Let {g 1 , . . ., g s } ⊂ I A be a generating set of I A .Assume that {(g 1 ) t , . . ., (g r ) t } generates I Aw , for some r ≤ s.Then {g 1 , . . ., g r } generates I A .
Proof.If r = s, there is nothing to prove.Suppose r < s and let j ∈ {r + 1, . . ., s}.By hypothesis, (g j ) t = r i=1 h i (x, t)(g i ) t .Making t = 1 it follows g j = r i=1 h i (x, 1)g i .
It is known that Example 12].Define Notice that the binomials f i , h i j , and g i j are S-homogeneous.In addition, {f 2 , . . ., f n } is a minimal generating set of I A .Indeed, assume, for instance, that f 2 can be written in terms of f 3 , . . ., f n .Then, evaluating at (0, 1, 0, . . ., 0) we obtain a contradiction.By the cardinality of {g i 2 , . . ., g in }, it follows that this set is also a minimal generating set of I A .
Let > denote the lexicographical order on K[x 1 , . . ., x n ] with the variables ordered as Then, for each j ∈ {2, . . ., n}, we have In particular, for any 1 < j < k ≤ n, the leading terms LT >w (g i j ) and LT >w (g i k ) are relatively prime.Hence, {g i 2 , . . ., g in } is a Gröbner basis of I A with respect to > w [9,Corollary 2.3.4].It follows that . We already proved that {g i 2 , . . ., g in } is a minimal generating set of I A .Therefore, {(g i 2 ) t , . . ., (g in ) t } is also a minimal generating set of I Aw (see Remark 1.1 and Lemma 4.3).We conclude that Betti(S w ) equals the set ) is well-defined and bijective.

Semigroups with unique minimal generating set
In this section we explore further consequences of Theorem 4.4 in the context of affine semigroups having a unique minimal generating set.Such semigroups have been studied, for instance, in [2,6,12].
We say that an affine semigroup S is uniquely presented if it has a unique minimal presentation.Notice that this is equivalent to the corresponding toric ideal having a unique minimal generating set of binomials, up to scalar multiplication.
Remark 4.9.The notion of uniquely presented is not to be confused with the previous notion of minimally presented.The former asks for a unique minimal generating set of a toric ideal whereas the latter just asks for a minimal generating set, up to scalar multiplication.
We define a partial order > S on S as follows (recall that we assume S ∩ (−S) = {0}): α > S β if α − β ∈ S. We say that α ∈ Betti(S) is Betti minimal if it is minimal with respect to the order > S .The set of such elements is denoted as Betti-min(S).Corollary 4.12.Let S be a uniquely presented affine semigroup.Let w ∈ N n be such that some minimal generating set of I A is also a Gröbner basis with respect to some refined order > w .Then all Betti elements of S w are Betti-minimal.In particular, I Aw is also uniquely presented.
Proof.Let {g 1 , . . ., g r } ⊂ I A be a minimal generating set that is also a Gröbner basis with respect to > w .Then {(g 1 ) t , . . ., (g r ) t } ⊂ I Aw is a generating set.In addition, by Lemma 4.3, it is minimal.Thus, every Betti element of S w is of the form (b, λ), for some λ ∈ N and b ∈ Betti(S).By Theorem 4.11, such Betti elements are Betti-minimal.The last statement of the corollary follows from Remark 4.10.
Let us look at an example where the conditions of the previous corollary are satisfied.
Example 4.13.Let S = a, a + 1, a + 2 ⊂ N, where a = 2q ≥ 4. The only minimal set of binomial generators of I A is {y 2 − xz, x q+1 − z q } [6, Theorem 15].Let w ∈ N 3 be such that 2w 2 > w 1 + w 3 .Then the leading terms of these two binomials with respect to any refined order > w are relatively prime.Hence, they form a Gröbner basis with respect to > w .By Corollary 4.12, I Aw is uniquely presented.Now we show an example of a uniquely presented affine semigroup such that any of its Gröbner degenerations is also uniquely presented.
As usual, let A = {a 1 , . . ., a n }.The Lawrence ideal of A, denoted I Λ(A) , is the ideal of K[x 1 , . . ., x n , y 1 , . . ., y n ] generated by This ideal is studied, for instance, in [14, Chapter 7].There, Lawrence ideals are used as an auxiliary tool to compute Graver bases.The relevant fact for us is that Lawrence ideals are uniquely presented [12,

A further example of Gröbner degenerations preserving the uniqueness of a presentation
In this section we present a further example showing that Gröbner degenerations preserve the property of being uniquely presented.We stress that, as opposed to Example 4.13 or Corollary 4.14, the results of this section do not rely on Corollary 4.12.
Proof.Let p 1 = y 2 − xz and p 2 = x q+2 − z q+1 .A straightforward computation shows that I A = p 1 , p 2 .More generally, generators for toric ideals of semigroups generated by intervals can be found in [8,Theorem 8].
1. Suppose 2w 2 ≥ w 1 + w 3 .Then the initial monomials of p 1 and p 2 are relatively prime.Hence, G = {p 1 , p 2 } is a Gröbner basis of I A .
By hypothesis, π A (α i ) = π A (α j ) for all i = j, and G 0 is a minimal generating set of I A .By Remark 4.2 β 1,π A (α i ) (K[x]/I A ) = 1, for each i ∈ {1, . . ., m}.This is a contradiction.Thus G 0 is the unique minimal generating set of binomials, up to scalar multiplication.
By the inequalities satisfied by w in this case, it follows that this set is a minimal generating set of the ideal it generates.It remains to prove that the A w -degrees are all different.It is enough to show this for the first entry of the A w -degrees.Indeed, the first entries are {2a + 2, (q + 2)a, . . ., (q + i + 3)a, . . ., (2q + 3)a}.
We conclude that I Aw is uniquely presented by Lemma 4.16.
Remark 4.18.By computing Gröbner bases and using Lemma 4.16, we verified that Proposition 4.17 also holds for other families of uniquely presented numerical semigroups generated by intervals.Moreover, we used the same method to study this property for other families of numerical semigroups.
Our computations give enough evidence to conjecture that the uniqueness of a presentation of a toric ideal is preserved under Gröbner degenerations.

Möbius functions
In this final section we study Möbius functions of affine semigroups.Several authors have provided explicit formulas for Möbius functions of some families of semigroups.In particular, the case of semigroups with a unique Betti element was studied in [1].As a final application of Theorem 1.2, we present some relations among the Möbius functions of S and S w , in the case where both S and S w have a unique Betti element.Let S ⊂ Z d be a pointed affine semigroup.As in previous sections, consider the following partial order: for x, y ∈ Z d , x < S y if y − x ∈ S.An interval on Z d with respect to < S is defined as The Möbius function of S, denoted µ S , is defined as This sum is always finite [1, Section 2].Notice that if y / ∈ S then µ S (y) = 0 (since, in this case, c l (0, y) = 0 for all l ≥ 0).Thus, we restrict the domain of µ S to S.
The following formula is the starting point of our discussion.
Proof.Let B ∈ b (z,λ) .We have two cases: • n + 1 / ∈ B. By (i) of Lemma 5.2, B = A j for some j.Hence, (z, λ) = i∈B a ′ i + k B (b, d w ) = i∈A j a ′ i + k B (b, d w ).We also know that k B = k A j .We conclude that λ = i∈B w i + k B d w = i∈A j w i + k A j d w = l j .

Definition 4 . 1 .
Let Betti(S) := {π A (α)|(α, β) ∈ ρ}, where ρ ⊂ N n × N n is any minimal presentation of S. The set Betti(S) does not depend on ρ [7, Chapter 9].It is called the set of Betti elements of S. The terminology in the previous definition comes from the Betti numbers of I A .Recall that I A is S-graded, where deg S (x i ) = π A (e i ), and e i is the i-th element of the canonical basis of N n .The first Betti number of degree a ∈ S of K[S] := K[x]/I A , denoted by β 1,a (K[S]) is the number of minimal generators of degree a of I A .It is well known that it does not depend on the minimal set of generators of I A .The first Betti number of K[S] is the cardinality of a minimal set of generators of I A , denoted by β 1 (K[S]), so β 1 (K[S]) = a∈S β 1,a (K[S]).

Remark 4 . 2 .
In view of the previous paragraph, for a ∈ N d , a ∈ Betti(S) if and only if β 1,a (K[S]) = 0.

Remark 4 . 5 .
A similar result to Theorem 4.4 was proved in [11, Theorem 8.29].In view of Theorem 4.4, several natural questions arise.Is the element λ unique?Does every Betti element of S w have as first coordinate a Betti element of S? In the following examples we show that every scenario could actually happen.Example 4.6 shows that λ may not be unique.Example 4.7 shows that there are Betti elements of S w whose first coordinate is not a Betti element of S. Finally, Example 4.8 exhibits an infinite family of numerical semigroups where the map Betti(S) → Betti(S w ), b → (b, λ) is well-defined and bijective.These examples show that Theorem 4.4 is the best result we can expect regarding Betti elements of S and S w .
[14,s known that any minimal binomial generating set {g 1 , ..., g r } of I Λ(A) is a reduced Gröbner basis with respect to any order[14, Theorem  7.1].By Corollary 4.12, any Gröbner degeneration of the Lawrence ideal is uniquely presented.