Computing graded Betti tables of toric surfaces

Castryck, Wouter; Cools, Filip; Demeyer, Jeroen; Lemmens, Alexander

doi:10.1090/tran/7643

Cited by 9 publications

(11 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…First, we show in Lemma 2.3 that each minimal polygon ∆ satisfies ls (∆) = lw(∆), where = conv{(0, 0), (1, 0), (1, 1), (0, 1)}. The latter can also be proven using results on lattice width directions of interior lattice polygons (see [4,Lemma 5.3]), but we choose to keep the paper self-contained and have provided a different proof. Moreover, we use the technical Lemma 2.2 in the proofs of both Lemma 2.3 and Theorem 2.4.…”

mentioning

confidence: 79%

“…In the joint paper [4] with Castryck and Demeyer, we study the Betti table of the toric surface Tor(∆) ⊂ P (∆∩Z 2 )−1 for lattice polygons ∆. In particular, we present a lower bound for the length of the linear strand of this Betti table in terms of lw(∆), which we conjecture to be sharp.…”

mentioning

confidence: 99%

“…In particular, we present a lower bound for the length of the linear strand of this Betti table in terms of lw(∆), which we conjecture to be sharp. For showing this conjecture for polygons of a fixed lattice width, it essentially suffices to prove it for the minimal polygons (see [4,Corollary 5.2]). Hence, Theorem 2.4 allows us to check the conjecture using a computer algebra system.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Minimal Polygons with Fixed Lattice Width

Cools

Lemmens

2019

Ann. Comb.

Self Cite

View full text Add to dashboard Cite

show abstract

mentioning

confidence: 79%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Minimal Polygons with Fixed Lattice Width

Cools

Lemmens

2019

Ann. Comb.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Additionally we produce an array of new conjectures related to Questions 0.1 and 0.3, including conjectures on: Boij-Söderberg coefficients; the number of (disctinct) Schur modules appearing in K p,q (P n , b; d); and a Schur functor interpretation of the conjecture of [CCDL,§8.3].…”

Section: # Of Relevantmentioning

confidence: 99%

Conjectures and Computations about Veronese Syzygies

Bruce

Erman

Goldstein

et al. 2018

Experimental Mathematics

View full text Add to dashboard Cite

We formulate several conjectures which shed light on the structure of Veronese syzygies of projective spaces. Our conjectures are based on experimental data that we derived by developing a numerical linear algebra and distributed computation technique for computing and synthesizing new cases of Veronese embeddings for P 2 .We analyze the Betti numbers of S(b; d), as well as multigraded and equivariant refinements. We write DateThus β p,p+q (P n , b; d) denotes the vector space dimension of K p,q (P n , b; d). The natural linear action of GL n+1 (C) on S induces an action on K p,q (P n , b; d), and so we can decompose this as a direct sum of Schur functors of total weight d(p + q) + b i.e.where S λ is the Schur functor corresponding to the partition λ [FH91, p. 76]. This is the Schur decomposition of K p,q (P n , b; d), and is the most compact way to encode the syzygies. Specializing to the action of (C * ) n+1 , gives a decomposition of K p,q (P n , b; d) into a sum of Z n+1 -graded vector spaces of total weight d(p + q) + b. Specifically, writing C(−a) for the vector space C together with the (C * ) n+1 -action given by (λ 0 , λ 1 , . . . , λ n )·µ = λ a 0 0 λ a 1 1 · · · λ an n µ we haveas a Z n+1 -graded vector spaces, or equivalently as (C * ) n+1 representations. This is referred to as the multigraded decomposition of K p,q (P n , b; d).We are motivated by three main questions. The most ambitious goal is to provide a full description of the Betti table of every Veronese embedding in terms of Schur modules.Question 0.1 (Schur Modules). Compute the Schur module decomposition of K p,q (P n , b; d).Almost nothing is known, or even conjectured, about this question, even in the case of P 2 . Our most significant conjecture provides a first step towards an answer to this question. Specifically, Conjecture 6.1 proposes an explicit prediction for the Schur modules S λ ⊆ K p,q (P n , b; d) with the most dominant weights.Our second question comes from Ein and Lazarsfeld's [EL12, Conjecture 7.5] and is related to more classical questions about Green's N p -condition for varieties [Gre84a, EL93]:Question 0.2 (Vanishing). When is K p,q (P n , b; d) = 0?Our Conjecture 6.1 would also imply [EL12, Conjecture 7.5], and thus it offers a new perspective on Question 0.2. Conjecture 6.1 is based on a construction of monomial syzygies, introduced in [EEL16]. Our new data suggests a surprisingly tight correspondence between the dominant weights of K p,q (P n , b; d) and the monomial syzygies constructed in [EEL16], and that there is much more to be understood from this simple monomial construction.Our third question is inspired by Ein, Erman, and Lazarsfeld's conjecture that each row of these Betti tables converges to a normal distribution [EEL15, Conjecture B]. Question 0.3 (Quantitative Behavior). Fix n, q and b.(1) Can one provide any reasonable quantitative description or bounds on K p,q (P n , b; d), either for a fixed d or as d → ∞? (2) More specifically, does the function p → dim K p,q (P n , b; d), when appropriately scaled...

show abstract

“…For a number of entries an explicit formula in terms of the defining lattice polytope was known before. Examples of this can be found in [3,11]. But for this paper the most relevant result is that of Schenck, who proved [16] that for projective toric surfaces coming from a lattice polygon with b lattice boundary points κ p,2 = 0 for all p ≤ b − 3.…”

Section: Introductionmentioning

confidence: 99%

On the n-th row of the graded Betti table of an n-dimensional toric variety

Lemmens

2017

J Algebr Comb

Self Cite

View full text Add to dashboard Cite

We prove an explicit formula for the first non-zero entry in the n-th row of the graded Betti table of an n-dimensional projective toric variety associated to a normal polytope with at least one interior lattice point. This applies to Veronese embeddings of P n . We also prove an explicit formula for the entire n-th row when the interior of the polytope is onedimensional. All results are valid over an arbitrary field k.

show abstract

Computing graded Betti tables of toric surfaces

Cited by 9 publications

References 32 publications

Minimal Polygons with Fixed Lattice Width

Minimal Polygons with Fixed Lattice Width

Conjectures and Computations about Veronese Syzygies

On the n-th row of the graded Betti table of an n-dimensional toric variety

Contact Info

Product

Resources

About