Asymptotics of random Betti tables

Ein, Lawrence; Erman, Daniel; Lazarsfeld, Robert

doi:10.1515/crelle-2013-0032

Cited by 25 publications

(20 citation statements)

References 14 publications

(24 reference statements)

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“…Research of the author partially supported by the Simons Foundation. 1 If F = [F 0 ← F 1 ← · · · ← F n ← 0] is a minimal graded free resolution of R(C, A d ), then we will use β i,j (O C , A d ) to denote the number of minimal generators of F i of degree j. Equivalently, we have…”

Section: Setupmentioning

confidence: 99%

The Betti table of a high-degree curve is asymptotically pure

Erman¹

2015

Recent Advances in Algebraic Geometry

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Section: Setupmentioning

confidence: 99%

The Betti table of a high-degree curve is asymptotically pure

Erman¹

2015

Recent Advances in Algebraic Geometry

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“…In [EEL15,§3], the Boij-Söderberg coefficients of Veronese varieties are shown to be closely connected to the conjectural "normal distribution" property discussed in the previous section, and thus Question 0.3 naturally raises the following question: Question 6.8. For fixed b and d → ∞, how do the Boij-Söderberg coefficients behave?…”

Section: Conjecturesmentioning

confidence: 97%

Conjectures and Computations about Veronese Syzygies

Bruce

Erman

Goldstein

et al. 2018

Experimental Mathematics

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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...

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“…It is also a fascinating problem to study the asymptotic behavior of the Betti numbers k p,q (X, B; L d ) := dim K p,q (X, B; L d ) when d is sufficiently large (see [12,Problem 7.3]). In this direction, Ein-Erman-Lazarsfeld conjectured that for each 1 ≤ q ≤ n, the Betti numbers k p,q (X, L d ) converge to a normal distribution (see [9,Conjecture B], [14,Conjecture 3.2]). This normal distribution conjecture has not been verified even for P 2 and P 1 × P 1 , and it seems that the conjecture is already very challenging for Veronese embeddings (cf.…”

Section: Open Problemsmentioning

confidence: 99%

Asymptotic vanishing of syzygies of algebraic varieties

Park¹

2021

Preprint

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The purpose of this paper is to prove Ein-Lazarsfeld's conjecture on asymptotic vanishing of syzygies of algebraic varieties. This result, together with Ein-Lazarsfeld's asymptotic nonvanishing theorem, describes the overall picture of asymptotic behaviors of the minimal free resolutions of the graded section rings of line bundles on a projective variety as the positivity of the line bundles grows. Previously, Raicu reduced the problem to the case of products of three projective spaces, and we resolve this case here.

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Asymptotics of random Betti tables

Cited by 25 publications

References 14 publications

The Betti table of a high-degree curve is asymptotically pure

The Betti table of a high-degree curve is asymptotically pure

Conjectures and Computations about Veronese Syzygies

Asymptotic vanishing of syzygies of algebraic varieties

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