Conjectures and Computations about Veronese Syzygies

Abstract

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...

Asymptotic vanishing of syzygies of algebraic varieties

“…There has been a great deal of attention to the syzygies of Veronese or Segre-Veronese embeddings, e.g. [4,6,7,8,19,25,26,28,30]. The syzygies of these varieties have connections to representation theory and combinatorics.…”

The Schur–Veronese package in Macaulay2

“…In [Bruce et al 2020] the authors used a combination of high-throughput and high-performance computation and numerical techniques to compute the Betti tables of ‫ސ‬ 2 under the d-fold Veronese embedding, as well as the Betti tables of the pushforwards of line bundles O ‫ސ‬ 2 (b) under that embedding, for a number of values of b and d. These computations resulted in new data, such as Betti tables, multigraded Betti numbers, and Schur Betti numbers. (For b = 0, most the cases had been previously computed in [Castryck et al].)…”

“…Our interest is in studying the syzygies of S(b; d). See the introduction of [Bruce et al 2020] for background on Veronese syzygies including a summary of known results. Throughout this paper, we set K p,q ‫ސ(‬ n , b; d) := Tor R p (S(b; d), ‫)ރ‬ p+q , which is isomorphic to the vector space of degree p +q syzygies of S(b; d) of homological degree p. Using the standard conventions for graded Betti numbers, the rank of the vector space K p,q corresponds to the Betti number β p, p+q , and we write β p, p+q (S(b; d)) := dim Tor R p (S(b; d), ‫)ރ‬ p+q = dim K p,q ‫ސ(‬ n , b; d).…”

“…The main data are for the cases n = 2 and 0 ≤ b < d ≤ 8. In [Bruce et al 2020], we obtained full computations for d ≤ 6; moreover since those algorithms worked in parallel with respect to multidegrees, we obtained incomplete data for some cases where d = 7, 8, and we have included these partial data in this package as well.…”

“…The algorithms in [Bruce et al 2020] are a mix of symbolic and numeric algebra. Thus some entries in the data are not provably correct, while others are.…”