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...
We introduce the VirtualResolutions package for the computer algebra system Macaulay2. This package has tools to construct, display, and study virtual resolutions for products of projective spaces. The package also has tools for generating curves in ސ 1 × ސ 2 , providing sources of interesting virtual resolutions.
We study systems of parameters over finite fields from a probabilistic perspective, and use this to give the first effective Noether normalization result over a finite field. Our central technique is an adaptation of Poonen's closed point sieve, where we sieve over higher dimensional subvarieties, and we express the desired probabilities via a zeta functionlike power series that enumerates higher dimensional varieties instead of closed points. This also yields a new proof of a recent result of Gabber-Liu-Lorenzini and Chinburg-Moret-Bailly-Pappas-Taylor on Noether normalizations of projective families over the integers.
We show that any abelian variety over a finite field is covered by a Jacobian whose dimension is bounded by an explicit constant. We do this by first proving an effective and explicit version of Poonen's Bertini theorem over finite fields, which allows us to show the existence of smooth curves arising as hypersurface sections of bounded degree and genus. Additionally, for simple abelian varieties we prove a better bound. As an application, we show that for any elliptic curve E over a finite field and any n ∈ N, there exist smooth curves of bounded genus whose Jacobians have a factor isogenous to E n . 13
We compute the top-weight rational cohomology of Ag for g = 5, 6, and 7, and we give some vanishing results for the top-weight rational cohomology of A8, A9, and A10. When g = 5 and g = 7, we exhibit nonzero cohomology groups of Ag in odd degree, thus answering a question highlighted by Grushevsky. Our methods develop the relationship between the top-weight cohomology of Ag and the homology of the link of the moduli space of principally polarized tropical abelian varieties of rank g. To compute the latter we use the Voronoi complexes used by Elbaz-Vincent-Gangl-Soulé. Our computations give natural candidates for compactly supported cohomology classes of Ag in weight 0 that produce the stable cohomology classes of the Satake compactification of Ag in weight 0, under the Gysin spectral sequence for the latter space.
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