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

Erman, Daniel

doi:10.1017/cbo9781107416000.012

Cited by 4 publications

(9 citation statements)

References 8 publications

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“…As pure diagrams are the building blocks of all Betti tables, Theorem 1.1 states that as the degree of the embedding grows, the Betti diagram of Σ k (C, L) becomes as simple as possible. Being a direct generalization of Erman's theorem [Erm15] for curves, this result further suggests that we should expect theorems about syzygies of curves to admit generalizations to secant varieties.…”

Section: Introductionmentioning

confidence: 64%

“…To prove the main theorem, we mimic the strategy of [Erm15]. In particular, we compare the coefficient of the highest degree term of the Hilbert numerator of β(Σ k (C, L d )) and of its Boij-Söderberg decomposition.…”

Section: Asymptotic Puritymentioning

confidence: 99%

“…As heuristic evidence for the conjecture, they demonstrate that the rows of a sequence of "randomly chosen" Betti tables become normally distributed as d → ∞. Some other special cases of normally distributed Betti numbers appear in [BEGY20,Bru,EY18] To prove Theorem 1.1, we use Erman's technique from [Erm15]. In particular, we study the limiting behavior of the Hilbert numerator of the homogeneous coordinate ring of Σ k (C, L d ).…”

Section: Introductionmentioning

confidence: 99%

“…Boij-Söderberg theory [ES09] shows that every Betti table is a nonnegative linear combination of pure diagrams, the simplest diagrams of S-modules. In [Erm15], Erman shows that as d → ∞, the Boij-Söderberg decomposition is dominated by a single pure diagram which only depends on the genus of C.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic syzygies of secant varieties of curves

Taylor¹

2021

Preprint

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Section: Introductionmentioning

confidence: 64%

Section: Asymptotic Puritymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Asymptotic syzygies of secant varieties of curves

Taylor¹

2021

Preprint

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“…Kişinin beden bütünlüğüne yönelik bütün tıbbi müdahalelerin tedavi amacına yönelmesi şart değildir [27] . Salt güzelleşme amaçlı estetik müdahaleler Munise Gülen KURT 111 2020/2 Ankara Barosu Dergisi HAK EMLİ sadece güzelleşme, şöhret kazanma, maddi kazanç elde etme ya da sosyal anlamda iyi bir konum elde etme amacıyla yapılan ve doğrudan ya da dolaylı olarak tedavi niteliği taşımayan müdahalelerdir.…”

Section: I-tibbi̇ Müdahale Olarak Esteti̇k Müdahale Kavrami Ve Hukuka Uygunluğunun Koşullariunclassified

Obligations of the Surgeon in Contr Acts Established for Aesthetic Interventions

Kurt

2020

Ankara Barosu Dergisi

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One of the new types of interventions that arise as a result of development and change in medical science is aesthetic interventions. Aesthetic interventions are one of the rare cases where the physician intervenes even though there is no medical indication. Although medical intervention must be based on a medical indication as a rule in order to be lawful, at the present time social or psychological indication situations can also be accepted as sufficient for the lawful of the intervention. The absence of a medical indication and the acceptance of the relationship between the patient and the physician in practice as a result guaranteed work contract distinguishes aesthetic interventions from other medical interventions. These situations, which make heavier the responsibility of the physician, also extend the scope of illumination. In this context, the obligation of illumination, which seems to be a burden to the physician, becomes an important issue.

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

Cited by 4 publications

References 8 publications

Asymptotic syzygies of secant varieties of curves

Asymptotic syzygies of secant varieties of curves

Obligations of the Surgeon in Contr Acts Established for Aesthetic Interventions

Conjectures and Computations about Veronese Syzygies

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