Interpolation for normal bundles of general curves

Atanasov, Atanas; Larson, Eric; Yang, David

doi:10.48550/arxiv.1509.01724

Cited by 3 publications

(4 citation statements)

References 2 publications

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“…In characteristic zero, we may also rely on interpolation theory to guarantee the existence (but not the precise number) of such a rational curve (see [ALY15]). Thus V 4,8 contains a general point of Y 4,8 and hence it contains all of Y 4,8 .…”

Section: The Veronese Compactification In Higher Dimensionsmentioning

confidence: 99%

Equations for point configurations to lie on a rational normal curve

Caminata

Giansiracusa

Moon

et al. 2018

Advances in Mathematics

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The parameter space of n ordered points in projective d-space that lie on a rational normal curve admits a natural compactification by taking the Zariski closure in (P d ) n . The resulting variety was used to study the birational geometry of the moduli space M 0,n of n-tuples of points in P 1 . In this paper we turn to a more classical question, first asked independently by both Speyer and Sturmfels: what are the defining equations? For conics, namely d = 2, we find scheme-theoretic equations revealing a determinantal structure and use this to prove some geometric properties; moreover, determining which subsets of these equations suffice set-theoretically is equivalent to a well-studied combinatorial problem. For twisted cubics, d = 3, we use the Gale transform to produce equations defining the union of two irreducible components, the compactified configuration space we want and the locus of degenerate point configurations, and we explain the challenges involved in eliminating this extra component. For d ≥ 4 we conjecture a similar situation and prove partial results in this direction.

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Section: The Veronese Compactification In Higher Dimensionsmentioning

confidence: 99%

Equations for point configurations to lie on a rational normal curve

Caminata

Giansiracusa

Moon

et al. 2018

Advances in Mathematics

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“…The first nontrivial case of interpolation in higher dimensional projective space is that rational normal curves satisfy interpolation, meaning there is one through a general collection of n+3 points in P n , see subsection 1.2.1. Interpolation of higher genus curves in projective space is extensively studied in [Ste89], [ALY15], [ALY15], and [Lar15]. We review interpolation for rational curves and results of interpolation for higher genus curves in subsection 1.2 below.…”

Section: Introductionmentioning

confidence: 99%

“…Around the same time, it was shown in [Ata14, Theorem 7.1] that nonspecial curves, apart from those of genus 2 and degree 5, in P 3 satisfy interpolation. This was generalized from P 3 to projective spaces of arbitrary dimension in the following comprehensive recent result of Atanasov-Larson-Yang: Theorem 1.3 (Theorem 1.3, [ALY15]). Strong interpolation holds for the main component of the Hilbert scheme parameterizing nonspecial curves of degree d, genus g in projective space P r , with d ≥ g + r unless (d, g, r) ∈ {(5, 2, 3), (6, 2, 4), (7, 2, 5)} .…”

Section: Introductionmentioning

confidence: 99%

Interpolation problems: Del Pezzo surfaces

Landesman¹,

Patel²

2019

Annali Scuola Normale Superiore - Classe Di Scienze

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“…Also see [LP16b, Section 1] for a more leisurely introduction to interpolation. It is known that nearly all nonspecial curves satisfy interpolation [ALY15], and that canonical curves in genus other than 6 and 4 satisfy weak interpolation as proven in [Ste03,Chapter 13]. Most recently, it was shown that smooth del Pezzo surfaces satisfy weak interpolation [LP16b,Theorem 1.1].…”

Section: Introductionmentioning

confidence: 99%