On the normality of secant varieties

Ullery, Brooke

doi:10.1016/j.aim.2015.10.025

Cited by 12 publications

(13 citation statements)

References 21 publications

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“…We claim that t * O P(E L ) (−Φ) I X/Σ , where I X/Σ denotes the ideal sheaf of X in Σ. By the assumption on L, the main result in [22] implies that t * O P(E L ) = O Σ . So from the exact sequence…”

Section: Then If Y F and W All Have Db Singularities Then So Does Xmentioning

confidence: 98%

“…Proof. By [22,Theorem D], it is sufficient to show that for each x ∈ X, b * x L(−2E x ) is normally generated on the blowup Bl x X. By pushing it down to X, it amounts to showing for each positive integer r…”

Section: Example Of Normal Secant Varietiesmentioning

confidence: 99%

“…Inspired by the paper [22], we study the singularities of Σ(X, L) from the cohomological point of view. To state results in a uniform way, throughout the paper we make the following assumption on L unless otherwise stated (In Section 5, the situation when L has weaker positivity is discussed).…”

Section: Introductionmentioning

confidence: 99%

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Singularities of Secant Varieties

Chou

Song

2017

Int Math Res Notices

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Section: Then If Y F and W All Have Db Singularities Then So Does Xmentioning

confidence: 98%

Section: Example Of Normal Secant Varietiesmentioning

confidence: 99%

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Singularities of Secant Varieties

Chou

Song

2017

Int Math Res Notices

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“…Among other things, the issue whether secant varieties are normal attracted special attention, as normality is critical in establishing many other important properties. Only for the first secant variety, the normality problem was settled by Ullery [22] fairly recently for a nonsingular projective variety of any dimension under suitable conditions on the embedding line bundle. Soon afterwards Chou and Song [2] further showed that the first secant variety has Du Bois singularities under the setting of Ullery's study.…”

Section: Introductionmentioning

confidence: 99%

“…which we prove to be a log resolution of the log pair (B k (L), Z k−1 ). Based on this setup, in Theorem 1.1, for instance, to prove the normality of Σ k at a point x, we adapt the strategy of Ullery in [22] to consider the unique minimal m-secant plane containing x. It cuts the curve along a degree m + 1 divisor ξ.…”

Section: Introductionmentioning

confidence: 99%

Singularities and syzygies of secant varieties of nonsingular projective curves

Ein,

Niu,

Park

2020

Preprint

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In recent years, the equations defining secant varieties and their syzygies have attracted considerable attention. The purpose of the present paper is to conduct a thorough study on secant varieties of curves by settling several conjectures and revealing interaction between singularities and syzygies. The main results assert that if the degree of the embedding line bundle of a nonsingular curve of genus g is greater than 2g + 2k + p for nonnegative integers k and p, then the k-th secant variety of the curve has normal Du Bois singularities, is arithmetically Cohen-Macaulay, and satisfies the property N k+2,p . In addition, the singularities of the secant varieties are further classified according to the genus of the curve, and the Castelnuovo-Mumford regularities are also obtained as well. As one of the main technical ingredients, we establish a vanishing theorem on the Cartesian products of the curve, which may have independent interests and may find applications elsewhere.

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Ideals of bounded rank symmetric tensors are generated in bounded degree

Sam

2016

Invent. math.

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Abstract. We study the syzygies of secant ideals of Veronese subrings of a fixed commutative graded algebra over a field of characteristic 0. One corollary is that the degrees of the minimal generators of the ith syzygy module of the coordinate ring of the rth secant variety of any Veronese embedding of a projective scheme X can be bounded by a constant that only depends on i, r, and X, and not on the choice of the Veronese embedding.

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On the normality of secant varieties

Cited by 12 publications

References 21 publications

Singularities of Secant Varieties

Singularities of Secant Varieties

Singularities and syzygies of secant varieties of nonsingular projective curves

Ideals of bounded rank symmetric tensors are generated in bounded degree

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