Secant varieties to high degree Veronese reembeddings, catalecticant matrices and smoothable Gorenstein schemes

Cactus 概形、catalecticant 子式与概形论方程

Cactus scheme, catalecticant minors, and scheme theoretic equations

本文通过非约化基上有限概形族的相对线性张成定义 cactus 概形，证明其与高次 Veronese 簇的 catalecticant 子式零概形在稠密开集上一致。

A comparison of different notions of ranks of symmetric tensors

We introduce various notions of rank for a symmetric tensor, namely: rank, border rank, catalecticant rank, generalized rank, scheme length, border scheme length, extension rank and smoothable rank. We analyze the stratification induced by these ranks. The mutual relations between these stratifications, allow us to describe the hierarchy among all the ranks. We show that strict inequalities are possible between rank, border rank, extension rank and catalecticant rank. Moreover we show that scheme length, generalized rank and extension rank coincide.

Finite schemes and secant varieties over arbitrary characteristic

We present scheme theoretic methods that apply to the study of secant varieties. This mainly concerns finite schemes and their smoothability. The theory generalises to the base fields of any characteristic, and even to non-algebraically closed fields. In particular, the smoothability of finite schemes does not depend on the embedding into a smooth variety or on base field extensions. Independent of the base field, secant varieties to high degree Veronese reembeddings behave well with respect to the intersection and they are defined by minors of catalecticants whenever a suitable smoothability …

VSPs of cubic fourfolds and the Gorenstein locus of the Hilbert scheme of 14 points on A^6

This paper is concerned with the geometry of the Gorenstein locus of the Hilbert scheme of $14$ points on $\mathbb{A}^6$. This scheme has two components: the smoothable one and an exceptional one. We prove that the latter is smooth and identify the intersection of components as a vector bundle over the Iliev-Ranestad divisor in the space of cubic fourfolds. The ninth secant variety of the triple Veronese reembedding lies inside the Iliev-Ranestad divisor. We point out that this secant variety is set-theoretically a codimension two complete intersection and discuss the degrees of its equations.