Chordal varieties of Veronese varieties and catalecticant matrices

3x3 Minors of Catalecticants

Secant varieties to Veronese embeddings of projective space are classical varieties whose equations are not completely understood. Minors of catalecticant matrices furnish some of their equations, and in some situations even generate their ideals. Geramita conjectured that this is the case for the secant line variety of the Veronese variety, namely that its ideal is generated by the 3x3 minors of any of the "middle" catalecticants. Part of this conjecture is the statement that the ideals of 3x3 minors are equal for most catalecticants, and this was known to hold set-theoretically. We prove the…

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

We study the secant varieties of the Veronese varieties and of Veronese reembeddings of a smooth projective variety. We give some conditions, under which these secant varieties are set-theoretically cut out by determinantal equations. More precisely, they are given by minors of a catalecticant matrix. These conditions include the case when the dimension of the projective variety is at most 3 and the degree of reembedding is sufficiently high. This gives a positive answer to a set-theoretic version of a question of Eisenbud in dimension at most 3. For dimension four and higher we produce plenty…

Equations for secant varieties to Veronese varieties

We define new classes of modules of equations for secant varieties of Veronese varieties using representation theory and geometry. We also revisit some old modules of equations (catalecticant minors) to determine when they are sufficient to give scheme-theoretic defining equations.

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。

Inverting catalecticants of ternary quartics

We study the reciprocal variety to the linear space of symmetric matrices (LSSM) of catalecticant matrices associated with ternary quartics. With numerical tools, we obtain 85 to be its degree and 36 to be the ML-degree of the LSSM. We provide a geometric explanation to why equality between these two invariants is not reached, as opposed to the case of binary forms, by describing the intersection of the reciprocal variety and the orthogonal of the LSSM in the rank loci. Moreover, we prove that only the rank-$1$ locus, namely the Veronese surface $ν_4(\mathbb{P}^2)$, contributes to the degree o…

Equations for secant varieties of Veronese and other varieties

New classes of modules of equations for secant varieties of Veronese varieties are defined using representation theory and geometry. Some old modules of equations (catalecticant minors) are revisited to determine when they are sufficient to give scheme-theoretic defining equations. An algorithm to decompose a general ternary quintic as the sum of seven fifth powers is given as an illustration of our methods. Our new equations and results about them are put into a larger context by introducing vector bundle techniques for finding equations of secant varieties in general. We include a few homoge…