[Paper Review] Secant Varieties of the Varieties of Reducible Hypersurfaces in ${\mathbb P}^n$

Maria Virginia Catalisano, Anthony V. Geramita+5 more|arXiv (Cornell University)|Jan 31, 2015

Commutative Algebra and Its Applications39 references3 citations

TL;DR

This paper proposes a conjectural formula for the dimension of secant varieties of varieties of reducible hypersurfaces in projective space, using a novel connection to the Weak Lefschetz Property in graded algebras. It proves the formula unconditionally in many cases, unifying prior results and verifying special cases of conjectures on secant varieties of Segre embeddings and Fröberg's conjecture on Hilbert functions.

ABSTRACT

Given the space $V={\mathbb P}^{\binom{d+n-1}{n-1}-1}$ of forms of degree $d$ in $n$ variables, and given an integer $\ell>1$ and a partition $λ$ of $d=d_1+\cdots+d_r$, it is in general an open problem to obtain the dimensions of the $\ell$-secant varieties $σ_\ell ({\mathbb X}_{n-1,λ})$ for the subvariety ${\mathbb X}_{n-1,λ} \subset V$ of hypersurfaces whose defining forms have a factorization into forms of degrees $d_1,\ldots,d_r$. Modifying a method from intersection theory, we relate this problem to the study of the Weak Lefschetz Property for a class of graded algebras, based on which we give a conjectural formula for the dimension of $σ_\ell({\mathbb X}_{n-1,λ})$ for any choice of parameters $n,\ell$ and $λ$. This conjecture gives a unifying framework subsuming all known results. Moreover, we unconditionally prove the formula in many cases, considerably extending previous results, as a consequence of which we verify many special cases of previously posed conjectures for dimensions of secant varieties of Segre varieties. In the special case of a partition with two parts (i.e., $r=2$), we also relate this problem to a conjecture by Fröberg on the Hilbert function of an ideal generated by general forms.

Motivation & Objective

• To determine the dimension of $(\ell-1)$-secant varieties $\sigma_\ell(\mathbb{X}_{n-1,\lambda})$ for varieties of reducible hypersurfaces in $\mathbb{P}^n$.
• To address the long-standing open problem of computing dimensions of secant varieties of reducible forms, especially when expected dimension is not achieved.
• To establish a unifying framework that subsumes known results on secant varieties of Veronese, Segre, and Grassmann varieties.
• To relate the problem to the Weak Lefschetz Property in graded algebras and to Fröberg's conjecture on Hilbert functions of ideals generated by general forms.
• To verify conjectures on the non-defectivity of secant varieties of Segre embeddings in new ranges of parameters.

Proposed method

• The authors use intersection theory to relate the secant variety dimension problem to the study of the Weak Lefschetz Property in a class of graded Artinian algebras associated with reducible forms.
• They introduce a method that transforms the geometric problem into an algebraic one by analyzing the structure of ideals generated by generic forms of specified degrees.
• The approach involves studying the failure of the Weak Lefschetz Property in these algebras to detect when secant varieties are defective.
• The paper proves the conjectural dimension formula in multiple cases by leveraging results from Hochster and Laksov, Anick, and complete intersection theory.
• For the case $r=2$ (binary forms), the problem is linked to Fröberg's conjecture on the Hilbert function of ideals generated by general forms.
• The authors use parameter counts and dimension bounds to derive the expected dimension and compare it with actual dimension to detect defectivity.

Experimental results

Research questions

• RQ1 What is the dimension of the $(\ell-1)$-secant variety $\sigma_\ell(\mathbb{X}_{n-1,\lambda})$ for the variety of $\lambda$-reducible forms in $\mathbb{P}^n$?
• RQ2 When is $\sigma_\ell(\mathbb{X}_{n-1,\lambda})$ defective, and what is its defect in such cases?
• RQ3 How does the Weak Lefschetz Property of associated graded algebras determine the dimension of secant varieties of reducible forms?
• RQ4 To what extent does the conjectural formula unify known results on secant varieties of Segre and Veronese varieties?
• RQ5 Can the connection to Fröberg's conjecture be used to prove new cases of the Hilbert function conjecture for general forms?

Key findings

• The paper proposes a conjectural formula for $\dim \sigma_\ell(\mathbb{X}_{n-1,\lambda})$ that unifies all known results on secant varieties of reducible forms.
• The formula is proven unconditionally in many cases, significantly extending previous results on secant varieties of Segre embeddings.
• The authors verify Conjecture 8.3 on non-defectivity of secant varieties of Segre embeddings in a new range of parameters.
• For the case $r=2$, the paper establishes a link between the secant variety dimension problem and Fröberg's conjecture on Hilbert functions of ideals generated by general forms.
• The paper shows that $\sigma_\ell(\mathbb{X}_{n-1,\lambda})$ is not defective when $d_2 = \cdots = d_r \gg 0$, and the Segre embedding is balanced in this limit.
• The authors demonstrate that the secant variety of the affine cone over $\mathbb{X}_{n-1,\lambda}$ cannot be defective, providing a key technical tool for the main results.