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Tangential varieties of Segre-Veronese varieties

Luke Oeding, Claudiu Raicu

TL;DR

This work gives a complete description of the tangential variety $ au(X)$ of a Segre--Veronese variety $X$ by determining its ideal generators up to degree $4$ and by decomposing the homogeneous coordinate ring into irreducible GL-representations. The authors develop a unified, generic framework using functorial generic algebras, Young tabloids, covariants, and MCB--graphs to describe the equations of $ au(X)$ and to control the generation of its ideal, showing that $I( au(X))$ is generated in degree $ obreak ext{at most }4$, with degree-$4$ generators appearing exactly when the multiset $oldsymbol{d}=iglrace d_1, frac{}{}igr.$ contains $oxed{3}$, $oxed{2,1}$, or $oxed{1,1,1}$. In the Segre case, these generators realize the Landsberg--Weyman conjecture, matching the predicted quadrics, cubics, and quartics in a representation-theoretic framework. The paper also provides an explicit description of the GL-representation content of $K[ au(X)]_r$, via $m_oldsymbol{ extlambda}$ determined by $e_oldsymbol{ extlambda}$ and $f_oldsymbol{ extlambda}$, and connects the quartic equations to classical invariants like Cayley’s hyperdeterminant. Overall, the results offer a robust, combinatorial-graphical approach to tangential varieties that parallels and extends known results for secant and Veronese varieties, with concrete generators and representation-theoretic data provided for practical computation and applications in algebraic geometry and tensorial geometry.

Abstract

We determine the minimal generators of the ideal of the tangential variety of a Segre-Veronese variety, as well as the decomposition into irreducible GL-representations of its homogeneous coordinate ring. In the special case of a Segre variety, our results confirm a conjecture of Landsberg and Weyman.

Tangential varieties of Segre-Veronese varieties

TL;DR

This work gives a complete description of the tangential variety of a Segre--Veronese variety by determining its ideal generators up to degree and by decomposing the homogeneous coordinate ring into irreducible GL-representations. The authors develop a unified, generic framework using functorial generic algebras, Young tabloids, covariants, and MCB--graphs to describe the equations of and to control the generation of its ideal, showing that is generated in degree , with degree- generators appearing exactly when the multiset contains , , or . In the Segre case, these generators realize the Landsberg--Weyman conjecture, matching the predicted quadrics, cubics, and quartics in a representation-theoretic framework. The paper also provides an explicit description of the GL-representation content of , via determined by and , and connects the quartic equations to classical invariants like Cayley’s hyperdeterminant. Overall, the results offer a robust, combinatorial-graphical approach to tangential varieties that parallels and extends known results for secant and Veronese varieties, with concrete generators and representation-theoretic data provided for practical computation and applications in algebraic geometry and tensorial geometry.

Abstract

We determine the minimal generators of the ideal of the tangential variety of a Segre-Veronese variety, as well as the decomposition into irreducible GL-representations of its homogeneous coordinate ring. In the special case of a Segre variety, our results confirm a conjecture of Landsberg and Weyman.

Paper Structure

This paper contains 14 sections, 18 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Equations of the tangential variety of a Segre--Veronese variety
  3. The generic case
  4. The generic equations of the tangential variety
  5. Young tabloids rai-yng
  6. Covariants associated to graphs
  7. MCB--graphs and graphs containing triangles
  8. A graphical description of the generic ideal of the tangential variety
  9. Case $e_{\lambda}=r$.
  10. Case $e_{\lambda}<r$.
  11. Minimal generators of the ideal of the tangential variety
  12. Proof that $I(\tau(X))$ is generated in degree at most $4$
  13. Minimal generators of degree $3$
  14. Minimal generators of degree $4$

Key Result

Theorem 1

Let $X=SV_{\underline{d}}(\mathbb{P}V_1^*\times\mathbb{P}V_2^*\times\cdots\times\mathbb{P}V_n^*)$ be a Segre--Veronese variety over a field $\mathbb{K}$ of characteristic zero. The degree $r$ part of the homogeneous coordinate ring of $\tau(X)$ decomposes as where $m_{\lambda}$ is either $0$ or $1$, obtained as follows. Set If some $\lambda^j$ has more than two parts, or $e_{\lambda}<2f_{\lambda

Theorems & Definitions (52)

  • Theorem 1
  • Conjecture 1.1: lan-wey
  • Theorem 2
  • Corollary
  • Example 2.1
  • Example 2.2
  • Proposition 2.3
  • Definition 3.1: The generic coordinate ring of $\mathbb{P}(\operatorname{Sym}^{\underline{d}}V$)
  • Example 3.2
  • Example 3.3: The generic ideal of the subspace variety, see also lan-wey-secant
  • ...and 42 more