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Quadratic equations of tangent varieties via four-way tensors of linear forms

Junho Choe

TL;DR

The paper develops a systematic method to produce quadratic equations for tangent varieties $\tau X$ and their higher secant variants $\sigma_q\tau^k X$ using $4$-way tensors of linear forms, thereby extending classical determinantal approaches. It establishes a surjective map $b_{q+1}(T)$ from suitably constructed tensor-derived spaces to $I(\sigma_q\tau^k X)_{q+1}$ for $X$-multiplicative tensors, and demonstrates how these quadrics and resulting syzygies align with Green–Lazarsfeld classes within a representation-theoretic framework, including Segre variety computations and matryoshka-structured results. A variant of the Eisenbud–Koh–Stillman conjecture is obtained for complete embeddings of smooth curves when the line bundle is sufficiently positive, and the work also yields new insights into the Koszul cohomology $K_{p,q+1}(I(\sigma_q\tau^k X))$, including explicit decompositions in the Segre case via Lascoux-type resolutions. Collectively, the results provide a unified approach to equations and linear syzygies of tangent and higher osculating varieties, linking determinantal equations, Koszul cohomology, and representation theory with geometric properties of secant varieties.

Abstract

In the present paper we construct quadratic equations and linear syzygies for tangent varieties using 4-way tensors of linear forms and generalize this method to higher secant varieties of higher osculating varieties. Such equations extend the classical determinantal ones of higher secant varieties and span all the equations of the same degree for smooth projective curves completely embedded by sufficiently positive line bundles, proving a variant of the Eisenbud-Koh-Stillman conjecture on determinantal equations. On the other hand, our syzygies are compatible with the Green-Lazarsfeld classes and generate the corresponding Koszul cohomology groups for Segre varieties with a prescribed number of factors. To obtain these results we describe the equations of minimal possible degrees and reinterpret the Green-Lazarsfeld classes from the perspective of representation theory.

Quadratic equations of tangent varieties via four-way tensors of linear forms

TL;DR

The paper develops a systematic method to produce quadratic equations for tangent varieties and their higher secant variants using -way tensors of linear forms, thereby extending classical determinantal approaches. It establishes a surjective map from suitably constructed tensor-derived spaces to for -multiplicative tensors, and demonstrates how these quadrics and resulting syzygies align with Green–Lazarsfeld classes within a representation-theoretic framework, including Segre variety computations and matryoshka-structured results. A variant of the Eisenbud–Koh–Stillman conjecture is obtained for complete embeddings of smooth curves when the line bundle is sufficiently positive, and the work also yields new insights into the Koszul cohomology , including explicit decompositions in the Segre case via Lascoux-type resolutions. Collectively, the results provide a unified approach to equations and linear syzygies of tangent and higher osculating varieties, linking determinantal equations, Koszul cohomology, and representation theory with geometric properties of secant varieties.

Abstract

In the present paper we construct quadratic equations and linear syzygies for tangent varieties using 4-way tensors of linear forms and generalize this method to higher secant varieties of higher osculating varieties. Such equations extend the classical determinantal ones of higher secant varieties and span all the equations of the same degree for smooth projective curves completely embedded by sufficiently positive line bundles, proving a variant of the Eisenbud-Koh-Stillman conjecture on determinantal equations. On the other hand, our syzygies are compatible with the Green-Lazarsfeld classes and generate the corresponding Koszul cohomology groups for Segre varieties with a prescribed number of factors. To obtain these results we describe the equations of minimal possible degrees and reinterpret the Green-Lazarsfeld classes from the perspective of representation theory.

Paper Structure

This paper contains 16 sections, 28 theorems, 164 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Multilinear algebra
  4. Higher secant varieties to higher osculating varieties
  5. Syzygies
  6. Tensors of linear forms
  7. Symmetric powers of smooth curves
  8. Green-Lazarsfeld classes revisited
  9. Proofs
  10. Proof of \ref{['Thm:SegreDecompo']}
  11. Proofs of \ref{['Thm:Main1', '', 'Thm:Main2']}
  12. Proof of \ref{['Thm:EKS']}
  13. Proof of \ref{['Cor:Byproduct']}
  14. Comments and questions
  15. Tangent varieties to smooth curves
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $X\subseteq\mathbb PV$ be a nondegenerate projective variety and $T:V_1\otimes\cdots\otimes V_{2k+2}\to V$ be an $X$-multiplicative$(2k+2)$-way tensor of linear forms for an integer $k\geq 0$. Then for every integer $q\geq 1$ the image of the map $b_{q+1}(T)$ lies in $I(\sigma_q\tau^kX)_{q+1}$, It is nontrivial whenever $\dim V_i\geq q+1$ for all $1\leq i\leq 2k+2$.

Theorems & Definitions (56)

  • Theorem 1.1
  • Example 1.2
  • Conjecture 1.3: Eisenbud-Koh-Stillman conjecture about determinantal equations
  • Theorem 1.4: Eisenbud-Koh-Stillman type result for tangent varieties
  • Corollary 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Conjecture 1.8: Green's conjecture for canonical curves, MR739785
  • Proposition 2.2
  • Proposition 2.3
  • ...and 46 more