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Geometry of first nonempty Terracini loci

Francesco Galuppi, Pierpaola Santarsiero, Douglas A. Torrance, Ettore Teixeira Turatti

TL;DR

This work investigates the geometry of Terracini loci Ter$_r(X)$, the loci where $r$ tangent spaces to a variety $X$ fail to span the expected dimension. It establishes precise descriptions for the first nonempty Terracini loci of Veronse and Segre–Veronese varieties and provides a complete description for del Pezzo surfaces embedded by the anticanonical system, using cohomological and projection techniques. It then connects these geometric loci to explicit combinatorial configurations (e.g., points on lines) and derives dimension counts, alongside a practical algorithm for computing Terracini loci via Jacobian/rank criteria, with a Macaulay2 implementation. The results extend the understanding of secant/tangent degeneracies in classical varieties and furnish computational tools for identifying Terracini loci in applied contexts.

Abstract

After a few results on curves, we characterize the smallest nonempty Terracini loci of Veronese and Segre-Veronese varieties. For del Pezzo surfaces, we give a full description of the Terracini loci. Moreover, we present an algorithm to explicitly compute the Terracini loci of a given variety.

Geometry of first nonempty Terracini loci

TL;DR

This work investigates the geometry of Terracini loci Ter, the loci where tangent spaces to a variety fail to span the expected dimension. It establishes precise descriptions for the first nonempty Terracini loci of Veronse and Segre–Veronese varieties and provides a complete description for del Pezzo surfaces embedded by the anticanonical system, using cohomological and projection techniques. It then connects these geometric loci to explicit combinatorial configurations (e.g., points on lines) and derives dimension counts, alongside a practical algorithm for computing Terracini loci via Jacobian/rank criteria, with a Macaulay2 implementation. The results extend the understanding of secant/tangent degeneracies in classical varieties and furnish computational tools for identifying Terracini loci in applied contexts.

Abstract

After a few results on curves, we characterize the smallest nonempty Terracini loci of Veronese and Segre-Veronese varieties. For del Pezzo surfaces, we give a full description of the Terracini loci. Moreover, we present an algorithm to explicitly compute the Terracini loci of a given variety.
Paper Structure (8 sections, 39 theorems, 73 equations)

This paper contains 8 sections, 39 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Definitions and preliminaries
  3. Basic geometric properties
  4. Terracini loci of curves
  5. Terracini loci of del Pezzo surfaces
  6. Veronese varieties
  7. Segre-Veronese varieties
  8. Algorithms

Key Result

Theorem 2.1

Let $X\subset\mathbb{P}^{N}$ be a nondegenerate irreducible variety. If $p_1,\ldots, p_r\in X$ are in general position and $z\in\langle p_1,\ldots, p_{r}\rangle$ is a general point, then the embedded tangent space to $\sigma_r(X)$ at $z$ is

Theorems & Definitions (82)

  • Theorem 2.1: Terracini's lemma
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Remark 3.3
  • ...and 72 more