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Four lectures on secant varieties

E. Carlini, N. Grieve, L. Oeding

TL;DR

This paper presents an accessible introduction to higher secant varieties, weaving together foundational definitions, computational tools, and key applications in algebraic geometry and tensor theory. It develops the core machinery, notably Terracini's Lemma and Veronese varieties, and explains the Veronese Waring problem and its resolution via the Alexander–Hirschowitz theorem, while also extending to non-Veronese contexts such as Segre varieties through flattenings and Strassen-type equations. Apolarity theory and Hilbert functions are employed to link rank questions with duality frameworks and to analyze exceptional cases, with concrete worked examples and exercises. Collectively, it surveys essential results, clarifies the methodological landscape for determining dimensions and equations of secant varieties, and highlights open problems and directions for future research.

Abstract

This paper is based on the first author's lectures at the 2012 University of Regina Workshop "Connections Between Algebra and Geometry". Its aim is to provide an introduction to the theory of higher secant varieties and their applications. Several references and solved exercises are also included.

Four lectures on secant varieties

TL;DR

This paper presents an accessible introduction to higher secant varieties, weaving together foundational definitions, computational tools, and key applications in algebraic geometry and tensor theory. It develops the core machinery, notably Terracini's Lemma and Veronese varieties, and explains the Veronese Waring problem and its resolution via the Alexander–Hirschowitz theorem, while also extending to non-Veronese contexts such as Segre varieties through flattenings and Strassen-type equations. Apolarity theory and Hilbert functions are employed to link rank questions with duality frameworks and to analyze exceptional cases, with concrete worked examples and exercises. Collectively, it surveys essential results, clarifies the methodological landscape for determining dimensions and equations of secant varieties, and highlights open problems and directions for future research.

Abstract

This paper is based on the first author's lectures at the 2012 University of Regina Workshop "Connections Between Algebra and Geometry". Its aim is to provide an introduction to the theory of higher secant varieties and their applications. Several references and solved exercises are also included.

Paper Structure

This paper contains 10 sections, 5 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Lecture One
  3. Lecture Two
  4. Lecture Three
  5. Lecture Four
  6. Dimension of secant varieties of Segre varieties
  7. Flattenings
  8. Equations of Secant varieties of Segre products
  9. Solution of the exercises
  10. Looking forward, further readings

Key Result

Lemma 2.15

Let $P_1,\ldots,P_s\in X$ be general points and $P\in\langle P_1,\ldots,P_s\rangle\subset\sigma_s(X)$ be a general point. Then the tangent space to $\sigma_s(X)$ in $P$ is

Theorems & Definitions (86)

  • Definition 2.1
  • Example 2.2
  • Remark 2.4
  • Definition 2.7
  • Definition 2.9
  • Remark 2.10
  • Example 2.11
  • Lemma 2.15: Terracini's Lemma
  • Remark 2.16
  • Example 2.17
  • ...and 76 more