[Paper Review] Four lectures on secant varieties

E. Carlini, N. Grieve+1 more|arXiv (Cornell University)|Sep 17, 2013

Tensor decomposition and applications4 citations

TL;DR

This paper provides a comprehensive introduction to higher secant varieties in algebraic geometry, focusing on foundational concepts, applications to Waring's problem, and connections to polynomial decomposition. It presents key tools like Terracini's Lemma and Apolarity Theory, with explicit examples, exercises, and solutions, offering a self-contained resource for researchers entering the field of secant varieties and their applications in algebraic geometry and tensor decomposition.

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.

Motivation & Objective

• To provide a foundational introduction to higher secant varieties for researchers new to the field.
• To explore connections between secant varieties and classical problems such as Waring's problem and polynomial decomposition.
• To present essential tools like Terracini's Lemma and Apolarity Theory for analyzing secant varieties.
• To offer solved exercises and references to guide further study in secant varieties and their applications.
• To bridge classical algebraic geometry with modern applications in tensor decomposition, signal processing, and algebraic statistics.

Proposed method

• Uses the definition of the $ s $-th secant variety $ \sigma_s(X) $ as the Zariski closure of the union of $ s $-secant linear spaces to an irreducible variety $ X \subset \mathbb{P}^N $.
• Applies Terracini's Lemma to compute the dimension of secant varieties via the span of tangent spaces at general points.
• Introduces Apolarity Theory through the Apolarity Lemma, linking forms to sets of points via annihilators.
• Employs Hilbert functions and zero-dimensional schemes to analyze Waring decompositions and identifiability of secant varieties.
• Analyzes Veronese varieties as a central class of examples, particularly for Waring problems and polynomial rank decomposition.
• Uses symbolic and numerical algebraic geometry tools to study equations and degrees of secant varieties, with references to computational software.

Experimental results

Research questions

• RQ1 What is the dimension of the $ s $-th secant variety $ \sigma_s(X) $ for a given variety $ X $, particularly for Veronese and Segre-Veronese varieties?
• RQ2 When is a general point in $ \sigma_s(X) $ uniquely representable as a sum of $ s $ points on $ X $, i.e., when is $ \sigma_s(X) $ generically identifiable?
• RQ3 What is the minimal $ s $ such that every homogeneous polynomial of degree $ d $ in $ n $ variables can be written as a sum of $ s $ $ d $-th powers of linear forms?
• RQ4 How can one compute defining equations for $ \sigma_s(X) $, and what is the degree of these varieties?
• RQ5 What are the structural properties of secant varieties, such as whether they are Cohen-Macaulay or defective?

Key findings

• The Alexander-Hirschowitz theorem resolves the Waring problem for generic forms: $ \sigma_s(\nu_d(\mathbb{P}^n)) $ has the expected dimension except in finitely many cases.
• Terracini's Lemma provides a method to compute the dimension of $ \sigma_s(X) $ by analyzing the span of tangent spaces at $ s $ general points on $ X $.
• Apolarity Theory establishes a duality between homogeneous polynomials and zero-dimensional schemes, enabling the study of Waring decompositions via linear algebra.
• For Veronese varieties, the secant varieties $ \sigma_s(\nu_d(\mathbb{P}^n)) $ are generically identifiable in most cases, except for known exceptional cases.
• The paper identifies key open problems in the theory, such as computing equations for secant varieties and determining their Cohen-Macaulay properties.
• The authors provide a curated list of references and software tools for further study, including symbolic computation and numerical algebraic geometry packages.