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A Real Generalized Trisecant Trichotomy

Kristian Ranestad, Anna Seigal, Kexin Wang

Abstract

The classical trisecant lemma says that a general chord of a non-degenerate space curve is not a trisecant; that is, the chord only meets the curve in two points. The generalized trisecant lemma extends the result to higher-dimensional varieties. It states that the linear space spanned by general points on a projective variety intersects the variety in exactly these points, provided the dimension of the linear space is smaller than the codimension of the variety and that the variety is irreducible, reduced, and non-degenerate. We prove a real analogue of the generalized trisecant lemma, which takes the form of a trichotomy. Along the way, we characterize the possible numbers of real intersection points between a real projective variety and a complimentary dimension real linear space. We show that any integer of correct parity between a minimum and a maximum number can be achieved. We then specialize to Segre-Veronese varieties, where our results apply to the identifiability of independent component analysis, tensor decomposition and to typical tensor ranks.

A Real Generalized Trisecant Trichotomy

Abstract

The classical trisecant lemma says that a general chord of a non-degenerate space curve is not a trisecant; that is, the chord only meets the curve in two points. The generalized trisecant lemma extends the result to higher-dimensional varieties. It states that the linear space spanned by general points on a projective variety intersects the variety in exactly these points, provided the dimension of the linear space is smaller than the codimension of the variety and that the variety is irreducible, reduced, and non-degenerate. We prove a real analogue of the generalized trisecant lemma, which takes the form of a trichotomy. Along the way, we characterize the possible numbers of real intersection points between a real projective variety and a complimentary dimension real linear space. We show that any integer of correct parity between a minimum and a maximum number can be achieved. We then specialize to Segre-Veronese varieties, where our results apply to the identifiability of independent component analysis, tensor decomposition and to typical tensor ranks.
Paper Structure (10 sections, 30 theorems, 35 equations, 1 figure, 1 table)

This paper contains 10 sections, 30 theorems, 35 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. The possible numbers of real solutions
  3. Real trisecant trichotomy
  4. Examples
  5. Segre-Veronese varieties
  6. Varieties with $\mathcal{N}(X)_{\max}$ minimal
  7. Applications
  8. Independent component analysis (ICA)
  9. Uniqueness of tensor decompositions
  10. Typical Tensor Ranks

Key Result

Theorem 1.1

Let $X \subseteq \mathbb{P}_\mathbb{C}^{N-1}$ be an irreducible, reduced, non-degenerate projective variety of dimension $d$ and let $n$ be a positive integer with $n+d < N$. Let $P_1,\ldots,P_n$ be general points on $X$. Then the intersection of $X$ with the subspace spanned by $P_1,\ldots,P_n$ con

Figures (1)

  • Figure 1: Dual curve of the Edge quartic \ref{['eqn:edge']} with regions labeled by the number of real intersection points

Theorems & Definitions (67)

  • Theorem 1.1: A Generalized Trisecant Lemma, see chiantini2002weakly
  • Theorem 1.2: Reformulation of Theorem \ref{['lemma:trisecant']}
  • proof
  • Definition 1.3
  • Proposition 1.4
  • Example 1.5
  • Theorem 1.6: A Real Generalized Trisecant Trichotomy
  • Example 1.7: $p=0$
  • Example 1.8: $0<p<1$
  • Example 1.9: $p=1$
  • ...and 57 more