A Real Generalized Trisecant Trichotomy

TL;DR

This paper extends the classical trisecant lemma to real projective varieties, providing a trichotomy and characterizing real intersection points, with applications to tensor decomposition and identifiability.

Contribution

It introduces a real analogue of the generalized trisecant lemma, including a trichotomy and intersection characterization, with specific focus on Segre-Veronese varieties.

Findings

Characterization of real intersection points between varieties and linear spaces.

Any integer of correct parity between minimum and maximum is achievable.

Applications to tensor decomposition and identifiability.

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.