A matryoshka structure of higher secant varieties and the generalized Bronowski's conjecture

TL;DR

This paper explores a nested matryoshka structure in higher secant varieties, extending classical results and proposing a weak form of the generalized Bronowski's conjecture linking secant variety identifiability to tangential projections.

Contribution

It introduces a new matryoshka framework for higher secant varieties and proves a weak form of the generalized Bronowski's conjecture in this context.

Findings

Established a matryoshka structure among higher secant varieties.

Proved a weak form of the generalized Bronowski's conjecture.

Provided geometric and syzygetic characterizations of minimal degree secant varieties.

Abstract

In projective algebraic geometry, there are classical and fundamental results that describe the structure of geometry and syzygies, and many of them characterize varieties of minimal degree and del Pezzo varieties. In this paper, we consider analogous objects in the category of higher secant varieties. Our main theorems say that there is a matryoshka structure among those basic objects including a generalized $K_{p,1}$ theorem, syzygetic and geometric characterizations of higher secant varieties of minimal degree and del Pezzo higher secant varieties, defined in this paper. For our purpose, we prove a weak form of the generalized Bronowski's conjecture raised by C. Ciliberto and F. Russo that relates the identifiability for higher secant varieties to the geometry of tangential projections.