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

Abstract

In this paper, we propose a matryoshka structure of higher secant varieties which includes many matryoshka phenomena in the category of higher secant varieties, and which generalizes certain celebrated classical results in the study of syzygy to higher secant varieties. For example, we prove a generalized $K_{p,1}$ theorem for higher secant varieties, the syzygetic and geometric characterizations of minimal degree higher secant varieties, defined by Ciliberto and Russo ([15]), and del Pezzo higher secant varieties, defined in this paper, and also give the determinantal presentation of higher secant varieties having minimal degree. These results come mainly from the structure of tangent cones to higher secant varieties and from inductive analyses of defining equations and their syzygies in relation to inner and tangential projections.

For our purposes, we prove a weak form of the generalized Bronowski's conjecture due to Ciliberto and Russo ([15]) that relates the identifiability for higher secant varieties to the geometry of tangential projections.