  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 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, and del Pezzo higher secant varieties, newly 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 the 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 that relates the identifiability for higher secant varieties to the
geometry of tangential projections.
