  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.
