 
Summary: ON THE SECANT DEFECTIVITY OF SEGREVERONESE VARIETIES
HIROTACHI ABO
1. Introduction
Let X PN be a nondegenerate projective variety of dimension d. Then the sth secant variety
of X, denoted s(X), is the Zariski closure of the union of linear spans of stuples of points lying
on X. The study of secant varieties has a long history. The interest in this subject goes back to
the Italian school at the turn of the 20th century. This topic has received renewed interest over
the past several decades, mainly due to its increasing importance in an ever widening collection
of disciplines including algebraic complexity theory [13, 26, 27], algebraic statistics [23, 22, 8], and
combinatorics [29, 30].
The major questions surrounding secant varieties center around finding invariants of those ob
jects such as dimension. A simple dimension count suggests that the expected dimension of s(X)
is min{s(d + 1)  1, N}. We say that X has a defective sth secant variety if s(X) does not have
the expected dimension. In particular, X is said to be defective if X has a defective sth secant
variety for some s. The paper explores problems related to the classification of defective secant
varieties of SegeVeronese varieties. This is analogous to the celebrated theorem of Alexander and
Hirschowitz [7], which asserts that higher secant varieties of Veronese varieties have the expected
dimension (modulo a fully described list of exceptions). This work completed the Waring problem
for polynomials which had remained unsolved for some time. There are corresponding conjecturally
complete lists of defective secant varieties for Segre varieties [5] and for Grassmann varieties [25, 10].
