 
Summary: NEW EXAMPLES OF DEFECTIVE SECANT VARIETIES OF
SEGREVERONESE VARIETIES
HIROTACHI ABO AND MARIA CHIARA BRAMBILLA
Abstract. We prove the existence of defective secant varieties of threefactor
and fourfactor SegreVeronese varieties embedded in certain multidegree.
These defective secant varieties were previously unknown and are of impor
tance in the classification of defective secant varieties of SegreVeronese vari
eties with three or more factors.
1. Introduction
Let X be a nondegenerate projective variety in projective space PN
and let
p1, . . . , ps be linearly independent points of X. Then the (s  1)plane p1, . . . , ps
spanned by p1, . . . , ps is called a secant (s  1)plane to X. The Zariski closure
of the union of all secant (s  1)planes to X is called the sth
secant variety of
X and denoted by s(X). A basic question about secant varieties is to find their
dimensions. A simple dimension count indicates
dim s(X) min{s(dim X + 1)  1, N}.
If equality holds, we say that s(X) has the expected dimension. Otherwise s(X)
is said to be defective.
