 
Summary: ON NONDEFECTIVITY OF CERTAIN SEGREVERONESE
VARIETIES
HIROTACHI ABO
Abstract. Let Xm,n be the SegreVeronese variety Pm ×Pn embedded by the
morphism given by O(1, 2) and let s(Xm,n) denote the sth secant variety to
Xm,n. In this paper, we prove that if m = n or m = n+1, then s(Xm,n) has
the expected dimension except for 6(X4,3). As an immediate consequence,
we will give function s1(m, n) s2(m, n) such that if s s1(m, n) or if
s s2(m, n), then s(Xm,n) has the expected dimension.
1. Introduction
Let X PN1
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 major questions surrounding secant
varieties center around finding invariants of those objects such as dimension. A
simple dimension count suggests that the expected dimension of s(X) is min{N 
1, s(d + 1)  1}. 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
