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ON NON-DEFECTIVITY OF CERTAIN SEGRE-VERONESE HIROTACHI ABO
 

Summary: ON NON-DEFECTIVITY OF CERTAIN SEGRE-VERONESE
VARIETIES
HIROTACHI ABO
Abstract. Let Xm,n be the Segre-Veronese 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 PN-1
be a non-degenerate 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 s-tuples 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

  

Source: Abo, Hirotachi - Department of Mathematics, University of Idaho

 

Collections: Mathematics