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NEW EXAMPLES OF DEFECTIVE SECANT VARIETIES OF SEGRE-VERONESE VARIETIES
 

Summary: NEW EXAMPLES OF DEFECTIVE SECANT VARIETIES OF
SEGRE-VERONESE VARIETIES
HIROTACHI ABO AND MARIA CHIARA BRAMBILLA
Abstract. We prove the existence of defective secant varieties of three-factor
and four-factor Segre-Veronese varieties embedded in certain multi-degree.
These defective secant varieties were previously unknown and are of impor-
tance in the classification of defective secant varieties of Segre-Veronese vari-
eties with three or more factors.
1. Introduction
Let X be a non-degenerate 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.

  

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

 

Collections: Mathematics