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NON-DEFECTIVITY OF GRASSMANNIANS OF PLANES HIROTACHI ABO, GIORGIO OTTAVIANI, AND CHRIS PETERSON
 

Summary: NON-DEFECTIVITY OF GRASSMANNIANS OF PLANES
HIROTACHI ABO, GIORGIO OTTAVIANI, AND CHRIS PETERSON
Abstract. Let Gr(k, n) be the Pl¨ucker embedding of the Grassmann variety
of projective k-planes in Pn. For a projective variety X, let s(X) denote the
variety of its s - 1 secant planes. More precisely, s(X) denotes the Zariski
closure of the union of linear spans of s-tuples of points lying on X. We
exhibit two functions s0(n) s1(n) such that s(Gr(2, n)) has the expected
dimension whenever n 9 and either s s0(n) or s1(n) s. Both s0(n) and
s1(n) are asymptotic to n2
18
. This yields, asymptotically, the typical rank of
an element of 3 Cn+1. Finally, we classify all defective s(Gr(k, n)) for s 6
and provide geometric arguments underlying each defective case.
1. Introduction
Let X PN
be a non-degenerate projective variety. The s-secant variety s(X)
is defined to be the Zariski closure of the union of linear spans of s-tuples of points
lying on X (see [Z]). Note that with this notation, 2(X) is the usual variety of
secant lines of X. There is a smallest s such that s(X) = PN
leading to a natural

  

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

 

Collections: Mathematics