 
Summary: NONDEFECTIVITY 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 kplanes 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 stuples 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 nondegenerate projective variety. The ssecant variety s(X)
is defined to be the Zariski closure of the union of linear spans of stuples 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
