Non-defectivity of Grassmannians of Planes

Let Gr(k, n) be the Plücker embedding of the Grassmann variety of projective k-planes in Pn. For a projective variety X, let σs(X) denote the variety of its s − 1 secant planes. More precisely, σs(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 σs(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 σs(Gr(k, n)) for s ≤ 6 and provide geometric arguments underlying each defective case.