Non-defectivity of Grassmannians of planes

Abstract

Let Gr(k, n) be the Plücker embedding of the Grassmann variety of projective k-planes in P n . For a projective variety X, let σ s (X) denote the variety of its secant (s − 1)-planes. More precisely, σ s (X) denotes the Zariski closure of the union of linear spans of s-tuples of points lying on X. We exhibit two functions s 0 (n) ≤ s 1 (n) such that σ s (Gr(2, n)) has the expected dimension whenever n ≥ 9 and either s ≤ s 0 (n) or s 1 (n) ≤ s. Both s 0 (n) and s 1 (n) are asymptotic to n 2 18 . This yields, asymptotically, the typical rank of an element of 3 C n+1 . Finally, we classify all defective σ s (Gr(k, n)) for s ≤ 6 and provide geometric arguments underlying each defective case.This leads to the following proposition: Proposition 4.3. Let X = Gr(2, n) with n ≥ 11. Let V = K n+1 , and let L and M be general codimension six subspaces of V . Let L (resp. M) be the Grassmann variety of 3-planes in L (resp. M ). Then (i) The system of hyperplanes in P( 3 V ) which contain L∪M and which contain the tangent spaces at 6n−49 9 general points on L, at 6n−49 9 general points on M, and at 4 general points on X has the expected dimension 36(n − 6) − 36 6n−49 9 − 4(3n − 5), which is ⎧ ⎨ ⎩ 20 if n = 0 (mod 3), 8 if n = 1 (mod 3), 32 if n = 2 (mod 3). (ii) There are no hyperplanes in P( 3 V ) which contain L ∪ M and which contain the tangent spaces at 6n−49 9 general points on L, at 6n−49 9 general points on M, and at 4 general points on X. Proof. We will let {p i } denote the 6n−49 9 (resp. 6n−49 9 ) general points on L, {q i } denotes the 6n−49 9 (resp. 6n−49 9 ) general points on M and {r i } denotes the four general points on X. The proof is by a 6-step induction Licensed to Univ of Mass Amherst. Prepared on Sun Jul 5

Fano congruences of index 3 and alternating 3-forms

“…-We observe that q η is obtained via composition from q [2] : η → q η = η • q [2] ; therefore the thesis follows from (2.1):…”

“…Clearly we have L ∧ L = 2L [2] (i.e. this definition does not depend on the chosen basis of V ), and [L] ∈ G if and only if L [2] = 0.…”

“…-G ⊂ P( 2 V ) is scheme-theoretically the zero locus of the quadratic map associated to the reduced square [2] .…”

“…-Let I r ⊂ P(I(G(2, 6)) 2 ) be the set of quadrics of rank r in the ideal of G (2,6). Then P(I(G(2, 6)) 2 ) = I 6 ∪ I 10 ∪ I 15 , and dim I 6 = 8, dim I 10 = 13, dim I 15 = 14.…”

“…of (3.9) is the projectivised map on global sections of the bundle map (2). The degeneracy locus M r defined at the beginning of Section 4.2 may therefore be interpreted as…”