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Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Non-defectivity of Grassmannians of planes

Authors: Hirotachi Abo, Giorgio Ottaviani and Chris Peterson
Journal: J. Algebraic Geom. 21 (2012), 1-20
Published electronically: January 3, 2011
MathSciNet review: 2846677
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Abstract | References | Additional Information

Abstract: Let $Gr(k,n)$ be the Plücker embedding of the Grassmann variety of projective $k$-planes in $\mathbb P^n$. For a projective variety $X$, let $\sigma _s(X)$ denote the variety of its secant $(s-1)$-planes. More precisely, $\sigma _s(X)$ denotes the Zariski closure of the union of linear spans of $s$-tuples of points lying on $X$. We exhibit two functions $s_0(n)\le s_1(n)$ such that $\sigma _s(Gr(2,n))$ has the expected dimension whenever $n\geq 9$ and either $s\le s_0(n)$ or $s_1(n)\le s$. Both $s_0(n)$ and $s_1(n)$ are asymptotic to $\frac {n^2}{18}$. This yields, asymptotically, the typical rank of an element of $\bigwedge \nolimits ^{3} {\mathbb C}^{n+1}$. Finally, we classify all defective $\sigma _s(Gr(k,n))$ for $s\le 6$ and provide geometric arguments underlying each defective case.

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Additional Information

Hirotachi Abo
Affiliation: Department of Mathematics, University of Idaho, Moscow, Idaho 83844
MR Author ID: 614361

Giorgio Ottaviani
Affiliation: Dipartimento di Matematica “Ulisse Dini”, Università degli Studi di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy
MR Author ID: 134700

Chris Peterson
Affiliation: Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523-1874
MR Author ID: 359254

Received by editor(s): January 8, 2009
Received by editor(s) in revised form: August 10, 2009
Published electronically: January 3, 2011