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
DOI: https://doi.org/10.1090/S1056-3911-2010-00540-1
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
Email: abo@uidaho.edu

Giorgio Ottaviani
Affiliation: Dipartimento di Matematica “Ulisse Dini”, Università degli Studi di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy
Email: ottavian@math.unifi.it

Chris Peterson
Affiliation: Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523-1874
Email: peterson@math.colostate.edu

DOI: https://doi.org/10.1090/S1056-3911-2010-00540-1
Received by editor(s): January 8, 2009
Received by editor(s) in revised form: August 10, 2009
Published electronically: January 3, 2011

American Mathematical Society