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Characterization of Veronese varieties via projection in Grassmannians

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Arrondo Esteban, Enrique and Paoletti, Raffaella (2005) Characterization of Veronese varieties via projection in Grassmannians. In Projective varieties with unexpected properties. Walter de Gruyter, Berlin, pp. 1-12. ISBN 978-3-11-018160-9

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Abstract

Let G(r,m) denote the Grassmann variety of r-dimensional linear subspaces of Pm. To any linear projection Pm⇢Pm′, m′<m, there corresponds a rational map G(r,m)⇢G(r,m′) which will also be called a projection. In [J. Algebraic Geom. 8 (1999), no. 1, 85–101; MR1658212 (99k:14083)], E. Arrondo started the study of smooth subvarieties of Grassmann varieties having "deep'' isomorphic projections and proved that, under a certain additional assumption, the only smooth n-dimensional subvariety of G(1,2n+1) isomorphically projectable to G(1,n+1) is the Veronese subvariety of G(1,2n+1), defined as the locus of lines joining the corresponding points of two disjoint n-dimensional linear subspaces in P2n+1. More generally, a smooth subvariety X⊂G(d−1,N) is said to be k-projectable to G(d−1,M), 0≤k≤d−1, if there exists a projection π:G(d−1,N)⇢G(d−1,M) such that dimL∩L′<k for any two subspaces L,L′∈π(X).
In the paper under review the authors extend this result to Grassmann varieties of higher-dimensional linear subspaces. To wit, they prove that, under certain assumptions, if X⊂G(d−1,nd+d−1) is 1-projectable to G(d−1,n+2d−3), then X is the d-tuple Veronese variety defined as the locus of Pd−1's spanned by the d-tuples of corresponding points of d copies of Pn in general position in Pnd+d−1. Unfortunately, the authors can only prove this under rather restrictive hypotheses, e.g. they assume that X has positive defect.


Item Type:Book Section
Additional Information:

A volume in memory of Giuseppe Veronese. Proceedings of the International Conference "Projective Varieties with Unexpected Properties'' held in Siena, June 8–13, 2004

Uncontrolled Keywords:linear subspaces;Low codimension problems
Subjects:Sciences > Mathematics > Algebraic geometry
ID Code:20855
Deposited On:16 Apr 2013 16:18
Last Modified:08 Jan 2016 10:09

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