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oai:docta.ucm.es:20.500.14352/532292023-09-07T15:27:24Zcom_20.500.14352_14col_20.500.14352_21
Docta Complutense author Arrondo Esteban, Enrique author Paoletti, Raffaella editor Ciliberto, C. editor Geramita, A.V. editor Harbourne, B. editor M. Miró-Roig, R.M. editor Ranestad, K. 2023-06-20T13:39:16Z 2023-06-20T13:39:16Z 2005 978-3-11-018160-9 https://hdl.handle.net/20.500.14352/53229 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. eng Characterization of Veronese varieties via projection in Grassmannians book part URL https://docta.ucm.es/bitstreams/eee7549f-bd42-437d-9144-9be4d89ce6bb/download File MD5 0cb4c339b3883aa40061a134b6d972d5 203628 application/pdf 01_ArrondoPaoletti.DG.pdf URL https://docta.ucm.es/bitstreams/ba3fac4c-758f-4207-bddc-02dc6a8c2fce/download File MD5 c555d05266b950e504705cceb211622c 29911 text/plain 01_ArrondoPaoletti.DG.pdf.txt