This article is cited in 1 paper

A differential-geometrical criterion for quadratic Veronese embeddings

V. V. Konnov

Moscow State Pedagogical University

Abstract: We obtain a criterion for quadratic Veronese varieties. We prove that in the set of smooth $n$-dimensional submanifolds of the projective space $P^N$ of dimension $N=n(n+3)/2$ only the Veronese varieties have the following two properties: (i) the tangent projective spaces at any two points intersect in a point, (ii) the osculating projective space at every point coincides with the ambient space. This result is a generalization to arbitrary $n$ of the criterion for two-dimensional Veronese surfaces in $P^5$ proved by Griffiths and Harris. We also find a criterion for a pair of submanifolds of $P^N$ to be contained in the same Veronese variety. We obtain calculation formulae that enable one to use these criteria in practice.

MSC: 53A20, 14C21, 53A60, 53C40

Received: 10.03.1999

DOI: 10.4213/im303

 English version: Izvestiya: Mathematics, 2000, 64:5, 891–914

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