Une nouvelle caract\'erisation des vari\'et\'es de Veronese

TL;DR

This paper characterizes Veronese varieties of order q by their regular osculation and the existence of rational normal curves in all tangent directions, generalizing classical results.

Contribution

It provides a new characterization of Veronese varieties based on local geometric conditions involving osculation and rational curves.

Findings

Veronese varieties are uniquely determined by their osculation properties.

Existence of rational normal curves in all tangent directions implies the variety is Veronese.

Generalizes classical results by Bompiani on characterizations of Veronese varieties.

Abstract

Let $X$ be the germ of a smooth complex variety at a given point $x\in {\mathbbb P}^N$ with regular osculation at order $q$ and suppose that, for any direction $v\in {\mathbbb P}T_xX$, there exists a rationnal normal curve locally contained in $X$ and passing through the point $x$ in direction $v$. We show that $X$ is necessarily a Veronese variety of order $q$. As a special case, we recover a classical result of Bompiani, in which it is assumed that the condition is verified by any point close to $x$.