Normalit\'e projective des var\'et\'es magnifiques de rang 1

TL;DR

This paper proves that for rank one wonderful varieties, the tensor product of global sections of two invertible sheaves surjects onto the global sections of their tensor product, ensuring normality of certain cones.

Contribution

It establishes the surjectivity of the multiplication map of global sections for invertible sheaves on rank one wonderful varieties, a result not previously known.

Findings

The multiplication map of global sections is surjective for rank one wonderful varieties.

The cone over such varieties defined by very ample sheaves is always normal.

Provides a new geometric criterion for normality in this class of varieties.

Abstract

Let $L$ and $L'$ be two invertible sheaves over a projective variety $X$. We suppose that $L$ and $L'$ are generated by their global section spaces $\Gamma(L)$ and $\Gamma(L')$. We prove in this article that the morphism : \[\Gamma(L) \otimes \Gamma(L') \to \Gamma(L \otimes L')\] is surjective, in the case where $X$ is a rank one wonderful variety. In particular, the cone over a rank one wonderful variety defined by a very ample invertible sheaf is always normal. Soient $L$ et $L'$ deux faisceaux inversibles sur une vari\'et\'e projective $X$. On suppose que $L$ et $L'$ sont engendr\'es par leurs espaces de sections globales $\Gamma(L)$ et $\Gamma(L')$. On d\'emontre dans cet article que le morphisme : \[\Gamma(L) \otimes \Gamma(L') \to \Gamma(L \otimes L')\] est surjectif, dans le cas o\`u $X$ est une vari\'et\'e magnifique de rang 1. En particulier, le c\^one au-dessus d'une vari\'et\'e magnifique $X$ de rang 1 d\'efini par un faisceau inversible tr\`es ample est toujours normal.