Some effects of Veronese map on syzygies of projective varieties

Details Some effects of Veronese map on syzygies of projective varieties Some effects of Veronese map on syzygies of projective varieties

2005-09-30

arxiv.org/abs/math/0509710v3

Mathematics

Algebraic Geometry

16 pages

Scientific paper

Let $X \subset \P^r$ be a nondegenerate projective variety and let $\nu_{\ell} : \P^r \to \P^N$ be the $\ell$-th Veronese embedding. In this paper we study the higher normality, defining equations and syzygies among them for the projective embedding $\nu_{\ell} (X) \subset \P^N$. We obtain that for a very ample line bundle $L \in {Pic}X$ such that $X \subset \P H^0 (X,L)$ is $m$-regular in the sense of Castelnuovo-Mumford, $(X,L^{\ell})$ satisfies property $N_{\ell}$ for all $\ell \geq m$ (Theorem \ref{thm:main1}). This is a generalization of M. Green's work that $(\P^r,\mathcal{O}_{\P^r} (\ell))$ satisfies property $N_{\ell}$. Also our result refines works of Ein-Lazarsfeld in \cite{EL}, Gallego-Purnaprajna in \cite{GP} and E. Rubei in \cite{Rubei1}, \cite{Rubei} and \cite{Ru1}.

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