Mathematical Research Letters

Volume 9, Issue 1, January 2002 pp. 1-15.

The Castelnuovo--Mumford regularity of an integral variety of a vector field on projective space

Authors: Eduardo Esteves
Author institution: Estrada Dona Castorina 110

Summary: The Castelnuovo--Mumford regularity $r$ of a variety $V\subseteq\text{\bf P}^n_{\hskip-0.1cm\text{\bf C}}$ is an upper bound for the degrees of the hypersurfaces necessary to cut out $V$. In this note we give a bound for $r$ when $V$ is left invariant by a vector field on $\text{\bf P}^n_{\hskip-0.1cm\text{\bf C}}$. More precisely, assume $V$ is arithmetically Cohen--Macaulay, for instance, a complete intersection. Assume as well that $V$ projects to a normal-crossings hypersurface, which is the case when $V$ is a curve with at most ordinary nodes. Then we show that $r\leq m+s+1$, where $s$ is the dimension of $V$ and $m$ is the degree of the vector field. Our method consists of using first central projections to reduce the problem to when $V$ is a hypersurface, and then using bounds given by Brunella and Mendes.

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