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

Abstract

The Castelnuovo-Mumford regularity r of a complex, projective variety V 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 the ambient projective space. 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<m+s+2, 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 bounds given by Brunella and Mendes.