Degrees of generators of ideals defining subschemes of projective space

Abstract

For an arithmetically Cohen--Macaulay subscheme of projective space, there is a well-known bound for the highest degree of a minimal generator for the defining ideal of the subscheme, in terms of the Hilbert function. We prove a natural extension of this bound for arbitrary locally Cohen--Macaulay subschemes. We then specialize to curves in $\pthree$, and show that the curves whose defining ideals have generators of maximal degree satisfy an interesting cohomological property. The even liaison classes possessing such curves are characterized, and we show that within an even liaison class, all curves with the property satisfy a strong Lazarsfeld--Rao structure theorem. This allows us to give relatively complete conditions for when a liaison class contains curves whose ideals have maximal degree generators, and where within the liaison class they occur.