Integral Subschemes of Codimension Two

Abstract

In this paper we study the problem of describing the integral subschemes within a fixed even linkage class $\L$ of subschemes in $\Pn$ of codimension two. In the case that $\L$ is not the class of arithmetically Cohen-Macaulay subschemes, we associate to any $X \in \L$ two invariants $\theta_X$ and $\eta_X$. When taken with the height $h_X$, each of these invariants determines the location of $X$ in $\L$, thought of as a poset under domination. In terms of these invariants, necessary conditions are given for integral subschemes. The necessary conditions are almost sufficient in the sense that if a subscheme $X$ satisfies the necessary conditions and dominates an integral subscheme $Y$, then $X$ can be deformed with constant cohomology through subschemes in $\L$ to an integral subscheme. In particular, if an even linkage class has a minimal element which is integral, then the conditions are both necessary and sufficient.