Integral Subschemes of Codimension Two

TL;DR

This paper investigates the structure of integral subschemes of codimension two within a fixed linkage class, introducing invariants that help characterize their positions and conditions for their deformation.

Contribution

It introduces invariants $ heta_X$ and $ta_X$ to describe subscheme positions and provides near-complete criteria for their integrality and deformation within linkage classes.

Findings

Invariants $ heta_X$ and $ta_X$ determine subscheme positions.

Necessary conditions for integrality are established.

Conditions are nearly sufficient for deformation to integral subschemes.

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.