Glicci ideals

TL;DR

This paper proves that under certain conditions, arithmetically Cohen-Macaulay subschemes can be linked to complete intersections via Gorenstein schemes, extending liaison theory results.

Contribution

It demonstrates that generically Gorenstein ACM schemes can be linked to complete intersections in higher-dimensional projective spaces, and shows unions of fat points can be linked to simple points.

Findings

ACM schemes linked to complete intersections if generically Gorenstein

Unions of fat points in P^3 linked to simple points

Links can be performed in higher-dimensional projective spaces

Abstract

A central problem in liaison theory is to decide whether every arithmetically Cohen-Macaulay subscheme of projective $n$-space can be linked by a finite number of arithmetically Gorenstein schemes to a complete intersection. We show that this can be indeed achieved if the given scheme is also generically Gorenstein and we allow the links to take place in an $(n+1)$-dimensional projective space. For example, this result applies to all reduced arithmetically Cohen-Macaulay subschemes. We also show that every union of fat points in projective 3-space can be linked in the same space to a union of simple points in finitely many steps, and hence to a complete intersection in projective 4-space.