# An application of Liaison theory to zero-dimensional schemes

## TL;DR

This paper applies Liaison theory to characterize the Cayley-Bacharach property of zero-dimensional schemes in projective space, extending previous results for rational points and analyzing related algebraic invariants.

## Contribution

It extends the characterization of the Cayley-Bacharach property to arbitrary zero-dimensional schemes using Liaison theory and investigates bounds on the Hilbert function and regularity index of the Dedekind different.

## Key findings

- Extended Cayley-Bacharach characterization to all zero-dimensional schemes
- Bounded the Hilbert function of the Dedekind different
- Analyzed the regularity index of the Dedekind different

## Abstract

Given a 0-dimensional scheme X in a n-dimensional projective space P^n_K over an arbitrary field K, we use Liaison theory to characterize the Cayley-Bacharach property of X. Our result extends the result for sets of K-rational points given in [7]. In addition, we examine and bound the Hilbert function and regularity index of the Dedekind different of X when X has the Cayley-Bacharach property.