# Syzygies in Hilbert schemes of complete intersections

## TL;DR

This paper investigates the maximum number of equations and syzygies for subschemes in Hilbert schemes of complete intersections, providing bounds and applications to broader classes of complete intersections.

## Contribution

It establishes sharp upper bounds on equations and syzygies in Hilbert schemes of points on complete intersections, extending to arbitrary cases.

## Key findings

- Derived sharp bounds on equations and syzygies
- Applied bounds to general complete intersections
- Enhanced understanding of Hilbert scheme structures

## Abstract

Let $ e_1, ..., e_c $ be positive integers and let $ Y \subseteq \mathbb{P}^n$ be the monomial complete intersection defined by the vanishing of $x_1^{e_1}, ..., x_c^{e_c}$. In this paper we study sharp upper bounds on the number of equations and syzygies of subschemes parametrized by the Hilbert scheme of points $Hilb^d(Y)$, and discuss applications to the Hilbert scheme of points $Hilb^d(X)$ of arbitrary complete intersections $X \subseteq \mathbb{P}^n$.