Cartwright-Sturmfels Hilbert Schemes

TL;DR

This paper investigates the Cartwright-Sturmfels Hilbert schemes associated with Cox rings of products of projective spaces, proving smoothness and irreducibility for low Picard rank cases, extending classical results to a multigraded setting.

Contribution

It establishes that these multigraded Hilbert schemes are always smooth and irreducible when the Picard rank is at most 2, generalizing known theorems to a new context.

Findings

Hilbert schemes are smooth and irreducible for Picard rank ≤ 2

Extension of Fogarty and Maclagan-Smith theorems to multigraded cases

Provides a multigraded analogue of classical Hilbert scheme results

Abstract

Let S be the Cox ring of a product of r projective spaces. In this paper, we study the Cartwright-Sturmfels Hilbert schemes of S, which are multigraded Hilbert schemes that only parametrize radical ideals. Our main result shows that these Hilbert schemes are always smooth and irreducible if the Picard rank r is at most 2. This result can be seen as a multigraded analogue of the famous theorems of Fogarty and Maclagan-Smith, where the Picard rank replaces the dimension of the ambient space.