Bornes pour la r\'egularit\'e de Castelnuovo-Mumford des sch\'emas non lisses

TL;DR

This paper establishes bounds on the Castelnuovo-Mumford regularity of singular schemes based on their defining equations, with improved bounds for isolated singularities and a doubly exponential bound in the general case.

Contribution

It provides new bounds on Castelnuovo-Mumford regularity for schemes with singularities, improving existing bounds for isolated singularities and establishing a doubly exponential bound generally.

Findings

Improved bound for schemes with isolated singularities.

Doubly exponential bound for general singular schemes.

Bounds depend on degrees of defining equations and dimensions.

Abstract

We show that bounds on the Castelnuovo-Mumford regularity of singular schemes, as a function of the degrees of the equations defining the shceme, of its dimension and of the dimension of their singular space. In the case where the singularities are isolated, we improve the bound given by Chardin and Ulrich, and in the general case we establish a bound doubly exponential in the dimension of the singular space. -- Nous montrons dans cet article des bornes pour la regularite de Castelnuovo-Mumford d'un schema admettant des singularites, en fonction des degres des equations definissant le schema, de sa dimension et de la dimension de son lieu singulier. Dans le cas ou les singularites sont isolees, nous ameliorons la borne fournie par Chardin et Ulrich et dans le cas general, nous etablissons une borne doublement exponentielle en la dimension du lieu singulier.