A Note on Equimultiple Deformations

TL;DR

This paper investigates the tangent space dimensions of families of curves with prescribed multiplicities at points on a smooth surface, providing formulas that depend on the singularity type and aiding in the study of special surface properties.

Contribution

It introduces a method to compute tangent space dimensions for families of curves with fixed multiplicities, extending previous descriptions to non-equisingular cases.

Findings

Derived explicit tangent space dimension formulas for families L_m.

Identified the expected dimension as dim|L|+2 - m*(m+1)/2.

Applied results to study triple-point defective surfaces.

Abstract

While the tangent space to an equisingular family of curves can be discribed by the sections of a twisted ideal sheaf, this is no longer true if we only prescribe the multiplicity which a singular point should have. However, it is still possible to compute the dimension of the tangent space with the aid of the equimulitplicity ideal. In this note we consider families L_m={(C,p) | mult_p(C)=m} with C in some linear system |L| on a smooth projective surface S and for a fixed positive integer m, and we compute the dimension of the tangent space to L_m at a point (C,p) depending on whether p is a unitangential singular point of C or not. We deduce that the expected dimension of L_m at (C,p) in any case is just dim|L|+2-m*(m+1)/2. The result is used in the study of triple-point defective surfaces in some joint papers with Luca Chiantini.