Cuspidal Multiple Structures on Smooth Algebraic Varieties as Support

TL;DR

This paper constructs specific nilpotent scheme structures on smooth varieties, characterized locally by particular ideal forms, advancing understanding of algebraic structures supported on smooth varieties.

Contribution

It introduces explicit local models for cuspidal multiple structures on smooth algebraic varieties, expanding the toolkit for studying nilpotent schemes in algebraic geometry.

Findings

Explicit local ideal forms for cuspidal structures

Construction of lci nilpotent schemes on smooth varieties

Enhanced understanding of scheme support structures

Abstract

We construct lci nilpotent scheme structures $Y \subset P$ on a smooth variety $X$ embedded in a smooth variety $P$, which are, locally, (i.e. in $\widehat{\mathcal O}_{p,P}$ ) given by ideals of the form $(y^2+x^n, xy, z_1,...,z_r)$, $(y^3+x^n, xy, z_1 ,...z_r)$