Deformations of pairs (X,L) when X is singular

TL;DR

This paper develops a new, elementary approach to the deformation theory of pairs (X, L) where X is a singular scheme, extending classical results from smooth cases and providing criteria for extending sections of L.

Contribution

It introduces a simplified tangent-obstruction theory for deformations of pairs with singular X and establishes criteria for extending sections of L.

Findings

Generalizes classical deformation theory to singular schemes

Provides criteria for extending sections of line bundles

Offers an elementary construction of the tangent-obstruction theory

Abstract

We give an elementary construction of the tangent-obstruction theory of the deformations of the pair $(X,L)$ with $X$ a reduced local complete intersection scheme and $L$ a line bundle on $X$. This generalizes the classical deformation theory of pairs in case $X$ is smooth. A criteria for sections of $L$ to extend is also given.