Smoothings of Fano schemes with normal crossing singularities of dimension at most three

TL;DR

This paper investigates the deformation theory of Fano varieties with normal crossing singularities up to dimension three, providing explicit formulas and criteria for their smoothability.

Contribution

It introduces a formula for the sheaf T^1(X) in a log resolution and establishes explicit conditions for smoothing such Fano varieties.

Findings

Derived a formula for T^1(X) in the context of normal crossing singularities.

Provided explicit criteria for the existence of smoothings.

Enhanced understanding of deformation theory for low-dimensional Fano varieties.

Abstract

We study the deformation theory of a Fano variety X with normal crossing singularities of dimension at most three. We obtain a formula for the sheaf T^1(X) of first order deformations of X in a suitable log resolution of X and its singular locus Z and we obtain explicit criteria for the existence of smoothings of X.