Pullback of varieties by finite maps

TL;DR

This paper investigates how certain geometric properties of varieties, like normality, prefactoriality, and smoothness, are preserved or reflected through finite holomorphic maps.

Contribution

It establishes conditions under which properties of the pullback variety imply the same properties for the original variety, including new results on smoothness under additional assumptions.

Findings

Normality and prefactoriality are preserved under pullback.

Smoothness of the original variety can be inferred from the pullback with extra conditions.

Provides criteria for property transfer via finite maps.

Abstract

We study the local geometry of the pullback of a variety via a finite holomorphic map. In particular, we are looking for properties of $V = F^{-1}(W)$ such that if $V$ has the property $A$, then $W$ must have the property $A$. We show that $A$ can be the property of normality or prefactoriality. We also show that $A$ can be the property of smoothness, under extra assumptions.