Finite morphisms and simultaneous reduction of the multiplicity

TL;DR

This paper investigates finite morphisms between singular algebraic varieties and their impact on the multiplicity reduction process, introducing the concept of strong transversality to relate the multiplicity strata of the varieties.

Contribution

It establishes conditions under which finite morphisms are strongly transversal, linking the multiplicity strata of the source and target varieties, and constructs such morphisms for given field extensions.

Findings

Strong link between multiplicity strata of related varieties

Definition and properties of strongly transversal morphisms

Construction of strongly transversal morphisms for field extensions

Abstract

Let $X$ be a singular algebraic variety defined over a field $k$, with quotient field $K(X)$. Let $s \geq 2$ be the highest multiplicity of $X$ and $F_s(X)$ the set of points of multiplicity $s$. If $Y\subset F_s(X)$ is a regular center and $X\leftarrow X_1$ is the blow up at $Y$, then the highest multiplicity of $X_1$ is less than or equal to $s$. A sequence of blow ups at regular centers $Y_i \subset F_s(X_i)$, say $X \leftarrow X_1 \leftarrow \dotsb \leftarrow X_n$, is said to be a {\em simplification} of the multiplicity if the maximum multiplicity of $X_n$ is strictly lower than that of $X$, that is, if $F_s(X_n) $ is empty. In characteristic zero there is an algorithm which assigns to each $X$ a unique simplification of the multiplicity. However, the problem remains open when the characteristic is positive. In this paper we will study finite dominant morphisms between singular varieties $\beta: X'\to X$ of generic rank $r \geq 1$ (i.e., $[K(X'):K(X)]=r$). We will see that, when imposing suitable conditions on $\beta$, there is a strong link between the strata of maximum multiplicity of $X$ and $X'$, say $F_{s}(X)$ and $F_{rs}(X')$ respectively. In such case, we will say that the morphism is strongly transversal. When $\beta: X'\to X$ is strongly transversal one can obtain information about the simplification of the multiplicity of $X$ from that of $X'$ and vice versa. Finally, we will see that given a singular variety $X$ and a finite field extension $L$ of $K(X)$ of rank $r \geq 1$, one can construct (at least locally, in \'etale topology) a strongly transversal morphism $\beta: X'\to X$, where $X'$ has quotient field $L$.