Finite morphisms and simultaneous reduction of the multiplicity

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 $L$.