Let $X$ be an algebraic variety with quotient field $K(X)$. Let $s$ be the highest multiplicity of $X$ and $F_s(X)$ the set of points of multiplicity $s$. A sequence of blow ups at regular centers $Y_i \subset F_s(X_i)$, $X \leftarrow X_1 \leftarrow \dotsb \leftarrow X_n$, is a reduction of the multiplicity if $F_s(X_n) $ is empty. In characteristic zero there is an algorithm which assigns to each $X$ a unique reduction of the multiplicity. Fix $K(X) \subset L$ a finite extension of fields of degree $r$. For a finite map $\beta : X' \to X$ with $K(X') = L$ the highest multiplicity of $X'$ is at most $r \cdot s$. When the bound is achieved we say that $\beta$ is transversal. We will see that, if $\beta$ is transversal, then $F_{rs}(X')$ is homeomorphic to its image and $\beta(F_{rs}(X'))\subset F_{s}(X)$. We will see that a blow up $X' \leftarrow X'_1$ along a regular center $Y'_1 \subset F_{rs}(X'_1)$ induces a blow up $X \leftarrow X_1$ along a center $Y_1 \subset F_s(X_1)$ and a fini 查看全文>>