Instituto de Investigación
en Matemáticas


Multiplicity and finite morphisms

Carlos Abad Reigadas (Universidad Autónoma de Madrid)

Fecha: 24/05/2017 18:00
Lugar: Seminario A125. Facultad de Ciencias

Let $X$ be a variety over a perfect field $k$. The multiplicity is an invariant that stratifies $X$ into locally closed sets. $X$ is regular at a point $\xi$ if and only if $\mathrm{mult}(\xi) = 1$. Dade proved that, if \begin{equation} (1)\qquad X \leftarrow X_1 \leftarrow \dotsb \leftarrow X_l \end{equation} is a sequence of blow-ups along closed regular equimultiple centers, then $$ \max\mathrm{mult}(X_l) \leq \max\mathrm{mult}(X). $$ If one can find a sequence like (1) so that $\max\mathrm{mult}(X_l) < \max\mathrm{mult}(X)$, then a resolution of singularities of $X$ can be constructed by iteration of this process. The study of the multiplicity has been historically linked to that of finite morphisms. Zariski's formula says that, if $\beta : X' \to X$ is a finite and dominant morphism of varieties of generic rank $r$, then \begin{equation} (2) \qquad \max\mathrm{mult}(X') \leq r \cdot \max\mathrm{mult}(X). \end{equation} Let $\underline{\mathrm{Max}}\mathrm{mult}(X)$ denote the stratum of maximum multiplicity of $X$. When the equality holds in (2), $\underline{\mathrm{Max}}\mathrm{mult}(X')$ is homeomorphic to its image in $X$, which is contained in $\underline{\mathrm{Max}}\mathrm{mult}(X)$. In our work, we study conditions under which \[ \beta \bigl( \underline{\mathrm{Max}}\mathrm{mult}(X') \bigr) \cong \underline{\mathrm{Max}}\mathrm{mult}(X), \] and such that this homeomorphism is preserved by permissible blow-ups. These conditions provide a relation between the processes of lowering of the maximum multiplicity of $X$ and $X'$ by blowing up equimultiple centers. This is a joint work with Ana Bravo and Orlando Villamayor.