Regular models of superelliptic curves via Mac Lane valuations

Abstract: Let $X \rightarrow \mathbb{P}^1$ be a $\mathbb{Z}/n$-branched cover over a complete discretely valued field $K$, where $n$ does not divide the residue characteristic of $K$. We explicitly construct the minimal regular normal crossings model of $X$ over the valuation ring of $K$. By “explicitly”, we mean that we construct a normal model of $\mathbb{P}^1$ whose normalization in $K(X)$ is the desired regular model. The normal model of $\mathbb{P}^1$ is fully encoded as a basket of finitely many discrete valuations on the rational function field $K(\mathbb{P}^1)$, each of which is given using Mac Lane’s 1936 notation involving finitely many polynomials and rational numbers. This is joint work with Padmavathi Srinivasan.

algebraic geometrynumber theory

Audience: researchers in the topic

Boston University Number Theory Seminar