Abstract
Let $C$ be a hyperelliptic curve $y^2 = f(x)$ over a discretely valued field $K$. The $p$-adic distances between the roots of $f(x)$ can be described by a completely combinatorial object known as the cluster picture. We show that the cluster picture of $C$, along with the leading coefficient of $f$ and the action of $\mathrm{Gal}(\bar{K}/K)$ on the roots of $f$, completely determines the combinatorics of the special fibre of the minimal strict normal crossings model of $C$. In particular, we give an explicit description of the special fibre in terms of this data.
Highlights
Let C be a hyperelliptic curve y2 = f (x) over a discretely valued field K
Using cluster pictures we will calculate a combinatorial description of the minimal SNC model X of C/K: a model whose singularities on the special fibre Xk are normal crossings, and where blowing down any exceptional component of Xk would result in a worse singularity
The minimal SNC model of a hyperelliptic curve has a rather straightforward description: it consists of a central component with some tails whose multiplicities can be explicitly described using the results of Section 3.2
Summary
For the convenience of the reader, the following two tables collate the general notation and terminology which we use throughout the paper. Whenever a component in a figure is drawn in bold it is a central component. In any figure describing the special fibre of a model numbers indicate multiplicities, except those preceded by g, which indicate the genus of a component.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
