STNB2022 (35th edition)

The special fibre of a semistable hyperelliptic curve


Nirvana Coppola


After having presented hyperelliptic curves $y^2 = f(x)$ over local fields of odd residue characteristic and having introduced the notion of a ``cluster picture" associated to the curve describing the $p$-adic distances between the roots of $f(x)$ in the first talk by A. Somoza, we will show that this elementary combinatorial object encodes whether the curve is semistable, and if so, the special fibre of its minimal regular model among other invariants. If time allows we will also see how these techniques can be used for the case of superelliptic curves.


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