Local invariants attached to Weierstrass points

Details Local invariants attached to Weierstrass points Local invariants attached to Weierstrass points

2007-10-29

arxiv.org/abs/0710.5464v2

manuscripta math. 129 (2009), 273--292

Mathematics

Algebraic Geometry

18 pages

Scientific paper

Let X/S be a hyperelliptic curve of genus g over the spectrum of a discrete valuation ring. Two fundamental numerical invariants are attached to X/S: the valuation of the hyperelliptic discriminant of X/S, and the valuation of the Mumford discriminant of X/S (equivalently, the Artin conductor). For a residue field of characteristic 0 as well as for X/S semistable these invariants are known to satisfy certain inequalities. We prove an exact formula relating the two invariants with intersection theoretic data determined by the distribution of Weierstrass points over the special fiber, in the semistable case. We also prove an exact formula for the stable Faltings height of an arbitrary curve over a number field, involving local contributions associated to its Weierstrass points.

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