Families of K3 surfaces over curves satisfying the equality of Arakelov-Yau's type and modularity

Details Families of K3 surfaces over curves satisfying the equality of Arakelov-Yau's type and modularity Families of K3 surfaces over curves satisfying the equality of Arakelov-Yau's type and modularity

2002-05-20

arxiv.org/abs/math/0205213v1

Mathematics

Algebraic Geometry

18 pages, Latex

Scientific paper

Let $f:X\to C$ be a family of semistable K3 surfaces with non-empty set $S$ of singular fibres having infinite local monodromy. Then, when the so called Arakelov-Yau inequality reaches equality, we prove that $C\setminus S$ is a modular curve and the family comes essentially from a family of elliptic curves through a so called Nikulin-Kummer construction. In particular, when $C=\BBb P^1$, the family of elliptic curves must be one of Beauville's 6 examples where Arakelov inequality reaches equality.

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