Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

DMV Tagung 2011 - Köln, 19. - 22. September
Hiro-o Tokunaga
Tokyo Metropolitan University
Splitting curves for double covers and the topology of the complements of certain curves on
rational ruled surfaces
Let Σ be a smooth projective surface. Let f ′ : Z ′ → Σ be a double cover, i.e, Z ′ : a normal projective surface,
f ′ : a finite surjective morphism of degree 2. Let µ : Z → Z ′ be the minimal resolution and we put f = µ ◦ f ′ .
1. An irreducible curve D on Σ is called a splitting curve with respect to f if f ∗ D is of the form
f ∗ D = D + + D − + E
where D + �= D − , f (D + ) = f (D − ) = D and Supp(E) is contained in the exceptional set of µ.
2. Let D be a splitting curve on Σ with respect to f : Z → Σ. If the double cover is determined by the
branch locus ∆( f ) (e.g., the case when Σ is simply connected), we say that
∆( f ) is a quadratic residue curve mod D.
In this talk, we discuss “reciprocity” for quadratic residue curves on rational ruled surfaces under some
special setting and consider its application to the topology of the complements of curves.
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