Stable reduction and topological invariants of complex polynomials

Details Stable reduction and topological invariants of complex polynomials Stable reduction and topological invariants of complex polynomials

2006-05-10

arxiv.org/abs/math/0605267v1

Mathematics

Algebraic Geometry

18 pages

Scientific paper

A topological invariant of a polynomial map $p:X\to B$ from a complex surface containing a curve $C\subset X$ to a one-dimensional base is given by a rational second homology class in the compactification of the moduli space of genus $g$ curves with $n$ labeled points $\modmgn$. Here the generic fibre of $p$ has genus $g$ and intersects $C$ in $n$ points. In this paper we give an efficient method to calculate this homology class. We apply this to any polynomial in two complex variables $p :\bc^2\to\bc$ where the $n$ points on a fibre are its points at infinity.

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