Decomposition into pairs-of-pants for complex algebraic hypersurfaces

2002/05/01 by Grigory Mikhalkin, Mikhalkin, Grigory · 1 citation
Computer Science · Mathematics · #Advanced Topics in Algebra #Algebraic Geometry and Number Theory #Polynomial and algebraic computation #math.AG #math.GT #math.SG #msc:14J70

paper · pdf · doi:10.48550/arxiv.math/0205011

35 pages, 9 figures, final version to appear in Topology

arxiv created 2003/11/19 · arxiv updated 2009/11/30

Abstract

It is well-known that a Riemann surface can be decomposed into the so-called pairs-of-pants. Each pair-of-pants is diffeomorphic to a Riemann sphere minus 3 points. We show that a smooth complex projective hypersurface of arbitrary dimension admits a similar decomposition. The n-dimensional pair-of-pants is diffeomorphic to the complex projective n-space minus n+2 hyperplanes. Alternatively, these decompositions can be treated as certain fibrations on the hypersurfaces. We show that there exists a singular fibration on the hypersurface with an n-dimensional polyhedral complex as its base and a real n-torus as its fiber. The base accomodates the geometric genus of a hypersurface V. Its homotopy type is a wedge of h^n,0(V) spheres Sⁿ.

Citations

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