Global equisingularity of families of affine hypersurfaces

Details Global equisingularity of families of affine hypersurfaces Global equisingularity of families of affine hypersurfaces

1998-05-18

arxiv.org/abs/math/9805075v1

Int.Math.Research Notices, 18 (1998), 979-990.

Mathematics

Algebraic Geometry

LaTeX, 12 pages; May 1997, revised Oct. 1997

Scientific paper

We consider an equisingularity problem for polynomial families of affine hypersurfaces $X_\tau \subset \bC^n$. We show that the constancy of the global polar invariants $\gamma^* (X_\tau)$ is equivalent to the $t$-equisingularity at infinity, an asymptotic-type equisingularity that we introduce. We prove that $\gamma^*$-constancy implies C$^\ity$-triviality in the neighbourhood of infinity. We show how the invariants $\gamma^*$ enter in the description of a CW-complex model of a hypersurface $X_\tau$ and therefore provide in particular new invariants at infinity for polynomial functions $f: \bC^n \to \bC$.

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