Skeleta of Affine Hypersurfaces

TL;DR

This paper presents a combinatorial method to construct a topological space that captures the homotopy type of smooth affine hypersurfaces, linking algebraic geometry with combinatorial topology.

Contribution

It introduces a purely combinatorial construction of a space homotopy equivalent to affine hypersurfaces using Newton polytope triangulations.

Findings

Constructs a space S from combinatorial data that models the hypersurface.

Proves S embeds into the hypersurface as a deformation retract.

Shows the hypersurface is homotopy equivalent to the constructed space.

Abstract

A smooth affine hypersurface Z of complex dimension n is homotopy equivalent to an n-dimensional cell complex. Given a defining polynomial f for Z as well as a regular triangulation of its Newton polytope, we provide a purely combinatorial construction of a compact topological space S as a union of components of real dimension n, and prove that S embeds into Z as a deformation retract. In particular, Z is homotopy equivalent to S.