#### Volume 18, issue 3 (2014)

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Skeleta of affine hypersurfaces

### Helge Ruddat, Nicolò Sibilla, David Treumann and Eric Zaslow

Geometry & Topology 18 (2014) 1343–1395
##### 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 ${\mathsc{T}}_{△}$ 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 $S$.

##### Keywords
skeleton, retraction, hypersurface, homotopy equivalence, affine, toric degeneration, Kato–Nakayama space, log geometry, Newton polytope, triangulation
Primary: 14J70
Secondary: 14R99
##### Publication
Revised: 19 December 2013
Accepted: 17 January 2014
Published: 7 July 2014
Proposed: Richard Thomas
Seconded: Benson Farb, Danny Calegari
##### Authors
 Helge Ruddat Mathematisches Institut Johannes Gutenberg-Universität Mainz Staudingerweg 9 D-55099 Mainz Germany Nicolò Sibilla Max Planck Institute for Mathematics Vivatsgasse 7 D-53111 Bonn Germany David Treumann Department of Mathematics Boston College Carney Hall, Room 301 Chestnut Hill Boston, MA 02467-3806 USA Eric Zaslow Department of Mathematics Northwestern University 2033 Sheridan Road Evanston, IL 60208-2730 USA