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 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
Mathematical Subject Classification 2010
Primary: 14J70
Secondary: 14R99
References
Publication
Received: 11 July 2013
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