### 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_{Δ} of its Newton polytope Δ, 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.

Original language | English (US) |
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Pages (from-to) | 1343-1395 |

Number of pages | 53 |

Journal | Geometry and Topology |

Volume | 18 |

Issue number | 3 |

DOIs | |

State | Published - Jul 7 2014 |

### Fingerprint

### Keywords

- Affine
- Homotopy equivalence
- Hypersurface
- Kato-Nakayama space
- Log geometry
- Newton polytope
- Retraction
- Skeleton
- Toric degeneration
- Triangulation

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Geometry and Topology*,

*18*(3), 1343-1395. https://doi.org/10.2140/gt.2014.18.1343

}

*Geometry and Topology*, vol. 18, no. 3, pp. 1343-1395. https://doi.org/10.2140/gt.2014.18.1343

**Skeleta of affine hypersurfaces.** / Ruddat, Helge; Sibilla, Nicolò; Treumann, David; Zaslow, Eric.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Skeleta of affine hypersurfaces

AU - Ruddat, Helge

AU - Sibilla, Nicolò

AU - Treumann, David

AU - Zaslow, Eric

PY - 2014/7/7

Y1 - 2014/7/7

N2 - 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 TΔ of its Newton polytope Δ, 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.

AB - 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 TΔ of its Newton polytope Δ, 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.

KW - Affine

KW - Homotopy equivalence

KW - Hypersurface

KW - Kato-Nakayama space

KW - Log geometry

KW - Newton polytope

KW - Retraction

KW - Skeleton

KW - Toric degeneration

KW - Triangulation

UR - http://www.scopus.com/inward/record.url?scp=84905053121&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84905053121&partnerID=8YFLogxK

U2 - 10.2140/gt.2014.18.1343

DO - 10.2140/gt.2014.18.1343

M3 - Article

VL - 18

SP - 1343

EP - 1395

JO - Geometry and Topology

JF - Geometry and Topology

SN - 1465-3060

IS - 3

ER -