The Cayley Trick, lifting subdivisions and the Bohne-Dress theorem on zonotopal tilings
Please always quote using this URN: urn:nbn:de:0297-zib-3874
- In 1994, Sturmfels gave a polyhedral version of the Cayley Trick of elimination theory: he established an order-preserving bijection between the posets of \emph{coherent} mixed subdivisions of a Minkowski sum $\mathcal{A}_1+\cdots+\mathcal{A}_r$ of point configurations and of \emph{coherent} polyhedral subdivisions of the associated Cayley embedding $\mathcal{C}(\mathcal{A}_1,\dots,\mathcal{A}_r)$. In this paper we extend this correspondence in a natural way to cover also \emph{non-coherent} subdivisions. As an application, we show that the Cayley Trick combined with results of Santos on subdivisions of Lawrence polytopes provides a new independent proof of the Bohne-Dress Theorem on zonotopal tilings. This application uses a combinatorial characterization of lifting subdivisions, also originally proved by Santos.
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| Author: | Birkett Huber, Jörg Rambau, Francisco Santos |
|---|---|
| Document Type: | ZIB-Report |
| Tag: | Bohne-Dress Theorem; Cayley Trick; Minkowski sum; Polyhedral subdivision; fiber polytope; lifting subdivision; mixed subdivision |
| MSC-Classification: | 14-XX ALGEBRAIC GEOMETRY / 14Mxx Special varieties / 14M25 Toric varieties, Newton polyhedra [See also 52B20] |
| 52-XX CONVEX AND DISCRETE GEOMETRY / 52Bxx Polytopes and polyhedra / 52B11 n-dimensional polytopes | |
| 52-XX CONVEX AND DISCRETE GEOMETRY / 52Bxx Polytopes and polyhedra / 52B20 Lattice polytopes (including relations with commutative algebra and algebraic geometry) [See also 06A11, 13F20, 13Hxx] | |
| Date of first Publication: | 1999/01/28 |
| Series (Serial Number): | ZIB-Report (SC-98-44) |
| ZIB-Reportnumber: | SC-98-44 |
| Published in: | Appeared in: "The Cayley Trick, lifting subdivisions and the Bohne-Dress theorem on zonotopal tilings", Journal of the European Mathematical Society, 2 (2000) 179-198 |
