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A bound for the splitting of smooth Fano polytopes with many vertices
Stockholm University, Faculty of Science, Department of Mathematics.
Number of Authors: 2
2016 (English)In: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192, Vol. 43, no 1, 153-172 p.Article in journal (Refereed) Published
Abstract [en]

The classification of toric Fano manifolds with large Picard number corresponds to the classification of smooth Fano polytopes with large number of vertices. A smooth Fano polytope is a polytope that contains the origin in its interior and is such that the vertex set of each facet forms a lattice basis. Casagrande showed that any smooth d-dimensional Fano polytope has at most 3d vertices. Smooth Fano polytopes in dimension d with at least vertices are completely known. The main result of this paper deals with the case of vertices for k fixed and d large. It implies that there is only a finite number of isomorphism classes of toric Fano d-folds X (for arbitrary d) with Picard number such that X is not a product of a lower-dimensional toric Fano manifold and the projective plane blown up in three torus-invariant points. This verifies the qualitative part of a conjecture in a recent paper by the first author, Joswig, and Paffenholz.

Place, publisher, year, edition, pages
2016. Vol. 43, no 1, 153-172 p.
Keyword [en]
Toric Fano varieties, Smooth Fano polytopes, Lattice polytopes, Reflexive polytopes
National Category
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-126368DOI: 10.1007/s10801-015-0630-1ISI: 000367611700009OAI: oai:DiVA.org:su-126368DiVA: diva2:903204
Available from: 2016-02-15 Created: 2016-02-01 Last updated: 2016-02-15Bibliographically approved

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