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Finiteness of the polyhedral $ \mathbb{Q}$-codegree spectrum


Author: Andreas Paffenholz
Journal: Proc. Amer. Math. Soc. 143 (2015), 4863-4873
MSC (2010): Primary 52B20; Secondary 14M25, 14C20
Published electronically: June 3, 2015
MathSciNet review: 3391044
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Abstract: In this note we prove Fujita's spectrum conjecture for polarized varieties in the case of $ \mathbb{Q}$-Gorenstein projective toric varieties of index $ r$. The theorem follows from a combinatorial result using the connection between lattice polytopes and polarized projective toric varieties. By this correspondence the spectral value of the polarized toric variety equals the $ \mathbb{Q}$-codegree of the polytope. Now the main theorem of the paper shows that the spectrum of the $ \mathbb{Q}$-codegree is finite above any positive threshold in the class of lattice polytopes with $ \alpha $-canonical normal fan for any fixed $ \alpha >0$.


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Additional Information

Andreas Paffenholz
Affiliation: TU Darmstadt, Fachbereich Mathematik, Dolivostr. 15, 64293 Darmstadt, Germany
Email: paffenholz@mathematik.tu-darmstadt.de

DOI: https://doi.org/10.1090/proc/12620
Keywords: Fujita's spectrum conjecture, toric varieties, spectral value, Kodaira energy, lattice polytopes, $\mathbb{Q}$-codegree
Received by editor(s): April 20, 2014
Published electronically: June 3, 2015
Additional Notes: The author has been supported by the Priority Program 1489 of the German Research Council (DFG)
Communicated by: Lev Borisov
Article copyright: © Copyright 2015 American Mathematical Society