Finiteness of the polyhedral $\mathbb {Q}$-codegree spectrum
HTML articles powered by AMS MathViewer
- by Andreas Paffenholz PDF
- Proc. Amer. Math. Soc. 143 (2015), 4863-4873 Request permission
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 $\alpha >0$.References
- Mauro C. Beltrametti and Andrew J. Sommese, Some effects of the spectral values on reductions, Classification of algebraic varieties (L’Aquila, 1992) Contemp. Math., vol. 162, Amer. Math. Soc., Providence, RI, 1994, pp. 31–48. MR 1272692, DOI 10.1090/conm/162/01526
- Mauro C. Beltrametti and Andrew J. Sommese, The adjunction theory of complex projective varieties, De Gruyter Expositions in Mathematics, vol. 16, Walter de Gruyter & Co., Berlin, 1995. MR 1318687, DOI 10.1515/9783110871746
- G. Di Cerbo. On Fujita’s log spectrum conjecture. preprint, October 2012, arxiv:1210.5324.
- David A. Cox, John B. Little, and Henry K. Schenck, Toric varieties, Graduate Studies in Mathematics, vol. 124, American Mathematical Society, Providence, RI, 2011. MR 2810322, DOI 10.1090/gsm/124
- Alicia Dickenstein, Sandra Di Rocco, and Ragni Piene, Classifying smooth lattice polytopes via toric fibrations, Adv. Math. 222 (2009), no. 1, 240–254. MR 2531373, DOI 10.1016/j.aim.2009.04.002
- Sandra Di Rocco, Christian Haase, Benjamin Nill, and Andreas Paffenholz, Polyhedral adjunction theory, Algebra Number Theory 7 (2013), no. 10, 2417–2446. MR 3194647, DOI 10.2140/ant.2013.7.2417
- Günter Ewald, On the classification of toric Fano varieties, Discrete Comput. Geom. 3 (1988), no. 1, 49–54. MR 918178, DOI 10.1007/BF02187895
- Takao Fujita, On Kodaira energy and adjoint reduction of polarized manifolds, Manuscripta Math. 76 (1992), no. 1, 59–84. MR 1171156, DOI 10.1007/BF02567747
- Takao Fujita, On Kodaira energy of polarized log varieties, J. Math. Soc. Japan 48 (1996), no. 1, 1–12. MR 1361544, DOI 10.2969/jmsj/04810001
- William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR 1234037, DOI 10.1515/9781400882526
- Douglas Hensley, Lattice vertex polytopes with interior lattice points, Pacific J. Math. 105 (1983), no. 1, 183–191. MR 688412
- Jeffrey C. Lagarias and Günter M. Ziegler, Bounds for lattice polytopes containing a fixed number of interior points in a sublattice, Canad. J. Math. 43 (1991), no. 5, 1022–1035. MR 1138580, DOI 10.4153/CJM-1991-058-4
- Oleg Pikhurko, Lattice points in lattice polytopes, Mathematika 48 (2001), no. 1-2, 15–24 (2003). MR 1996360, DOI 10.1112/S0025579300014339
- Andrew John Sommese, On the adjunction theoretic structure of projective varieties, Complex analysis and algebraic geometry (Göttingen, 1985) Lecture Notes in Math., vol. 1194, Springer, Berlin, 1986, pp. 175–213. MR 855885, DOI 10.1007/BFb0077004
Additional Information
- Andreas Paffenholz
- Affiliation: TU Darmstadt, Fachbereich Mathematik, Dolivostr. 15, 64293 Darmstadt, Germany
- MR Author ID: 745378
- Email: paffenholz@mathematik.tu-darmstadt.de
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 4863-4873
- MSC (2010): Primary 52B20; Secondary 14M25, 14C20
- DOI: https://doi.org/10.1090/proc/12620
- MathSciNet review: 3391044