Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-23T18:23:36.133Z Has data issue: false hasContentIssue false
  • English
  • Français

On Fano Varieties with Torus Action of Complexity 1

Published online by Cambridge University Press:  16 April 2014

Elaine Herppich
Elaine Herppich*
Affiliation:
Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany, (xlink:href="[email protected]">[email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this work we provide effective bounds and classification results for rational ℚ-factorial Fano varieties with a complexity-one torus action and Picard number 1 depending on the two invariants dimension and Picard index. This complements earlier work by Hausen et al., where the case of a free divisor class group of rank 1 was treated.

Keywords

Fano varietiesgroup actions on varietiesfactorially graded ringsPrimary 14J4514L3013A0213F15
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 57 , Issue 3 , October 2014 , pp. 737 - 753
Copyright
Copyright © Edinburgh Mathematical Society 2014 

References

1.Arzhantsev, I., Derenthal, U., Hausen, J. and Laface, A., Cox rings, eprint (arXiv: 1003.4229, 2010; see also extended version on the authors' webpages.)Google Scholar
2.Batyrev, V. V., Toric Fano threefolds, Izv. Akad. Nauk SSSR Ser. Mat. 45(4) (1981), 704–717.Google Scholar
3.Ewald, G., Combinatorial convexity and algebraic geometry, Graduate Texts in Mathematics, Volume 168 (Springer, 1996).CrossRefGoogle Scholar
4.Hausen, J., Cox rings and combinatorics, II, Mosc. Math. J. 8(4) (2008), 711–757.Google Scholar
5.Hausen, J., Three lectures on Cox rings, in Torsors, étale homotopy and applications to rational points, London Mathematical Society Lecture Note Series, Volume 405, pp. 3–60 (Cambridge University Press, 2013).CrossRefGoogle Scholar
6.Hausen, J. and Herppich, E., Factorially graded rings of complexity one, in Torsors, etale homotopy and applications to rational points, London Mathematical Society Lecture Note Series, Volume 405, pp. 414–428 (Cambridge University Press, 2013).CrossRefGoogle Scholar
7.Hausen, J. and Süss, H., The Cox ring of an algebraic variety with torus action, Adv. Math. 225 (2010), 977–1012.Google Scholar
8.Hausen, J., Herppich, E. and Süss, H., Multigraded factorial rings and Fano varieties with torus action, Documenta Math. 16 (2011), 71–109.CrossRefGoogle Scholar
9.Iano-Fletcher, A. R., Working with weighted complete intersections, in Explicit birational geometry of 3-folds, London Mathematical Society Lecture Note Series, Volume 281, pp. 101–173 (Cambridge University Press, 2000).CrossRefGoogle Scholar
10.Iskovskih, V. A., Fano threefolds, II, Izv. Akad. Nauk SSSR Ser. Mat. 42(3) (1978), 506–549.Google Scholar
11.Kasprzyk, A., Bounds on fake weighted projective spaces, Kodai Math. J. 32(2) (2009), 197–208.Google Scholar
12.Kasprzyk, A., Canonical toric Fano threefolds, Can. J. Math. 62(6) (2010), 1293–1309.Google Scholar
13.Kasprzyk, A., Kreuzer, M. and Nill, B., On the combinatorial classification of toric log del Pezzo surfaces, LMS J. Comput. Math. 13 (2010), 33–46.Google Scholar
14.Mori, S. and Mukai, S., Classification of Fano 3-folds with B2 > 2, Manuscr. Math. 36(2) (1981), 147–162.Google Scholar
15.Sumihiro, H., Equivariant completion, J. Math. Kyoto Univ. 14 (1974), 1–28.Google Scholar
16.Süss, H., Canonical divisors on T-varieties, eprint (arXiv:0811.0626v1, 2008).Google Scholar