Deformations of rational $T$-varieties
Authors:
Nathan Owen Ilten and Robert Vollmert
Journal:
J. Algebraic Geom. 21 (2012), 531-562
DOI:
https://doi.org/10.1090/S1056-3911-2011-00585-7
Published electronically:
September 12, 2011
MathSciNet review:
2914803
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Abstract |
References |
Additional Information
Abstract: We show how to construct certain homogeneous deformations for rational normal varieties with codimension one torus action. This can then be used to construct homogeneous deformations of any toric variety in arbitrary degree. For locally trivial deformations coming from this construction, we calculate the image of the Kodaira-Spencer map. We then show that for a smooth complete toric variety, our homogeneous deformations span the space of first-order deformations.
References
- Klaus Altmann and Jürgen Hausen, Polyhedral divisors and algebraic torus actions, Math. Ann. 334 (2006), no. 3, 557–607. MR 2207875, DOI https://doi.org/10.1007/s00208-005-0705-8
- Klaus Altmann, Jürgen Hausen, and Hendrik Süss, Gluing affine torus actions via divisorial fans, Transform. Groups 13 (2008), no. 2, 215–242. MR 2426131, DOI https://doi.org/10.1007/s00031-008-9011-3
- Klaus Altmann, Minkowski sums and homogeneous deformations of toric varieties, Tohoku Math. J. (2) 47 (1995), no. 2, 151–184. MR 1329519, DOI https://doi.org/10.2748/tmj/1178225590
- Klaus Altmann, The versal deformation of an isolated toric Gorenstein singularity, Invent. Math. 128 (1997), no. 3, 443–479. MR 1452429, DOI https://doi.org/10.1007/s002220050148
- Klaus Altmann, One parameter families containing three-dimensional toric-Gorenstein singularities, Explicit birational geometry of 3-folds, London Math. Soc. Lecture Note Ser., vol. 281, Cambridge Univ. Press, Cambridge, 2000, pp. 21–50. MR 1798979
- David Cox, John Little, and Hal Schenck, Toric varieties, Available online at http://www.cs.amherst.edu/~dac/toric.html, 2010.
- Nathan Owen Ilten, One-parameter toric deformations of cyclic quotient singularities, J. Pure Appl. Algebra 213 (2009), no. 6, 1086–1096. MR 2498799, DOI https://doi.org/10.1016/j.jpaa.2008.11.010
- Nathan Ilten, Deformations of rational varieties with codimension-one torus action, Ph.D. thesis, Freie Universität, Berlin, 2010, urn:nbn:de:kobv:188-fudissthesis000000018440-0.
- Nathan Ilten, Deformations of smooth toric surfaces, Manuscripta Math. Online first (2010), http://dx.doi.org/10.1007/s00229-010-0386-9.
- Nathan Ilten and Hendrik Süß, Polarized complexity-one T-varieties, arXiv: 0910.5919v1 [math.AG], 2009.
- Alvaro Liendo and Hendrik Süß, Normal singularities with torus actions, arXiv:10052462v2 [math.AG], 2010.
- Anvar R. Mavlyutov, Deformations of Calabi-Yau hypersurfaces arising from deformations of toric varieties, Invent. Math. 157 (2004), no. 3, 621–633. MR 2092771, DOI https://doi.org/10.1007/s00222-004-0362-7
- Anvar R. Mavlyutov, Embedding of Calabi-Yau deformations into toric varieties, Math. Ann. 333 (2005), no. 1, 45–65. MR 2169828, DOI https://doi.org/10.1007/s00208-005-0664-0
- Anvar Mavlyutov, Deformations of toric varieties via Minkowski sum decompositions of polyhedral complexes, arXiv:0902.0967v2 [math.AG], 2009.
- Edoardo Sernesi, Deformations of algebraic schemes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 334, Springer-Verlag, Berlin, 2006. MR 2247603
- Hendrik Süß, Canonical divisors on T-varieties, arXiv:0811.0626v1 [math.AG], 2008.
- Robert Vollmert, Deformations of T-varieties, Ph.D. thesis, Freie Universität, Berlin, 2011. In preparation.
References
- Klaus Altmann and Jürgen Hausen, Polyhedral divisors and algebraic torus actions, Math. Ann. 334 (2006), no. 3, 557–607. MR 2207875 (2006m:14062)
- Klaus Altmann, Jürgen Hausen, and Hendrik Süss, Gluing affine torus actions via divisorial fans, Transform. Groups 13 (2008), no. 2, 215–242. MR 2426131
- Klaus Altmann, Minkowski sums and homogeneous deformations of toric varieties, Tohoku Math. J. (2) 47 (1995), no. 2, 151–184. MR 1329519 (96f:14063)
- ---, The versal deformation of an isolated toric Gorenstein singularity, Invent. Math. 128 (1997), no. 3, 443–479. MR 1452429 (98g:14006)
- ---, One parameter families containing three-dimensional toric-Gorenstein singularities, Explicit birational geometry of 3-folds, London Math. Soc. Lecture Note Ser., vol. 281, Cambridge Univ. Press, Cambridge, 2000, pp. 21–50. MR 1798979 (2001j:14006)
- David Cox, John Little, and Hal Schenck, Toric varieties, Available online at http://www.cs.amherst.edu/~dac/toric.html, 2010.
- Nathan Ilten, One-parameter toric deformations of cyclic quotient singularities, J. Pure Appl. Algebra 213 (2009), no. 6, 1086–1096. MR 2498799 (2010g:14002)
- Nathan Ilten, Deformations of rational varieties with codimension-one torus action, Ph.D. thesis, Freie Universität, Berlin, 2010, urn:nbn:de:kobv:188-fudissthesis000000018440-0.
- Nathan Ilten, Deformations of smooth toric surfaces, Manuscripta Math. Online first (2010), http://dx.doi.org/10.1007/s00229-010-0386-9.
- Nathan Ilten and Hendrik Süß, Polarized complexity-one T-varieties, arXiv: 0910.5919v1 [math.AG], 2009.
- Alvaro Liendo and Hendrik Süß, Normal singularities with torus actions, arXiv:10052462v2 [math.AG], 2010.
- Anvar Mavlyutov, Deformations of Calabi-Yau hypersurfaces arising from deformations of toric varieties, Invent. Math. 157 (2004), no. 3, 621–633. MR 2092771 (2006e:14057)
- ---, Embedding of Calabi-Yau deformations into toric varieties, Math. Ann. 333 (2005), no. 1, 45–65. MR 2169828 (2007b:14112)
- Anvar Mavlyutov, Deformations of toric varieties via Minkowski sum decompositions of polyhedral complexes, arXiv:0902.0967v2 [math.AG], 2009.
- Edoardo Sernesi, Deformations of algebraic schemes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 334, Springer-Verlag, Berlin, 2006. MR 2247603 (2008e:14011)
- Hendrik Süß, Canonical divisors on T-varieties, arXiv:0811.0626v1 [math.AG], 2008.
- Robert Vollmert, Deformations of T-varieties, Ph.D. thesis, Freie Universität, Berlin, 2011. In preparation.
Additional Information
Nathan Owen Ilten
Affiliation:
Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
Address at time of publication:
Department of Mathematics, University of California, Berkeley, 970 Evans Hall #3840, Berkeley, California 94720-3840
Email:
nilten@cs.uchicago.edu
Robert Vollmert
Affiliation:
Mathematisches Institut, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany
Email:
vollmert@math.fu-berlin.de
Received by editor(s):
September 21, 2009
Received by editor(s) in revised form:
December 3, 2010, and January 3, 2011
Published electronically:
September 12, 2011
