publication . Preprint . 2014

Class of the affine line is a zero divisor in the Grothendieck ring

Borisov, Lev;
Open Access English
  • Published: 18 Dec 2014
Abstract
We show that the class of the affine line is a zero divisor in the Grothendieck ring of algebraic varieties over complex numbers. The argument is based on the Pfaffian-Grassmannian double mirror correspondence.
Subjects
arXiv: Mathematics::Algebraic GeometryMathematics::Commutative Algebra
free text keywords: Mathematics - Algebraic Geometry, 14A10
Related Organizations
Funded by
NSF| Derived equivalences inspired by mirror symmetry
Project
  • Funder: National Science Foundation (NSF)
  • Project Code: 1201466
  • Funding stream: Directorate for Mathematical & Physical Sciences | Division of Mathematical Sciences
Download from

[1] D. Abramovich, K. Karu, K. Matsuki, J. Wlodarsczyk, Torification and Factorization of Birational Maps. J. Amer. Math. Soc. 15 (2002), no. 3, 531-572.

[2] L. Borisov, A. Caˇldˇararu, The Pfaffian-Grassmannian derived equivalence. J. Algebraic Geom. 18 (2009), no. 2, 201-222.

[3] J. Denef, F. Loeser, On some rational generating series occurring in arithmetic geometry. Geometric Aspects of Dwork Theory, edited by A. Adolphson, F. Baldassarri, P. Berthelot, N. Katz and F. Loeser, volume 1, de Gruyter, 509-526 (2004).

[4] S. Galkin, E. Shinder, The Fano variety of lines and rationality problem for a cubic hypersurface. preprint arXiv:1405.5154.

[5] I. Karzhemanov, On the cut-and-paste property of algebraic varieties, preprint arXiv:1411.6084.

[6] J. Koll´ar, Y. Miyaoka, S. Mori, Rationally connected varieties. J. Algebraic Geom. 1 (1992), no. 3, 429-448.

[7] A. Kuznetsov, Lefschetz decompositions and categorical resolutions of singularities. Selecta Math. (N.S.) 13 (2008), no. 4, 661-696.

[8] M. Larsen, V. Lunts, Motivic measures and stable birational geometry. Moscow Mathematical Journal, Vol.3 (1), Jan-Mar. 2003, 85-95.

[9] M. Larsen, V. Lunts, Rationality of motivic zeta function and cut-and-paste problem, preprint arXiv:1410.7099.

[10] D. Litt, Symmetric powers do not stabilize. Proc. Amer. Math. Soc. 142 (2014), no. 12, 4079-4094.

[11] B. Poonen, The Grothendieck ring of varieties is not a domain. Math. Res. Lett. 9 (2002), no. 4, 493-497. [OpenAIRE]

[12] E. Rødland, The Pfaffian Calabi-Yau, its mirror, and their link to the Grassmannian G(2,7). Compositio Math. 122 (2000), no. 2, 135-149.

Any information missing or wrong?Report an Issue