Skip to main content
×
×
Home

Cremona transformations and derived equivalences of K3 surfaces

  • Brendan Hassett (a1) and Kuan-Wen Lai (a2)
Abstract

We exhibit a Cremona transformation of $\mathbb{P}^{4}$ such that the base loci of the map and its inverse are birational to K3 surfaces. The two K3 surfaces are derived equivalent but not isomorphic to each other. As an application, we show that the difference of the two K3 surfaces annihilates the class of the affine line in the Grothendieck ring of varieties.

Copyright
References
Hide All
[Bea96] Beauville, A., Complex algebraic surfaces, London Mathematical Society Student Texts, vol. 34, second edition (Cambridge University Press, Cambridge, 1996); translated from the 1978 French original by R. Barlow, with assistance from N. I. Shepherd-Barron and M. Reid.
[Bor15] Borisov, L., The class of the affine line is a zero divisor in the Grothendieck ring, Preprint (2015), arXiv:1412.6194v3.
[CG72] Clemens, C. H. and Griffiths, P. A., The intermediate Jacobian of the cubic threefold , Ann. of Math. (2) 95 (1972), 281356.
[CK89] Crauder, B. and Katz, S., Cremona transformations with smooth irreducible fundamental locus , Amer. J. Math. 111 (1989), 289307.
[DGPS15] Decker, W., Greuel, G.-M., Pfister, G. and Schönemann, H., Singular 4-0-2 — A computer algebra system for polynomial computations, http://www.singular.uni-kl.de, 2015.
[Dol12] Dolgachev, I. V., Classical algebraic geometry: A modern view (Cambridge University Press, Cambridge, 2012).
[Ful98] Fulton, W., Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 2, second edition (Springer, Berlin, 1998).
[GS14] Galkin, S. and Shinder, E., The Fano variety of lines and rationality problem for a cubic hypersurface, Preprint (2014), arXiv:1405.5154v2.
[Has16] Hassett, B., Cubic fourfolds, K3 surfaces, and rationality questions , in Rationality problems in algebraic geometry, Lecture Notes in Mathematics, vol. 2172 (Springer, Cham, 2016).
[HLOY03] Hosono, S., Lian, B. H., Oguiso, K. and Yau, S.-T., Fourier–Mukai partners of a K3 surface of Picard number one , in Vector bundles and representation theory (Columbia, MO, 2002), Contemporary Mathematics, vol. 322 (American Mathematical Society, Providence, RI, 2003), 4355.
[IM04] Iliev, A. and Markushevich, D., Elliptic curves and rank-2 vector bundles on the prime Fano threefold of genus 7 , Adv. Geom. 4 (2004), 287318.
[IMOU16a] Ito, A., Miura, M., Okawa, S. and Ueda, K., The class of the affine line is a zero divisor in the Grothendieck ring: via $G_{2}$ -Grassmannians, Preprint (2016), arXiv:1606.04210v2.
[IMOU16b] Ito, A., Miura, M., Okawa, S. and Ueda, K., Derived equivalence and Grothendieck ring of varieties: via K3 surfaces of degree 12, Preprint (2016), arXiv:1612.08497v1.
[KS17] Kuznetsov, A. and Shinder, E., Grothendieck ring of varieties, D- and L-equivalence, and families of quadrics , Selecta Math. (N.S.) (2017), doi:10.1007/s00029-017-0344-4.
[Kuz06] Kuznetsov, A. G., Hyperplane sections and derived categories , Izv. Math. 70 (2006), 447547.
[Kuz16] Kuznetsov, A., Derived equivalence of Ito–Miura–Okawa–Ueda Calabi–Yau 3-folds, J. Math. Soc. Japan, to appear. Preprint (2016), arXiv:1611.08386v1.
[LL03] Larsen, M. and Lunts, V. A., Motivic measures and stable birational geometry , Mosc. Math. J. 3 (2003), 8595; 259.
[Mar11] Markman, E., A survey of Torelli and monodromy results for holomorphic-symplectic varieties , in Complex and differential geometry, Springer Proceedings in Mathematics, vol. 8 (Springer, Heidelberg, 2011), 257322.
[Mar16] Martin, N., The class of the affine line is a zero divisor in the Grothendieck ring: an improvement , C. R. Math. Acad. Sci. Paris 354 (2016), 936939.
[Muk87] Mukai, S., On the moduli space of bundles on K3 surfaces. I , in Vector bundles on algebraic varieties (Bombay, 1984), Tata Institute of Fundamental Research Studies in Mathematics, vol. 11 (Tata Institute of Fundamental Research, Bombay, 1987), 341413.
[Muk88] Mukai, S., Curves, K3 surfaces and Fano 3-folds of genus ⩽10 , in Algebraic geometry and commutative algebra, Vol. I (Kinokuniya, Tokyo, 1988), 357377.
[Muk99] Mukai, S., Duality of polarized K3 surfaces , in New trends in algebraic geometry (Warwick, 1996), London Mathematical Society Lecture Note Series, vol.  264 (Cambridge University Press, Cambridge, 1999), 311326.
[Ogu02] Oguiso, K., K3 surfaces via almost-primes , Math. Res. Lett. 9 (2002), 4763.
[Orl97] Orlov., D. O., Equivalences of derived categories and K3 surfaces , J. Math. Sci. (N.Y.) 84 (1997), 13611381.
[Som81] Sommese, A. J., Hyperplane sections , in Algebraic geometry (Chicago, Ill., 1980), Lecture Notes in Mathematics, vol. 862 (Springer, Berlin, 1981), 232271.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Compositio Mathematica
  • ISSN: 0010-437X
  • EISSN: 1570-5846
  • URL: /core/journals/compositio-mathematica
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords:

Metrics

Full text views

Total number of HTML views: 2
Total number of PDF views: 19 *
Loading metrics...

Abstract views

Total abstract views: 66 *
Loading metrics...

* Views captured on Cambridge Core between 28th May 2018 - 18th June 2018. This data will be updated every 24 hours.