Hostname: page-component-848d4c4894-wzw2p Total loading time: 0 Render date: 2024-05-13T04:54:50.970Z Has data issue: false hasContentIssue false
  • English
  • Français

On the non-divisorial base locus of big and nef line bundles on K3[2]-type varieties

Part of: Special varieties

Published online by Cambridge University Press:  20 February 2020

Ulrike Rieß
Ulrike Rieß*
Affiliation:
ETH Zürich, Institute of Theoretical Studies, Clausisusstrasse 47, 8092 Zürich, Switzerland (ulrike.riess@eth-its.ethz.ch)
Get access Rights & Permissions [Opens in a new window]

Abstract

We approach non-divisorial base loci of big and nef line bundles on irreducible symplectic varieties. While for K3 surfaces, only divisorial base loci can occur, nothing was known about the behaviour of non-divisorial base loci for more general irreducible symplectic varieties. We determine the base loci of all big and nef line bundles on the Hilbert scheme of two points on very general K3 surfaces of genus two and on their birational models. Remarkably, we find an ample line bundle with a non-trivial base locus in codimension two. We deduce that, generically in the moduli spaces of polarized K3[2]-type varieties, the polarization is base point free.

Keywords

Irreducible symplectic varietieshyperkhler manifoldsbase locusFujita's conjecture

MSC classification

Primary: 14M99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 1 , February 2021 , pp. 52 - 78
Check for updates
Copyright
Copyright © The Author(s) 2020. Published by The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Apostolov, Apostol. Moduli spaces of polarized irreducible symplectic manifolds are not necessarily connected. Ann. Inst. Fourier 64 (2014), 189–202.CrossRefGoogle Scholar
2Artin, Michael. Algebraization of formal moduli: II.Existence of modifications. Ann. Math. 91 (1970), 88–135.CrossRefGoogle Scholar
3Bayer, Arend and Macrì, Emanuele. MMP for moduli of sheaves on K3s via wall-crossing: Nef and movable cones, Lagrangian fibrations. Invent. Math. 198 (2014), 505–590.CrossRefGoogle Scholar
4Beauville, Arnaud. Variétés kähleriennes dont la première classe de Chern est nulle. J. Differ. Geom. 18 (1983), 755–782.CrossRefGoogle Scholar
5Beauville, Arnaud and Donagi, Ron. La variété des droites d'une hypersurface cubique de dimension 4. C. R. Acad. Sci., Paris, Sér. I 301 (1985), 703–706.Google Scholar
6Fantechi, Barbara, Göttsche, Lothar, Illusie, Luc, Kleiman, Steven L., Nitsure, Nitin and Vistoli, Angelo, Fundamental algebraic geometry: Grothendieck's FGA explained, volume 123 of Mathematical Surveys and Monographs (American Mathematical Society (AMS), 2005).CrossRefGoogle Scholar
7Gross, Mark, Huybrechts, Daniel and Joyce, Dominic, Calabi–Yau Manifolds and Related Geometries. Lectures at a Summer School in Nordfjordeid, Norway, June 2001. Universitext (Berlin: Springer, 2003).CrossRefGoogle Scholar
8Hartshorne, Robin Algebraic geometry. Graduate Texts in Mathematics (New York: Springer, 1977).CrossRefGoogle Scholar
9Hu, Yi and Yau, Shing-Tung. HyperKähler manifolds and birational transformations. Adv. Theor. Math. Phys. 6 (2002), 557–574.CrossRefGoogle Scholar
10Huybrechts, Daniel. Birational symplectic manifolds and their deformations. J. Differ. Geom. 45 (1997), 488–513.CrossRefGoogle Scholar
11Huybrechts, Daniel. Compact hyperKähler manifolds: Basic results. Invent. Math. 135 (1999), 63–113.CrossRefGoogle Scholar
12Huybrechts, Daniel. Erratum to: Compact hyperKähler manifolds: basic results. Invent. Math. 152 (2003), 209–212.CrossRefGoogle Scholar
13Huybrechts, Daniel. Lectures on K3 surfaces Camb. Stud. Adv. Math. 158 (Cambridge University Press, 2016).CrossRefGoogle Scholar
14Lazarsfeld, Robert, Positivity in algebraic geometry. II. Positivity for vector bundles, and multiplier ideals, volume 49 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge (Springer, 2004).CrossRefGoogle Scholar
15Lehn, Manfred. Chern classes of tautological sheaves on Hilbert schemes of points on surfaces. Invent. Math. 136 (1999), 157–207.CrossRefGoogle Scholar
16Matsushita, Daisuke, On isotropic divisors on irreducible symplectic manifolds. arXiv:1310.0896v2 (2013).Google Scholar
17Mayer, Alan L.. Families of K-3 surfaces. Nagoya Math. J. 48 (1972), 1–17.CrossRefGoogle Scholar
18Mongardi, Giovanni. A note on the Kähler and Mori cones of hyperKähler manifolds. Asian J. Math. 19 (2015), 583–592.CrossRefGoogle Scholar
19Mukai, Shigeru. Symplectic structure of the moduli space of sheaves on an abelian or K3 surface. Invent. Math. 77 (1984), 101–116.CrossRefGoogle Scholar
20Mumford, David, Abelian varieties. With appendices by C. P. Ramanujam and Yuri Manin. Corrected reprint of the 2nd ed. 1974. New Delhi: Hindustan Book Agency/distrib. by American Mathematical Society (AMS); Bombay: Tata Institute of Fundamental Research, corrected reprint of the 2nd ed. 1974 edition, 2008.Google Scholar
21O'Grady, Kieran G., Compact hyperKähler manifolds: An introduction. available online (2013).Google Scholar
22Rieß, Ulrike, Base divisors of big and nef line bundles on irreducible symplectic varieties. arXiv:1807.05192v1 (2018).Google Scholar
23Saint-Donat, Bernard. Projective models of K-3 surfaces. Am. J. Math. 96 (1974), 602–639.CrossRefGoogle Scholar
24Wierzba, Jan, Birational geometry of symplectic 4-folds. Unpublished preprint: http://www.mimuw.edu.pl/~jarekw/postscript/bir4fd.ps, 2002.Google Scholar