Skip to main content Accessibility help
×
×
Home

Double Solids, Categories and Non-Rationality

  • Atanas Iliev (a1), Ludmil Katzarkov (a2) and Victor Przyjalkowski (a3)

Abstract

This paper suggests a new approach to questions of rationality of 3-folds based on category theory. Following work by Ballard et al., we enhance constructions of Kuznetsov by introducing Noether–Lefschetz spectra: an interplay between Orlov spectra and Hochschild homology. The main goal of this paper is to suggest a series of interesting examples where the above techniques might apply. We start by constructing a sextic double solid X with 35 nodes and torsion in H3(X, ℤ). This is a novelty: after the classical example of Artin and Mumford, this is the second example of a Fano 3-fold with a torsion in the third integer homology group. In particular, X is non-rational. We consider other examples as well: V10 with 10 singular points, and the double covering of a quadric ramified in an octic with 20 nodal singular points. After analysing the geometry of their Landau–Ginzburg models, we suggest a general non-rationality picture based on homological mirror symmetry and category theory.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Double Solids, Categories and Non-Rationality
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Double Solids, Categories and Non-Rationality
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Double Solids, Categories and Non-Rationality
      Available formats
      ×

Copyright

References

Hide All
1.Artin, M. and Mumford, D., Some elementary examples of unirational varieties which are not rational, Proc. Lond. Math. Soc. 25(3) (1972), 7595.
2.Aspinwall, P., Morrison, D. and Gross, M., Stable singularities in string theory, Commun. Math. Phys. 178(1) (1996), 115134.
3.Ballard, M., Favero, D. and Katzarkov, L., A category of kernels for graded matrix factorizations and its implications for Hodge theory, Publ. IHES (in press).
4.Ballard, M., Favero, D. and Katzarkov, L., Orlov spectra: bounds and gaps, Invent. Math. 189(2) (2012), 359430.
5.Beauville, A., Variétés rationnelles et unirationnelles, Lect. Notes Math. 997 (1983), 1633.
6.Bott, R. and Tu, L., Differential forms in algebraic topology, Graduate Texts in Mathematics, Volume 82 (Springer, 1982).
7.Cheltsov, I. and Park, J., Sextic double solids, in Cohomological and geometric approaches to rationality problems: new perspectives, Progress in Mathematics, Volume 282, pp. 75132 (Birkhäuser, 2010).
8.Cheltsov, I., Katzarkov, L. and Przyjalkowski, V., Birational geometry via moduli spaces, in Birational geometry, rational curves, and arithmetic (Springer, 2013).
9.Cossec, F., Reye congruences, Trans. Am. Math. Soc. 280(2) (1983), 737751.
10.Endrass, S., On the divisor class group of double solids, Manuscr. Math. 99(3) (1999), 341358.
11.Favero, D., Iliev, A. and Katzarkov, L., On the Griffiths groups of Fano manifolds of Calabi–Yau Hodge type, Pure Appl. Math. Quart. (in press).
12.Garavuso, R., Katzarkov, L., Kreuzer, M. and Noll, A., Super Landau–Ginzburg mirrors and algebraic cycles, J. High. Energy. Phys. 2011(3) (2011), 17.
13.Gross, M., Katzarkov, L. and Rudat, H., Towards mirror symmetry for varieties of general type, Duke Math. J. (in press).
14.Grothendieck, A., Le groupe de Brauer I, II, III, in Dix exposés sur la cohomologie des schemas, Advanced Studies in Pure Mathematics, Volume 3, pp. 46188 (North-Holland, Amsterdam, 1968).
15.Harris, J. and Tu, L., On symmetric and skew-symmetric determinantal varieties, Topology 23 (1984), 7184.
16.Hirzebruch, F., Topological methods in algebraic geometry, Die Grundlehren der Mathematischen Wissenschaften, Volume 131 (Springer, 1978).
17.Iliev, A., Katzarkov, L. and Scheidegger, E., 4-dimensional cubics, Noether–Lefschetz loci and gaps (in preparation).
18.Ilten, N., Lewis, J. and Przyjalkowski, V., Toric degenerations of Fano threefolds giving weak Landau–Ginzburg models, J. Alg. 374 (2013), 104121.
19.Ingalls, C. and Kuznetsov, A., On nodal Enriques surfaces and quartic double solids, Math. Ann. (in press).
20.Iskovskikh, V. A., Birational automorphisms of three-dimensional algebraic varieties, J. Sov. Math. 13 (1980), 815868.
21.Iskovskikh, V. A. and Prokhorov, Yu., Algebraic geometries V: Fano varieties, Encyclopaedia of Mathematical Sciences, Volume 47 (Springer, 1999).
22.Jozefiak, T., Lascoux, A. and Pragacz, P., Classes of determinantal varieties associated with symmetric and skew-symmetric matrices, Math. USSR Izv. 18 (1982), 575586.
23.Katzarkov, L., Homological mirror symmetry and algebraic cycles, in Riemannian topology and geometric structures on manifolds (ed. Galicki, K. and Simanca, S. R.), Progress in Mathematics, Volume 271 (Birkhäuser, 2009).
24.Katzarkov, L. and Kerr, G., Orlov spectra as a filtered cohomology theory, Adv. Math. 243 (2013), 232261.
25.Katzarkov, L. and Przyjalkowski, V., Landau–Ginzburg models: old and new, in Proc. 18th Gokova Geometry–Topology Conf. (Gokova, Turkey), pp. 97124 (International Press, Somerville, MA, 2011).
26.Katzarkov, L., Kontsevich, M. and Pantev, T., Hodge theoretic aspects of mirror symmetry, in From Hodge theory to integrability and TQFT: tt* -geometry (ed. Donagi, R. and Wendlang, K.), Proceedings of Symposia in Pure Mathematics, Volume 78, pp. 87174 (American Mathematical Society, Providence, RI, 2008).
27.Katzarkov, L., Nemethi, A. and Stepanov, D., Four-dimensional cubics spectra and monodromy (in preparation).
28.Kuznetsov, A., Derived categories of Fano threefolds, Proc. Steklov Inst. Math. 264(1) (2009), 110122.
29.Oliva, C., Algebraic cycles and Hodge theory on generalized Reye congruences, Compositio Math. 92(1) (1994), 122.
30.Orlov, D., Remarks on generators and dimensions of triangulated categories, Moscow Math. J. 9(1) (2009), 143149.
31.Przyjalkowski, V., Weak Landau–Ginzburg models for smooth Fano threefolds, Izv. Math. 77(4) (2013), 135160.
32.Rouquier, R., Dimensions of triangulated categories, J. K-Theory 1(2) (2008), 193256.
33.Zagorskii, A., Three-dimensional conical fibrations, Math. Notes 21 (1977), 420427.
Recommend this journal

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

Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
  • URL: /core/journals/proceedings-of-the-edinburgh-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed