Skip to main content
SpringerLink
Log in

Scheme of lines on a family of 2-dimensional quadrics: geometry and derived category

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

Given a generic family \(Q\) of 2-dimensional quadrics over a smooth 3-dimensional base \(Y\) we consider the relative Fano scheme \(M\) of lines of it. The scheme \(M\) has a structure of a generically conic bundle \(M \rightarrow X\) over a double covering \(X \rightarrow Y\) ramified in the degeneration locus of \(Q \rightarrow Y\). The double covering \(X \rightarrow Y\) is singular in a finite number of points (corresponding to the points \(y \in Y\) such that the quadric \(Q_y\) degenerates to a union of two planes), the fibers of \(M\) over such points are unions of two planes intersecting in a point. The main result of the paper is a construction of a semiorthogonal decomposition for the derived category of coherent sheaves on \(M\). This decomposition has three components, the first is the derived category of a small resolution \(X^+\) of singularities of the double covering \(X \rightarrow Y\), the second is a twisted resolution of singularities of \(X\) (given by the sheaf of even parts of Clifford algebras on \(Y\)), and the third is generated by a completely orthogonal exceptional collection.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Artin, M.: Brauer-Severi varieties. In: van Oystaeyen, F.M.J., Verschoren, A.H.M.J. (eds.) Brauer Groups in Ring Theory and Algebraic Geometry, pp. 194–210. Springer, Berlin (1982)

  2. Artin, M., Mumford, D.: Some elementary examples of unirational varieties which are not rational. Proc. Lond. Math. Soc. 25(3), 75–95 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bernardara, M.: A semiorthogonal decomposition for Brauer-Severi schemes. Math. Nach. 282(10), 1406–1413 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bondal, A., Orlov, D.: Semiorthogonal decomposition for algebraic varieties. preprint math.AG/9506012

  5. Debarre, O.: Higher-Dimensional Algebraic Geometry, Universitext, vol. 4. Springer, New York (2001)

    Book  Google Scholar 

  6. Ingalls, C., Kuznetsov, A.: On nodal Enriques surfaces and quartic double solids, preprint math.AG/1012.3530

  7. Kawamata, Yu.: Small contractions of four-dimensional algebraic manifolds. Math. Ann. 284, 595–600 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  8. Kollár, J., Mori, S.: Birational Geometry of Algebraic Varieties, vol. 134. Cambridge University Press, Cambridge (2008)

    MATH  Google Scholar 

  9. Kuznetsov, A.: Base change for semiorthogonal decompositions. Compos. Math. 147(3), 852–876 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  10. Kuznetsov, A.: Derived categories of quadric fibrations and intersections of quadrics. Adv. Math. 218(5), 1340–1369 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. Milne, J.: Étale Cohomology, vol. 33. Princeton university press, Princeton (1980)

    MATH  Google Scholar 

  12. Nakayama, N.: The lower semicontinuity of the plurigenera of complex varieties. In: Algebraic Geometry, Sendai, 1985, Advanced Studies in Pure Mathematics, vol. 10, pp. 551–590. North-Holland, Amsterdam (1987)

  13. Tetsuya, A.: On extremal rays of the higher dimensional varieties. Invent. math. 81, 347–357 (1985)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

I would like to thank L.Katzarkov, D.Orlov, and Yu.Prokhorov for helpful discussions. I am also grateful to the referee for valuable comments.

Author information

Authors and Affiliations

  1. Department of Algebraic Geometry, Steklov Mathematical Institute, 8 Gubkin str., 119991,  Moscow, Russia

    Alexander Kuznetsov

  2. The Poncelet Laboratory, Independent University of Moscow, Moscow, Russia

    Alexander Kuznetsov

  3. Laboratory of Algebraic Geometry, SU-HSE, Moscow, Russia

    Alexander Kuznetsov

Authors
  1. Alexander Kuznetsov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Kuznetsov.

Additional information

I was partially supported by RFFI grants 11-01-00393, 11-01-00568, 12-01-33024, NSh-5139.2012.1, the grant of the Simons foundation, and by AG Laboratory SU-HSE, RF government grant, ag.11.G34.31.0023.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kuznetsov, A. Scheme of lines on a family of 2-dimensional quadrics: geometry and derived category. Math. Z. 276, 655–672 (2014). https://doi.org/10.1007/s00209-013-1217-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-013-1217-y

Keywords

Search

Navigation