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Rank-two stable sheaves with odd determinant on Fano threefolds of genus nine

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Abstract

According to Mukai and Iliev, a smooth prime Fano threefold \(X\) of genus \(9\) is associated with a surface \(\mathbb{P }(\mathcal{V })\), ruled over a smooth plane quartic \(\varGamma \), and the derived category of \(\varGamma \) embeds into that of \(X\) by a theorem of Kuznetsov. We use this setup to study the moduli spaces of rank-\(2\) stable sheaves on \(X\) with odd determinant. For each \(c_2 \ge 7\), we prove that a component of their moduli space \(\mathsf{M}_X(2,1,c_2)\) is birational to a Brill–Noether locus of vector bundles with fixed rank and degree on \(\varGamma \), having enough sections when twisted by \(\mathcal{V }\). For \(c_2=7\), we prove that \(\mathsf{M}_X(2,1,7)\) is isomorphic to the blow-up of the Picard variety \(\text{ Pic}^{2}({\varGamma })\) along the curve parametrizing lines contained in \(X\).

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References

  1. Ancona, V., Ottaviani, G.: Stability of special instanton bundles on \({ P}^{2n+1}\). Trans. Am. Math. Soc. 341(2), 677–693 (1994)

    MathSciNet  MATH  Google Scholar 

  2. Arbarello, E., Cornalba, M., Griffiths, P.A., Harris, J.: Geometry of algebraic curves, vol. I. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267. Springer, New York (1985)

  3. Atiyah, M.F., Hitchin, N.J., Drinfel’d, V.G., Manin, Y.I.: Construction of instantons. Phys. Lett. A 65(3), 185–187 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  4. Atiyah, M.F., Ward, R.S.: Instantons and algebraic geometry. Commun. Math. Phys. 55(2), 117–124 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  5. Barth, W.: Some properties of stable rank-\(2\) vector bundles on \({ P}_{n}\). Math. Ann. 226(2), 125–150 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  6. Barth, W., Hulek, K.: Monads and moduli of vector bundles. Manuscripta Math. 25(4), 323–347 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bondal, A.I.: Representations of associative algebras and coherent sheaves. Izv. Akad. Nauk SSSR Ser. Mat. 53(1), 25–44 (1989)

    MathSciNet  Google Scholar 

  8. Bondal, A.I., Orlov, D.O.: Semiorthogonal decomposition for algebraic varieties. ArXiv:alg-geom/9506012 (1995)

  9. Brambilla, M.C., Faenzi, D.: Vector bundles on Fano threefolds of genus 7 and Brill-Noether loci. ArXiv:math.AG/0810.3138 (2008)

  10. Brambilla, M.C., Faenzi, D.: Moduli spaces of rank-2 ACM bundles on prime Fano threefolds. Mich. Math. J. 60(1), 113–148 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Căldăraru, A.: Derived categories of sheaves: a skimming. In: Snowbird Lectures in Algebraic Geometry. Contemp. Math., vol. 388, pp. 43–75. American Mathematical Society, Providence (2005)

  12. Casanellas, M., Drozd, E., Hartshorne, R.: Gorenstein liaison and ACM sheaves. J. Reine Angew. Math. 584, 149–171 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Druel, S.: Espace des modules des faisceaux de rang 2 semi-stables de classes de Chern \(c_1=0, c_2=2\) et \(c_3=0\) sur la cubique de \({ P}^4\). Int. Math. Res. Not. 19, 985–1004 (2000)

    Article  MathSciNet  Google Scholar 

  14. Faenzi, D.: Even and odd instanton bundles on Fano threefolds of Picard number 1. ArXiv:math.AG/1109.3858 (2011)

  15. Gelfand, S.I., Manin, Y.I.: Methods of Homological Algebra. Springer, Berlin (1996). Translated from the 1988 Russian original

  16. Gorodentsev, A.L.: Exceptional objects and mutations in derived categories. In: Helices and Vector Bundles. London Math. Soc. Lecture Note Ser., vol. 148, pp. 57–73. Cambridge University Press, Cambridge (1990)

  17. Gruson, L., Laytimi, F., Nagaraj, D.S.: On prime Fano threefolds of genus 9. Int. J. Math. 17(3), 253–261 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hartshorne, R.: Residues and duality. Lecture Notes of a Seminar on the Work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. Lecture Notes in Mathematics, No. 20. Springer, Berlin (1966)

  19. Hartshorne, R.: Stable reflexive sheaves. Math. Ann. 254(2), 121–176 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hoppe, H.J.: Generischer Spaltungstyp und zweite Chernklasse stabiler Vektorraumbündel vom Rang \(4\) auf \({ P}_{4}\). Math. Z. 187(3), 345–360 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  21. Huybrechts, D., Lehn, M.: The geometry of moduli spaces of sheaves. Aspects of Mathematics, E31. Friedr. Vieweg & Sohn, Braunschweig (1997)

  22. Iliev, A.: The \({\rm Sp}_3\)-Grassmannian and duality for prime Fano threefolds of genus 9. Manuscripta Math. 112(1), 29–53 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. Iliev, A., Markushevich, D.G.: The Abel-Jacobi map for a cubic threefold and periods of Fano threefolds of degree 14. Doc. Math. 5, 23–47 (2000)

    MathSciNet  MATH  Google Scholar 

  24. Iliev, A., Ranestad, K.: Geometry of the Lagrangian Grassmannian \({ LG}(3,6)\) with applications to Brill-Noether loci. Michigan Math. J. 53(2), 383–417 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Iskovskikh, V.A., Prokhorov, Y.G.: Fano varieties. In: Algebraic Geometry, V. Encyclopaedia Math. Sci., vol. 47, pp. 1–247. Springer, Berlin (1999)

  26. Jardim, M., Verbitsky, M.: Trihyperkahler reduction and instanton bundles on \(CP^3\). ArXiv:math.AG/1103.4431 (2011)

  27. Kuznetsov, A.: Hyperplane sections and derived categories. Izv. Ross. Akad. Nauk Ser. Mat. 70(3), 23–128 (2006)

    Article  MathSciNet  Google Scholar 

  28. Kuznetsov, A.: Instanton bundles on Fano threefolds. Cent. Eur. J. Math. 10(4), 1198–1231 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Markushevich, D.G., Tikhomirov, A.S.: The Abel-Jacobi map of a moduli component of vector bundles on the cubic threefold. J. Algebraic Geom. 10(1), 37–62 (2001)

    MathSciNet  MATH  Google Scholar 

  30. Maruyama, M.: Openness of a family of torsion free sheaves. J. Math. Kyoto Univ. 16(3), 627–637 (1976)

    MathSciNet  MATH  Google Scholar 

  31. Mukai, S.: Curves, \(K3\) surfaces and Fano \(3\)-folds of genus \(\le 10\). In: Algebraic Geometry and Commutative Algebra, vol. I, pp. 357–377. Kinokuniya, Tokyo (1988)

  32. Mukai, S.: Biregular classification of Fano \(3\)-folds and Fano manifolds of coindex \(3\). Proc. Natl. Acad. Sci. USA 86(9), 3000–3002 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  33. Mukai, S.: Non-abelian Brill-Noether theory and Fano 3-folds [translation of Sūgaku 49(1), 1–24 (1997); MR 99b:14012]. Sugaku Expositions 14(2), 125–153 (2001)

    Google Scholar 

  34. Tikhomirov, A.S.: Moduli of mathematical instanton vector bundles with odd \(c_2\) on projective space. ArXiv:math.AG/1101.3016 (2011)

  35. Weyman, J.: Cohomology of vector bundles and syzygies. Cambridge Tracts in Mathematics, vol. 149. Cambridge University Press, Cambridge (2003)

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Acknowledgments

We would like to thank the referee for many useful comments that helped us to correct some arguments and simplify some of the proofs.

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Authors and Affiliations

  1. Università Politecnica delle Marche, Via Brecce Bianche, 60131 , Ancona, Italia

    Maria Chiara Brambilla

  2. Université de Pau et des Pays de l’Adour, Av. de l’Université, BP 576, 64012, PAU Cedex, France

    Daniele Faenzi

Authors
  1. Maria Chiara Brambilla
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  2. Daniele Faenzi
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Correspondence to Daniele Faenzi.

Additional information

M.C. Brambilla was partially supported by INDAM and MIUR. D. Faenzi was partially supported by GRIFGA, ANR-09-JCJC-0097-0 INTERLOW and ANR GEOLMI.

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Brambilla, M.C., Faenzi, D. Rank-two stable sheaves with odd determinant on Fano threefolds of genus nine. Math. Z. 275, 185–210 (2013). https://doi.org/10.1007/s00209-012-1131-8

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