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Revista Matemática Complutense

, Volume 32, Issue 2, pp 475–529 | Cite as

The blow-up of \(\mathbb {P}^4\) at 8 points and its Fano model, via vector bundles on a del Pezzo surface

  • Cinzia CasagrandeEmail author
  • Giulio Codogni
  • Andrea Fanelli
Article
  • 68 Downloads

Abstract

Building on the work of Mukai, we explore the birational geometry of the moduli spaces \(M_{S,L}\) of semistable rank two torsion-free sheaves, with \(c_1=-K_S\) and \(c_2=2\), on a polarized degree one del Pezzo surface (SL); this is related to the birational geometry of the blow-up X of \(\mathbb {P}^4\) in 8 points. Our analysis is explicit and is obtained by looking at the variation of stability conditions. Then we provide a careful investigation of the blow-up X and of the moduli space \(Y=M_{S,-K_S}\), which is a remarkable family of smooth Fano fourfolds. In particular we describe the relevant cones of divisors of Y, the group of automorphisms, and the base loci of the anticanonical and bianticanonical linear systems.

Keywords

Moduli of vector bundles Del Pezzo surfaces, Birational geometry Fano varieties 

Mathematics Subject Classification

14J60 14J35 14J45 14E30 

Notes

Acknowledgements

We would like to thank Paolo Cascini, Ana-Maria Castravet, Daniele Faenzi, Emanuele Macrì, John Ottem, Zsolt Patakfalvi, and Filippo Viviani for interesting discussions related to this work, and the referees for useful comments. The first-named author has been partially supported by the PRIN 2015 “Geometria delle Varietà Algebriche”. The second-named author has been supported by the FIRB 2012 “Moduli spaces and their applications”. The third-named author has been supported by the SNF grant “Algebraic subgroups of the Cremona groups” and the DFG grant “Gromov-Witten Theorie, Geometrie und Darstellungen” (PE 2165/1-2). The second-named and third-named authors are grateful to the University of Torino for the warm hospitality provided during part of the preparation of this work.

References

  1. 1.
    Araujo, C., Casagrande, C.: On the Fano variety of linear spaces contained in two odd-dimensional quadrics. Geom. Topol. 21, 3009–3045 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Araujo, C., Massarenti, A.: Explicit log Fano structures on blow-ups of projective spaces. Proc. Lond. Math. Soc. 113, 445–473 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Batyrev, V.V., Popov, O.N.: The Cox ring of a del Pezzo surface, arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002). Progr. Math. 226, 85–103 (2004)CrossRefGoogle Scholar
  4. 4.
    Bauer, T., Pokora, P., Schmitz, D.: On the boundedness of the denominators in the Zariski decomposition on surfaces. J. Reine Angew. Math. 733, 251–259 (2017)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bayer, A., Macrì, E.: MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones. Lagrangian fibrations. Invent. Math. 198, 505–590 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Beltrametti, M.C., Sommese, A.J.: The Adjunction Theory of Complex Projective Varieties, de Gruyter Expositions in Mathematics, vol. 16. Walter de Gruyter & Co., Berlin (1995)CrossRefGoogle Scholar
  7. 7.
    Bertram, A., Martinez, C., Wang, J.: The birational geometry of moduli spaces of sheaves on the projective plane. Geom. Dedicata 173, 37–64 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Birkar, C., Cascini, P., Hacon, C.D., McKernan, J.: Existence of minimal models for varieties of log general type. J. Am. Math. Soc. 23, 405–468 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Blanc, J., Lamy, S.: Weak Fano threefolds obtained by blowing-up a space curve and construction of Sarkisov links. Proc. Lond. Math. Soc. 105, 1047–1075 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Casagrande, C.: Fano 4-folds, flips, and blow-ups of points. J. Algebra 483, 362–414 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Castravet, A.-M., Tevelev, J.: Hilbert’s 14th problem and Cox rings. Compos. Math. 142, 1479–1498 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Coble, A.B.: The ten nodes of the rational sextic and of the Cayley symmetroid. Am. J. Math. 41, 243–265 (1919)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Codogni, G., Fanelli, A., Svaldi, R., Tasin, L.: Fano varieties in Mori fibre spaces. Int. Math. Res. Not. 2016, 2026–2067 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Costa, L., Miró-Roig, R.M.: On the rationality of moduli spaces of vector bundles on Fano surfaces. J. Pure Appl. Algebra 137, 199–220 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dolgachev, I.V.: Weyl groups and Cremona transformations, singularities. Proc. Symp. Part 1 (Arcata, Calif., 1981, Pure Math.) 40, 283–294 (1983)Google Scholar
  16. 16.
    Dolgachev, I.V.: On certain families of elliptic curves in projective space. Ann. Math. Pura Appl. 183, 317–331 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Dolgachev, I.V.: Classical Algebraic Geometry–A Modern View. Cambridge University Press, Cambridge (2012)CrossRefzbMATHGoogle Scholar
  18. 18.
    Dolgachev, I.V., Ortland, D.: Point sets in projective spaces and theta functions, Astérisque, vol. 165. Société Mathématique de France, Marseille (1988)zbMATHGoogle Scholar
  19. 19.
    Du Val, P.: Crystallography and Cremona Transformations, The Geometric Vein—The Coxeter Festschrift, pp. 191–201. Springer, Berlin (1981)Google Scholar
  20. 20.
    Dumitrescu, O., Postinghel, E.: Positivity of divisors on blown-up projective spaces, I. preprint (2015). arXiv:1506.04726
  21. 21.
    Eisenbud, D., Popescu, S.: The projective geometry of the Gale transform. J. Algebra 230, 127–173 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ellingsrud, G., Göttsche, L.: Variation of moduli space and Donaldson invariants under change of polarization. J. Reine Angew. Math. 467, 1–49 (1995)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Friedman, R.: Algebraic Surfaces and Holomorphic Vector Bundles. Universitext. Springer, Berlin (1998)CrossRefzbMATHGoogle Scholar
  24. 24.
    Friedman, R., Qin, Z.: Flips of moduli spaces and transition formulas for Donaldson polynomial invariants of rational surfaces. Commun. Anal. Geom. 3, 11–83 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Gómez, T.L.: Irreducibility of the moduli space of vector bundles on surfaces and Brill-Noether theory on singular curves. preprint (2000). arXiv:alg-geom/9710029v2 (Revised PhD thesis (Princeton, 1997))
  26. 26.
    Harris, J.: Algebraic Geometry-A First Course, Graduate Texts in Mathematics, vol. 133. Springer, Berlin (1992)Google Scholar
  27. 27.
    Hu, Y., Keel, S.: Mori dream spaces and GIT. Michigan Math. J. 48, 331–348 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Huybrechts, D., Lehn, M.: The Geometry of the Moduli Space of Sheaves, 2nd edn. Cambridge University Press, Cambridge (2010)CrossRefzbMATHGoogle Scholar
  29. 29.
    Kollár, J., Mori, S.: Birational Geometry of Algebraic Varieties, Cambridge Tracts in Mathematics, vol. 134. Cambridge University Press, Cambridge (1998)CrossRefGoogle Scholar
  30. 30.
    Lazarsfeld, R.: Positivity in Algebraic Geometry II. Springer, Berlin (2004)CrossRefzbMATHGoogle Scholar
  31. 31.
    Le Potier, J.: (1992) Fibré déterminant et courbes de saut sur les surfaces algébriques, Complex projective geometry (Trieste, 1989, Bergen, 1989) London Mathematical Society Lecture Note Series, vol. 179, pp. 213–240. Cambridge University Press, Cambridge (1989)Google Scholar
  32. 32.
    Lesieutre, J., Park, J.: Log Fano structures and Cox rings of blow-ups of products of projective spaces. Proc. Am. Math. Soc. 145, 4201–4209 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Li, J.: Algebraic geometric interpretation of Donaldson’s polynomial invariants. J. Differ. Geom. 37, 417–466 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Matsuki, K., Wentworth, R.: Mumford-Thaddeus principle on the moduli space of vector bundles on an algebraic surface. Int. J. Math. 8, 97–148 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Moody, E.I.: Notes on the Bertini involution. Bull. Am. Math. Soc. 49, 433–436 (1943)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Mukai, S.: Counterexample to Hilbert’s fourteenth problem for the \(3\)-dimensional additive group. RIMS Preprint n. 1343, Kyoto (2001)Google Scholar
  37. 37.
    Mukai, S.: Geometric Realization of \(T\)-Shaped Root Systems and Counterexamples to Hilbert’s Fourteenth Problem, Algebraic Transformation Groups and Algebraic Varieties, Encyclopaedia of Mathematical Sciences, vol. 132, pp. 123–129. Springer, Berlin (2004)Google Scholar
  38. 38.
    Mukai, S.: Finite generation of the Nagata invariant rings in \(A\)-\(D\)-\(E\) cases. RIMS Preprint n. 1502, Kyoto (2005)Google Scholar
  39. 39.
    Okawa, S.: On images of Mori dream spaces. Math. Ann. 364, 1315–1342 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Yu, G. Prokhorov, Shokurov, V.V.: Towards the second main theorem on complements. J. Algebr. Geom. 18, 151–199 (2009)Google Scholar
  41. 41.
    Qin, Z.: Equivalence classes of polarizations and moduli spaces of sheaves. J. Differ. Geom. 37, 397–415 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Ranestad, K., Schreyer, F.-O.: Varieties of sums of powers. J. Reine Angew. Math. 2000, 147–181 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Voisin, C.: Théorie de Hodge et géométrie algébrique complexe, Cours Spécialisés, vol. 10. Société Mathématique de France, Marseille (2002)zbMATHGoogle Scholar

Copyright information

© Universidad Complutense de Madrid 2018

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di TorinoTurinItaly
  2. 2.EPFL SB MATHGEOM CAGLausanneSwitzerland
  3. 3.Mathematisches InstitutHeinrich-Heine-UniversitätDüsseldorfGermany

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