Skip to main content
SpringerLink
Log in

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

  • Published:
Revista Matemática Complutense Aims and scope Submit manuscript

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 (S, L); 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.

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

Fig. 1

Similar content being viewed by others

Notes

  1. A contraction is a surjective map with connected fibers \(\varphi :Y\rightarrow Z\), where Z is normal and projective.

  2. A prime divisor E is fixed if it is a fixed component of the linear system |mE| for every \(m\ge 1\).

  3. A facet is a face of codimension one.

  4. A birational map \(\varphi :X_1\dasharrow X_2\) is contracting if there exist open subsets \(U_1\subseteq X_1\) and \(U_2\subseteq X_2\) such that \(\varphi \) yields an isomorphism between \(U_1\) and \(U_2\), and \({\text {codim}}(X_2\smallsetminus U_2)\ge 2\).

  5. Note that the points \(p_i\) here depend on \(h'\), as they are the images of the exceptional divisors of \(\eta _{h'}:Y\dasharrow \mathbb {P}^4\). For simplicity we still denote them by \(p_1,\cdots ,p_8\), and similarly in the sequel of the proof.

References

  1. Araujo, C., Casagrande, C.: On the Fano variety of linear spaces contained in two odd-dimensional quadrics. Geom. Topol. 21, 3009–3045 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  2. Araujo, C., Massarenti, A.: Explicit log Fano structures on blow-ups of projective spaces. Proc. Lond. Math. Soc. 113, 445–473 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Book  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  10. Casagrande, C.: Fano 4-folds, flips, and blow-ups of points. J. Algebra 483, 362–414 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  11. Castravet, A.-M., Tevelev, J.: Hilbert’s 14th problem and Cox rings. Compos. Math. 142, 1479–1498 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Coble, A.B.: The ten nodes of the rational sextic and of the Cayley symmetroid. Am. J. Math. 41, 243–265 (1919)

    Article  MathSciNet  MATH  Google Scholar 

  13. Codogni, G., Fanelli, A., Svaldi, R., Tasin, L.: Fano varieties in Mori fibre spaces. Int. Math. Res. Not. 2016, 2026–2067 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  15. Dolgachev, I.V.: Weyl groups and Cremona transformations, singularities. Proc. Symp. Part 1 (Arcata, Calif., 1981, Pure Math.) 40, 283–294 (1983)

  16. Dolgachev, I.V.: On certain families of elliptic curves in projective space. Ann. Math. Pura Appl. 183, 317–331 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Dolgachev, I.V.: Classical Algebraic Geometry–A Modern View. Cambridge University Press, Cambridge (2012)

    Book  MATH  Google Scholar 

  18. Dolgachev, I.V., Ortland, D.: Point sets in projective spaces and theta functions, Astérisque, vol. 165. Société Mathématique de France, Marseille (1988)

    MATH  Google Scholar 

  19. Du Val, P.: Crystallography and Cremona Transformations, The Geometric Vein—The Coxeter Festschrift, pp. 191–201. Springer, Berlin (1981)

  20. Dumitrescu, O., Postinghel, E.: Positivity of divisors on blown-up projective spaces, I. preprint (2015). arXiv:1506.04726

  21. Eisenbud, D., Popescu, S.: The projective geometry of the Gale transform. J. Algebra 230, 127–173 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ellingsrud, G., Göttsche, L.: Variation of moduli space and Donaldson invariants under change of polarization. J. Reine Angew. Math. 467, 1–49 (1995)

    MathSciNet  MATH  Google Scholar 

  23. Friedman, R.: Algebraic Surfaces and Holomorphic Vector Bundles. Universitext. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Harris, J.: Algebraic Geometry-A First Course, Graduate Texts in Mathematics, vol. 133. Springer, Berlin (1992)

    Google Scholar 

  27. Hu, Y., Keel, S.: Mori dream spaces and GIT. Michigan Math. J. 48, 331–348 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  28. Huybrechts, D., Lehn, M.: The Geometry of the Moduli Space of Sheaves, 2nd edn. Cambridge University Press, Cambridge (2010)

    Book  MATH  Google Scholar 

  29. Kollár, J., Mori, S.: Birational Geometry of Algebraic Varieties, Cambridge Tracts in Mathematics, vol. 134. Cambridge University Press, Cambridge (1998)

    Book  Google Scholar 

  30. Lazarsfeld, R.: Positivity in Algebraic Geometry II. Springer, Berlin (2004)

    Book  MATH  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  33. Li, J.: Algebraic geometric interpretation of Donaldson’s polynomial invariants. J. Differ. Geom. 37, 417–466 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  35. Moody, E.I.: Notes on the Bertini involution. Bull. Am. Math. Soc. 49, 433–436 (1943)

    Article  MathSciNet  MATH  Google Scholar 

  36. Mukai, S.: Counterexample to Hilbert’s fourteenth problem for the \(3\)-dimensional additive group. RIMS Preprint n. 1343, Kyoto (2001)

  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)

  38. Mukai, S.: Finite generation of the Nagata invariant rings in \(A\)-\(D\)-\(E\) cases. RIMS Preprint n. 1502, Kyoto (2005)

  39. Okawa, S.: On images of Mori dream spaces. Math. Ann. 364, 1315–1342 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  40. Yu, G. Prokhorov, Shokurov, V.V.: Towards the second main theorem on complements. J. Algebr. Geom. 18, 151–199 (2009)

  41. Qin, Z.: Equivalence classes of polarizations and moduli spaces of sheaves. J. Differ. Geom. 37, 397–415 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  42. Ranestad, K., Schreyer, F.-O.: Varieties of sums of powers. J. Reine Angew. Math. 2000, 147–181 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  43. Voisin, C.: Théorie de Hodge et géométrie algébrique complexe, Cours Spécialisés, vol. 10. Société Mathématique de France, Marseille (2002)

    MATH  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, 10123, Turin, Italy

    Cinzia Casagrande

  2. EPFL SB MATHGEOM CAG, MA B3 444 (Batiment MA) Station 8, 1015, Lausanne, Switzerland

    Giulio Codogni

  3. Mathematisches Institut, Heinrich-Heine-Universität, 40204, Düsseldorf, Germany

    Andrea Fanelli

Authors
  1. Cinzia Casagrande
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Giulio Codogni
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Andrea Fanelli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cinzia Casagrande.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Casagrande, C., Codogni, G. & Fanelli, A. The blow-up of \(\mathbb {P}^4\) at 8 points and its Fano model, via vector bundles on a del Pezzo surface. Rev Mat Complut 32, 475–529 (2019). https://doi.org/10.1007/s13163-018-0282-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13163-018-0282-5

Keywords

Mathematics Subject Classification

Search

Navigation