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


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.


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

Mathematics Subject Classification

14J60 14J35 14J45 14E30 



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.


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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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