Algebra & Number Theory

Fano 4-folds with rational fibrations

Cinzia Casagrande

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Abstract

We study (smooth, complex) Fano 4-folds X having a rational contraction of fiber type, that is, a rational map X −−→Y that factors as a sequence of flips followed by a contraction of fiber type. The existence of such a map is equivalent to the existence of a nonzero, nonbig movable divisor on X. Our main result is that if Y is not 1 or 2, then the Picard number ρX of X is at most 18, with equality only if X is a product of surfaces. We also show that if a Fano 4-fold X has a dominant rational map X −−→Z, regular and proper on an open subset of X, with dim(Z)=3, then either X is a product of surfaces, or ρX is at most 12. These results are part of a program to study Fano 4-folds with large Picard number via birational geometry.

Article information

Source
Algebra Number Theory, Volume 14, Number 3 (2020), 787-813.

Dates
Received: 27 February 2019
Revised: 17 September 2019
Accepted: 8 November 2019
First available in Project Euclid: 2 July 2020

Permanent link to this document
https://projecteuclid.org/euclid.ant/1593655272

Digital Object Identifier
doi:10.2140/ant.2020.14.787

Mathematical Reviews number (MathSciNet)
MR4113781

Subjects
Primary: 14J45: Fano varieties
Secondary: 14E30: Minimal model program (Mori theory, extremal rays) 14J35: $4$-folds

Keywords
Fano 4-folds Mori dream spaces birational geometry MMP

Citation

Casagrande, Cinzia. Fano 4-folds with rational fibrations. Algebra Number Theory 14 (2020), no. 3, 787--813. doi:10.2140/ant.2020.14.787. https://projecteuclid.org/euclid.ant/1593655272


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References

  • M. Andreatta and J. A. Wiśniewski, “A view on contractions of higher-dimensional varieties”, pp. 153–183 in Algebraic geometry (Santa Cruz, CA, 1995), edited by J. Kollár et al., Proc. Sympos. Pure Math. 62, Amer. Math. Soc., Providence, RI, 1997.
  • M. Andreatta, E. Ballico, and J. Wiśniewski, “Vector bundles and adjunction”, Int. J. Math. 3:3 (1992), 331–340.
  • C. Birkar, P. Cascini, C. D. Hacon, and J. McKernan, “Existence of minimal models for varieties of log general type”, J. Amer. Math. Soc. 23:2 (2010), 405–468.
  • C. Casagrande, “Quasi-elementary contractions of Fano manifolds”, Compos. Math. 144:6 (2008), 1429–1460.
  • C. Casagrande, “On the Picard number of divisors in Fano manifolds”, Ann. Sci. École Norm. Sup. $(4)$ 45:3 (2012), 363–403.
  • C. Casagrande, “On the birational geometry of Fano 4-folds”, Math. Ann. 355:2 (2013), 585–628.
  • C. Casagrande, “Numerical invariants of Fano 4-folds”, Math. Nachr. 286:11-12 (2013), 1107–1113.
  • C. Casagrande, “Fano 4-folds, flips, and blow-ups of points”, J. Algebra 483 (2017), 362–414.
  • C. Casagrande, G. Codogni, and A. Fanelli, “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. 32:2 (2019), 475–529.
  • S. Cutkosky, “Elementary contractions of Gorenstein threefolds”, Math. Ann. 280:3 (1988), 521–525.
  • G. Della Noce, “On the Picard number of singular Fano varieties”, Int. Math. Res. Not. 2014:4 (2014), 955–990.
  • S. Druel, “Codimension one foliations with numerically trivial canonical class on singular spaces”, preprint, 2018.
  • O. Fujino, “Applications of Kawamata's positivity theorem”, Proc. Japan Acad. Ser. A Math. Sci. 75:6 (1999), 75–79.
  • Y. Hu and S. Keel, “Mori dream spaces and GIT”, Michigan Math. J. 48 (2000), 331–348.
  • V. A. Iskovskikh and Y. G. Prokhorov, Algebraic geometry, V: Fano varieties, Encycl. Math. Sci. 47, Springer, 1999.
  • J. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Math. 134, Cambridge Univ. Press, 1998.
  • S. Mori, “Threefolds whose canonical bundles are not numerically effective”, Ann. of Math. $(2)$ 116:1 (1982), 133–176.
  • S. Okawa, “On images of Mori dream spaces”, Math. Ann. 364:3-4 (2016), 1315–1342.
  • W. Ou, “Fano varieties with ${\rm Nef}(X)={\rm Psef}(X)$ and $\rho (X)={\rm dim} X-1$”, Manuscripta Math. 157:3-4 (2018), 551–587.
  • Y. G. Prokhorov, “The degree of Fano threefolds with canonical Gorenstein singularities”, Mat. Sb. 196:1 (2005), 81–122. In Russian; translated in Sb. Math. 196:1 (2005), 77–114.
  • Y. G. Prokhorov and V. V. Shokurov, “Towards the second main theorem on complements”, J. Algebraic Geom. 18:1 (2009), 151–199.
  • E. A. Romano, “Non-elementary Fano conic bundles”, Collect. Math. 70:1 (2019), 33–50.