Algebra & Number Theory

Positivity of anticanonical divisors and $F$-purity of fibers

Sho Ejiri

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/ant.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We prove that given a flat generically smooth morphism between smooth projective varieties with F - pure closed fibers, if the source space is Fano, weak Fano or a variety with nef anticanonical divisor, respectively, then so is the target space. We also show that, in arbitrary characteristic, a generically smooth surjective morphism between smooth projective varieties cannot have nef and big relative anticanonical divisor, if the target space has positive dimension.

Article information

Source
Algebra Number Theory, Volume 13, Number 9 (2019), 2057-2080.

Dates
Received: 22 May 2018
Revised: 9 May 2019
Accepted: 13 June 2019
First available in Project Euclid: 14 December 2019

Permanent link to this document
https://projecteuclid.org/euclid.ant/1576292485

Digital Object Identifier
doi:10.2140/ant.2019.13.2057

Mathematical Reviews number (MathSciNet)
MR4039496

Zentralblatt MATH identifier
07141309

Subjects
Primary: 14D06: Fibrations, degenerations
Secondary: 14J45: Fano varieties

Keywords
Fano variety weak Fano variety anticanonical divisor restricted base locus augmented base locus

Citation

Ejiri, Sho. Positivity of anticanonical divisors and $F$-purity of fibers. Algebra Number Theory 13 (2019), no. 9, 2057--2080. doi:10.2140/ant.2019.13.2057. https://projecteuclid.org/euclid.ant/1576292485


Export citation

References

  • D. Abramovich and K. Karu, “Weak semistable reduction in characteristic 0”, Invent. Math. 139:2 (2000), 241–273.
  • M. Artin, “Algebraic approximation of structures over complete local rings”, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 23–58.
  • C. Birkar and Y. Chen, “Images of manifolds with semi-ample anti-canonical divisor”, J. Algebraic Geom. 25:2 (2016), 273–287.
  • M. Chen and Q. Zhang, “On a question of Demailly–Peternell–Schneider”, J. Eur. Math. Soc. 15:5 (2013), 1853–1858.
  • Y. Chen and L. Zhang, “The subadditivity of the Kodaira dimension for fibrations of relative dimension one in positive characteristics”, Math. Res. Lett. 22:3 (2015), 675–696.
  • A. Corti, “Adjunction of log divisors”, pp. 171–182 in Flips and abundance for algebraic threefolds (Salt Lake City, UT, 1991), edited by J. Kollár, Astérisque 211, Soc. Math. France, Paris, 1992.
  • O. Das and K. Schwede, “The $F$-different and a canonical bundle formula”, Ann. Sc. Norm. Super. Pisa Cl. Sci. $(5)$ 17:3 (2017), 1173–1205.
  • O. Debarre, Higher-dimensional algebraic geometry, Springer, 2001.
  • A. Grothendieck, “Eléments de géométrie algébrique, IV: Étude locale des schémas et des morphismes de schémas, III”, Inst. Hautes Études Sci. Publ. Math. 28 (1966), 5–255.
  • L. Ein, R. Lazarsfeld, M. Mustaţă, M. Nakamaye, and M. Popa, “Asymptotic invariants of base loci”, Ann. Inst. Fourier $($Grenoble$)$ 56:6 (2006), 1701–1734.
  • S. Ejiri, “Weak positivity theorem and Frobenius stable canonical rings of geometric generic fibers”, J. Algebraic Geom. 26:4 (2017), 691–734.
  • O. Fujino and Y. Gongyo, “On images of weak Fano manifolds”, Math. Z. 270:1-2 (2012), 531–544.
  • O. Fujino and Y. Gongyo, “On images of weak Fano manifolds, II”, pp. 201–207 in Algebraic and complex geometry (Hannover, Germany, 2012), edited by A. Frühbis-Krüger et al., Springer Proc. Math. Stat. 71, Springer, 2014.
  • N. Hara and K.-I. Watanabe, “F-regular and F-pure rings vs. log terminal and log canonical singularities”, J. Algebraic Geom. 11:2 (2002), 363–392.
  • R. Hartshorne, Algebraic geometry, Graduate Texts in Math. 52, Springer, 1977.
  • R. Hartshorne, “Generalized divisors on Gorenstein schemes”, $K$-Theory 8:3 (1994), 287–339.
  • Y. Kawamata, “On algebraic fiber spaces”, pp. 135–154 in Contemporary trends in algebraic geometry and algebraic topology (Tianjin, 2000), edited by S.-S. Chern et al., Nankai Tracts Math. 5, World Sci., River Edge, NJ, 2002.
  • D. S. Keeler, “Ample filters of invertible sheaves”, J. Algebra 259:1 (2003), 243–283.
  • J. Kollár, Y. Miyaoka, and S. Mori, “Rational connectedness and boundedness of Fano manifolds”, J. Differential Geom. 36:3 (1992), 765–779.
  • R. Lazarsfeld, Positivity in algebraic geometry, II: Positivity for vector bundles, and multiplier ideals, Ergebnisse der Mathematik $($3$)$ 49, Springer, 2004.
  • H. Matsumura, Commutative ring theory, Cambridge Studies in Adv. Math. 8, Cambridge Univ. Press, 1986.
  • L. E. Miller and K. Schwede, “Semi-log canonical vs $F$-pure singularities”, J. Algebra 349:1 (2012), 150–164.
  • Y. Miyaoka, “Relative deformations of morphisms and applications to fibre spaces”, Comment. Math. Univ. St. Paul. 42:1 (1993), 1–7.
  • M. Mustaţă, “The non-nef locus in positive characteristic”, pp. 535–551 in A celebration of algebraic geometry (Cambridge, MA, 2011), edited by B. Hassett et al., Clay Math. Proc. 18, Amer. Math. Soc., Providence, RI, 2013.
  • Z. Patakfalvi, “Semi-positivity in positive characteristics”, Ann. Sci. École Norm. Sup. $(4)$ 47:5 (2014), 991–1025.
  • Z. Patakfalvi, K. Schwede, and W. Zhang, “$F$-singularities in families”, Algebr. Geom. 5:3 (2018), 264–327.
  • Y. G. Prokhorov and V. V. Shokurov, “Towards the second main theorem on complements”, J. Algebraic Geom. 18:1 (2009), 151–199.
  • K. Schwede, “Generalized test ideals, sharp $F$-purity, and sharp test elements”, Math. Res. Lett. 15:6 (2008), 1251–1261.
  • S. Takagi, “An interpretation of multiplier ideals via tight closure”, J. Algebraic Geom. 13:2 (2004), 393–415.
  • E. Viehweg, “Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces”, pp. 329–353 in Algebraic varieties and analytic varieties (Tokyo, 1981), edited by S. Iitaka, Adv. Stud. Pure Math. 1, North-Holland, Amsterdam, 1983.
  • E. Viehweg, Quasi-projective moduli for polarized manifolds, Ergebnisse der Mathematik $($3$)$ 30, Springer, 1995.