Algebra & Number Theory

Effective Matsusaka's theorem for surfaces in characteristic $p$

Gabriele Di Cerbo and Andrea Fanelli

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

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


We obtain an effective version of Matsusaka’s theorem for arbitrary smooth algebraic surfaces in positive characteristic, which provides an effective bound on the multiple that makes an ample line bundle D very ample. The proof for pathological surfaces is based on a Reider-type theorem. As a consequence, a Kawamata–Viehweg-type vanishing theorem is proved for arbitrary smooth algebraic surfaces in positive characteristic.

Article information

Algebra Number Theory, Volume 9, Number 6 (2015), 1453-1475.

Received: 24 February 2015
Revised: 16 April 2015
Accepted: 17 May 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14J25: Special surfaces {For Hilbert modular surfaces, see 14G35}

effective Matsusaka surfaces in positive characteristic Fujita's conjectures Bogomolov's stability Reider's theorem bend-and-break effective Kawamata–Viehweg vanishing


Di Cerbo, Gabriele; Fanelli, Andrea. Effective Matsusaka's theorem for surfaces in characteristic $p$. Algebra Number Theory 9 (2015), no. 6, 1453--1475. doi:10.2140/ant.2015.9.1453.

Export citation


  • M. F. Atiyah, “Vector bundles over an elliptic curve”, Proc. London Math. Soc. $(3)$ 7 (1957), 414–452.
  • E. Ballico, “A positive characteristic extension of a result of del Busto on line bundles on an algebraic surface”, Rend. Circ. Mat. Palermo $(2)$ 45:3 (1996), 473–478.
  • F. A. Bogomolov, “Holomorphic tensors and vector bundles on projective manifolds”, Izv. Akad. Nauk SSSR Ser. Mat. 42:6 (1978), 1227–1287. In Russian; translated in Math. USSR Izv. 13:3 (1979) 499–555.
  • G. Fernández del Busto, “A Matsusaka-type theorem on surfaces”, J. Algebraic Geom. 5:3 (1996), 513–520.
  • J.-P. Demailly, “Effective bounds for very ample line bundles”, Invent. Math. 124:1-3 (1996), 243–261.
  • J.-P. Demailly, “$L\sp 2$ vanishing theorems for positive line bundles and adjunction theory”, pp. 1–97 in Transcendental methods in algebraic geometry (Cetraro, 1994), edited by F. Catanese and C. Ciliberto, Lecture Notes in Math. 1646, Springer, Berlin, 1996.
  • G. Di Cerbo, “A cohomological interpretation of Bogomolov's instability”, Proc. Amer. Math. Soc. 141:9 (2013), 3049–3053.
  • T. Ekedahl, “Canonical models of surfaces of general type in positive characteristic”, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 97–144.
  • J. Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 32, Springer, Berlin, 1996.
  • J. Kollár and T. Matsusaka, “Riemann–Roch type inequalities”, Amer. J. Math. 105:1 (1983), 229–252.
  • J. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics 134, Cambridge University Press, 1998.
  • A. Langer, “The Bogomolov–Miyaoka–Yau inequality for logarithmic surfaces in positive characteristic”, preprint, 2014.
  • R. Lazarsfeld, Positivity in algebraic geometry, I: Classical setting: line bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 48, Springer, Berlin, 2004.
  • R. Lazarsfeld, Positivity in algebraic geometry, II: Positivity for vector bundles, and multiplier ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 49, Springer, Berlin, 2004.
  • C. Liedtke, “Algebraic surfaces in positive characteristic”, pp. 229–292 in Birational geometry, rational curves, and arithmetic, edited by F. Bogomolov et al., Springer, New York, 2013.
  • T. Matsusaka, “Polarized varieties with a given Hilbert polynomial”, Amer. J. Math. 94 (1972), 1027–1077.
  • S. Mukai, “Counterexamples to Kodaira's vanishing and Yau's inequality in positive characteristics”, Kyoto J. Math. 53:2 (2013), 515–532.
  • I. Reider, “Vector bundles of rank $2$ and linear systems on algebraic surfaces”, Ann. of Math. $(2)$ 127:2 (1988), 309–316.
  • F. Sakai, “Reider–Serrano's method on normal surfaces”, pp. 301–319 in Algebraic geometry (L'Aquila, 1988), edited by A. J. Sommese et al., Lecture Notes in Math. 1417, Springer, Berlin, 1990.
  • N. I. Shepherd-Barron, “Unstable vector bundles and linear systems on surfaces in characteristic $p$”, Invent. Math. 106:2 (1991), 243–262.
  • N. I. Shepherd-Barron, “Geography for surfaces of general type in positive characteristic”, Invent. Math. 106:2 (1991), 263–274.
  • Y.-T. Siu, “Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semipositively twisted plurigenera for manifolds not necessarily of general type”, pp. 223–277 in Complex geometry (Göttingen, 2000), edited by I. Bauer et al., Springer, Berlin, 2002.
  • Y.-T. Siu, “A new bound for the effective Matsusaka big theorem”, Houston J. Math. 28:2 (2002), 389–409.
  • L. Szpiro, “Sur le théorème de rigidité de Parsin et Arakelov”, pp. 169–202 in Journées de Géométrie Algébrique de Rennes, II (Rennes, 1978), Astérisque 64, Soc. Math. France, Paris, 1979.
  • S. Takagi, “Fujita's approximation theorem in positive characteristics”, J. Math. Kyoto Univ. 47:1 (2007), 179–202.
  • H. Tanaka, “The X-method for klt surfaces in positive characteristic”, preprint, 2012.
  • H. Tanaka, “Minimal models and abundance for positive characteristic log surfaces”, Nagoya Math. J. 216 (2014), 1–70.
  • H. Tango, “On the behavior of extensions of vector bundles under the Frobenius map”, Nagoya Math. J. 48 (1972), 73–89.
  • H. Terakawa, “The $d$-very ampleness on a projective surface in positive characteristic”, Pacific J. Math. 187:1 (1999), 187–199.
  • Q. Xie, “Counterexamples to the Kawamata–Viehweg vanishing on ruled surfaces in positive characteristic”, J. Algebra 324:12 (2010), 3494–3506.