On birational boundedness of foliated surfaces

Christopher D. Hacon 1  and Adrian Langer 2
  • 1 Department of Mathematics, University of Utah, 155 S 1400 E, JWB 233, Salt Lake City, USA
  • 2 Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097, Warsaw, Poland
Christopher D. Hacon
  • Corresponding author
  • Department of Mathematics, University of Utah, 155 S 1400 E, JWB 233, Salt Lake City, UT 84112, USA
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and Adrian Langer


In this paper we prove a result on the effective generation of pluri-canonical linear systems on foliated surfaces of general type. Fix a function P:0, then there exists an integer N>0 such that if (X,) is a canonical or nef model of a foliation of general type with Hilbert polynomial χ(X,𝒪X(mK))=P(m) for all m0, then |mK| defines a birational map for all mN.

On the way, we also prove a Grauert–Riemenschneider-type vanishing theorem for foliated surfaces with canonical singularities.

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The Journal für die reine und angewandte Mathematik is the oldest mathematics periodical still in existence. Founded in 1826 by August Leopold Crelle and edited by him until his death in 1855, it soon became widely known under the name of Crelle’s Journal. In the 190 years of its existence, Crelle’s Journal has developed to an outstanding scholarly periodical with one of the worldwide largest circulations among mathematics journals.