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ON THE BASE-POINT-FREE THEOREM OF 3-FOLDS IN POSITIVE CHARACTERISTIC

Published online by Cambridge University Press:  31 March 2014

Chenyang Xu
Chenyang Xu*
Affiliation:
Beijing International Center of Mathematical Research, 5 Yiheyuan Road, Beijing, 100871, China ([email protected])
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Abstract

Let $(X,{\rm\Delta})$ be a projective klt (standing for Kawamata log terminal) three-dimensional pair defined over an algebraically closed field $k$ with $\text{char}(k)>5$. Let $L$ be a nef (numerically eventually free) and big line bundle on $X$ such that $L-K_{X}-{\rm\Delta}$ is big and nef. We show that $L$ is indeed semi-ample.

Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 14 , Issue 3 , July 2015 , pp. 577 - 588
Copyright
© Cambridge University Press 2014 

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References

Abhyankar, S. S., Resolution of singularities of embedded algebraic surfaces, Second edn., Springer Monographs in Mathematics, (Springer-Verlag, Berlin, 1998).Google Scholar
Alexeev, Valery, Boundedness and K 2 for log surfaces, Int. J. Math. 5(6) (1994), 779–810.CrossRefGoogle Scholar
Artin, M., Algebraization of formal moduli. II. Existence of modifications, Ann. of Math. (2) 91 (1970), 88–135.Google Scholar
Birkar, C., Ascending chain condition for log canonical thresholds and termination of log flips, Duke Math. J. 136(1) (2007), 173–180.CrossRefGoogle Scholar
Birkar, C., Existence of flips and minimal models for 3-folds in char p, 2013 (arXiv:1311.3098).Google Scholar
Cossart, V. and Piltant, O., Resolution of singularities of threefolds in positive characteristic. I. Reduction to local uniformization on Artin–Schreier and purely inseparable coverings, J. Algebra 320(3) (2008), 1051–1082.Google Scholar
Cossart, V. and Piltant, O., Resolution of singularities of threefolds in positive characteristic. II, J. Algebra 321(7) (2009), 1836–1976.Google Scholar
Cascini, P., Tanaka, H. and Xu, Chenyang, On base point freeness in positive characteristic, Ann. Sci. École Norm. Sup. (2013), (to appear) (arXiv:1305.3502).Google Scholar
Cutkosky, S. D., Resolution of singularities for 3-folds in positive characteristic, Amer. J. Math. 131(1) (2009), 59–127.Google Scholar
Grothendieck, A., Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes, Inst. Hautes Études Sci. Publ. Math.(8) (1961), 222.Google Scholar
Fujino, O., Special termination and reduction to pl flips, in Flips for 3-folds and 4-folds, Oxford Lecture Ser. Math. Appl., Volume 35, pp. 63–75 (Oxford Univ. Press, Oxford, 2007).Google Scholar
Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, No. 52, pp. xvi+496 (Springer-Verlag, New York, 1977).Google Scholar
Hochster, M. and Huneke, C., Tight closure, invariant theory, and the Briançon–Skoda theorem, J. Amer. Math. Soc. 3(1) (1990), 31–116.Google Scholar
Hacon, C. D., McKernan, J. and Xu, C., ACC for log canonical thresholds, Ann. of Math. (2012), (to appear) (arXiv:1208.4150).Google Scholar
Hacon, C. D. and Xu, C.,On the three dimensional minimal model program in positive characteristic, 2013 arXiv:1302.0298.Google Scholar
Keel, S., Basepoint freeness for nef and big line bundles in positive characteristic, Ann. of Math. (2) 149(1) (1999), 253–286.Google Scholar
J. Kollár and 14 coauthors, Flips and abundance for algebraic threefolds, Société Mathématique de France, Paris, 1992. Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991, Astérisque No. 211 (1992).Google Scholar
Kollár, J., Quotient spaces modulo algebraic groups, Ann. of Math. (2) 145(1) (1997), 33–79.Google Scholar
Kollár, J., Singularities of the minimal model program, Cambridge Tracts in Mathematics, Volume 200 (Cambridge University Press, Cambridge, 2013). With a collaboration of Sándor Kovács.Google Scholar
Kollár, J. and Mori, S., Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, Volume 134, pp. viii+254 (Cambridge University Press, Cambridge, 1998). With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original.Google Scholar
Schwede, K., A canonical linear system associated to adjoint divisors in characteristic p > 0, J. Reine Angew. Math. (2011), (to appear) (arXiv:1107.3833v2).Google Scholar
Shokurov, V. V., Three-dimensional log perestroikas, Izv. Ross. Akad. Nauk Ser. Mat. 56(1) (1992), 105–203.Google Scholar
Tanaka, H., Minimal models and abundance for positive characteristic log surfaces, Nagoya Math. J. (2012), (to appear) (arXiv:1201.5699).Google Scholar