Abstract
We define and study a version of Seshadri constant for ample line bundles in positive characteristic. We prove that lower bounds for this constant imply the global generation or very ampleness of the corresponding adjoint line bundle. As a consequence, we deduce that the criterion for global generation and very ampleness of adjoint line bundles in terms of usual Seshadri constants holds also in positive characteristic.
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References
Bauer, T., Di Rocco, S., Harbourne, B., Kapustka, M., Knutsen, A., Syzdek, W., Szemberg, T.: A primer on Seshadri constants. In: Interactions of classical and numerical algebraic geometry, 3370, Contemp. Math., vol. 496. American Mathematical Society, Providence (2009)
Brion, M., Kumar, S.: Frobenius splitting methods in geometry and representation theory, Progress in Mathematics, 231. Birkhäuser Boston Inc., Boston (2005)
Demailly, J.-P.: Singular Hermitian metrics on positive line bundles. In: Complex algebraic varieties (Bayreuth, 1990), pp. 87–104. Springer, Berlin (1992)
Di Rocco, S.: Generation of k-jets on toric varieties. Math. Z. 231, 169–188 (1999)
Ein, L., Küchle, O., Lazarsfeld, R.: Local positivity of ample line bundles. J. Differ. Geom. 42, 193–219 (1995)
Ein, L., Lazarsfeld, R.: Seshadri constants on smooth surfaces. Astérisque 218, 177–186 (1993)
Fujita, T.: Vanishing theorems for semipositive line bundles. In: Algebraic geometry (Tokyo/Kyoto, 1982), Lecture Notes in Mathematics, vol. 1016, pp. 519–528. Springer, Berlin (1983)
Fulton, W.: Introduction to toric varieties, Ann. of Math. Stud. 131, The William H. Rover Lectures in Geometry. Princeton University Press, Princeton (1993)
Kunz, E.: Characterizations of regular local rings for characteristic \(p\). Am. J. Math. 91, 772–784 (1969)
Ito, A.: Seshadri constants via toric degenerations, arXiv:1202.6664 (2011, preprint)
Lazarsfeld, R.: Positivity in algebraic geometry I, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 48. Springer, Berlin (2004)
Acknowledgments
This project originated in discussions held during the AIM workshop “Relating test ideals and multiplier ideals”. We are indebted to AIM for organizing this event. We would also like to thank Bhargav Bhatt and Rob Lazarsfeld for discussions related to this work.
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M. Mustaţă was partially supported by NSF research Grant No: 1068190 and by a Packard Fellowship. K. Schwede was partially support by NSF research Grant No: 1064485 and by a Sloan Fellowship.