Bott vanishing for algebraic surfaces
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Abstract:
Bott proved a strong vanishing theorem for sheaf cohomology on projective space. It holds for toric varieties, but not for most other varieties.
We prove Bott vanishing for the quintic del Pezzo surface, also known as the moduli space $\overline {M_{0,5}}$ of 5-pointed stable curves of genus zero. This is the first non-toric Fano variety for which Bott vanishing has been shown, answering a question by Achinger, Witaszek, and Zdanowicz.
In another direction, we prove sharp results on which K3 surfaces satisfy Bott vanishing. For example, a K3 surface with Picard number 1 satisfies Bott vanishing if and only if the degree is 20 or at least 24. For K3 surfaces of any Picard number, we have complete results when the degree is big enough. We build on Beauville, Mori, and Mukai’s work on moduli spaces of K3 surfaces, as well as recent advances by Arbarello-Bruno-Sernesi, Ciliberto-Dedieu-Sernesi, and Feyzbakhsh.
The most novel aspect of the paper is our analysis of Bott vanishing for K3 surfaces with an elliptic curve of low degree. (In other terminology, this concerns K3 surfaces that are monogonal, hyperelliptic, trigonal, or tetragonal.) It turns out that the crucial issue is whether an elliptic fibration has a certain special type of singular fiber.
References
- P. Achinger, J. Witaszek, and M. Zdanowicz, Liftability of the Frobenius morphism and images of toric varieties, arXiv:1708.03777
- E. Arbarello, A. Bruno, and E. Sernesi, Mukai’s program for curves on a K3 surface, Algebr. Geom. 1 (2014), no. 5, 532–557. MR 3296804, DOI 10.14231/AG-2014-023
- Enrico Arbarello, Andrea Bruno, and Edoardo Sernesi, On hyperplane sections of K3 surfaces, Algebr. Geom. 4 (2017), no. 5, 562–596. MR 3710056, DOI 10.14231/2017-028
- Victor V. Batyrev and David A. Cox, On the Hodge structure of projective hypersurfaces in toric varieties, Duke Math. J. 75 (1994), no. 2, 293–338. MR 1290195, DOI 10.1215/S0012-7094-94-07509-1
- Arnaud Beauville, Fano threefolds and $K3$ surfaces, The Fano Conference, Univ. Torino, Turin, 2004, pp. 175–184. MR 2112574
- James N. Brawner, Tetragonal curves, scrolls, and $K3$ surfaces, Trans. Amer. Math. Soc. 349 (1997), no. 8, 3075–3091. MR 1401515, DOI 10.1090/S0002-9947-97-01811-4
- Anders Buch, Jesper F. Thomsen, Niels Lauritzen, and Vikram Mehta, The Frobenius morphism on a toric variety, Tohoku Math. J. (2) 49 (1997), no. 3, 355–366. MR 1464183, DOI 10.2748/tmj/1178225109
- C. Ciliberto, T. Dedieu, C. Galati, and A. Knutsen, Moduli of curves on Enriques surfaces, arXiv:1902.07142
- C. Ciliberto, T. Dedieu, and E. Sernesi, Wahl maps and extensions of canonical curves and K3 surfaces, J. Reine Angew. Math., to appear. https://doi.org/10.1515/crelle-2018-0016
- Ciro Ciliberto, Angelo Felice Lopez, and Rick Miranda, Classification of varieties with canonical curve section via Gaussian maps on canonical curves, Amer. J. Math. 120 (1998), no. 1, 1–21. MR 1600256
- François R. Cossec and Igor V. Dolgachev, Enriques surfaces. I, Progress in Mathematics, vol. 76, Birkhäuser Boston, Inc., Boston, MA, 1989. MR 986969, DOI 10.1007/978-1-4612-3696-2
- Olivier Debarre, Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York, 2001. MR 1841091, DOI 10.1007/978-1-4757-5406-3
- Pierre Deligne and Luc Illusie, Relèvements modulo $p^2$ et décomposition du complexe de de Rham, Invent. Math. 89 (1987), no. 2, 247–270 (French). MR 894379, DOI 10.1007/BF01389078
- S. Feyzbakhsh, Mukai’s program (reconstructing a K3 surface from a curve) via wall-crossing, J. Reine Angew. Math., to appear. https://doi.org/10.1515/crelle-2019-0025
- Osamu Fujino, Multiplication maps and vanishing theorems for toric varieties, Math. Z. 257 (2007), no. 3, 631–641. MR 2328817, DOI 10.1007/s00209-007-0140-5
- William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR 1234037, DOI 10.1515/9781400882526
- Mark Green and Robert Lazarsfeld, Special divisors on curves on a $K3$ surface, Invent. Math. 89 (1987), no. 2, 357–370. MR 894384, DOI 10.1007/BF01389083
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
- Daniel Huybrechts, Lectures on K3 surfaces, Cambridge Studies in Advanced Mathematics, vol. 158, Cambridge University Press, Cambridge, 2016. MR 3586372, DOI 10.1017/CBO9781316594193
- Andreas Leopold Knutsen, On $k$th-order embeddings of $K3$ surfaces and Enriques surfaces, Manuscripta Math. 104 (2001), no. 2, 211–237. MR 1821184, DOI 10.1007/s002290170040
- Andreas Leopold Knutsen and Angelo Felice Lopez, A sharp vanishing theorem for line bundles on $K3$ or Enriques surfaces, Proc. Amer. Math. Soc. 135 (2007), no. 11, 3495–3498. MR 2336562, DOI 10.1090/S0002-9939-07-08968-X
- Robert Lazarsfeld, Positivity in algebraic geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 49, Springer-Verlag, Berlin, 2004. Positivity for vector bundles, and multiplier ideals. MR 2095472, DOI 10.1007/978-3-642-18808-4
- Yu. I. Manin, Cubic forms, 2nd ed., North-Holland Mathematical Library, vol. 4, North-Holland Publishing Co., Amsterdam, 1986. Algebra, geometry, arithmetic; Translated from the Russian by M. Hazewinkel. MR 833513
- G. Martens, On curves on $K3$ surfaces, Algebraic curves and projective geometry (Trento, 1988) Lecture Notes in Math., vol. 1389, Springer, Berlin, 1989, pp. 174–182. MR 1023397, DOI 10.1007/BFb0085931
- Shigefumi Mori, On degrees and genera of curves on smooth quartic surfaces in $\textbf {P}^3$, Nagoya Math. J. 96 (1984), 127–132. MR 771073, DOI 10.1017/S0027763000021188
- Shigefumi Mori and Shigeru Mukai, The uniruledness of the moduli space of curves of genus $11$, Algebraic geometry (Tokyo/Kyoto, 1982) Lecture Notes in Math., vol. 1016, Springer, Berlin, 1983, pp. 334–353. MR 726433, DOI 10.1007/BFb0099970
- Shigeru Mukai, Fano $3$-folds, Complex projective geometry (Trieste, 1989/Bergen, 1989) London Math. Soc. Lecture Note Ser., vol. 179, Cambridge Univ. Press, Cambridge, 1992, pp. 255–263. MR 1201387, DOI 10.1017/CBO9780511662652.018
- Mircea Mustaţă, Vanishing theorems on toric varieties, Tohoku Math. J. (2) 54 (2002), no. 3, 451–470. MR 1916637
- Yu. G. Prokhorov, The degree of Fano threefolds with canonical Gorenstein singularities, Mat. Sb. 196 (2005), no. 1, 81–122 (Russian, with Russian summary); English transl., Sb. Math. 196 (2005), no. 1-2, 77–114. MR 2141325, DOI 10.1070/SM2005v196n01ABEH000873
- Yu. G. Prokhorov, On Fano-Enriques varieties, Mat. Sb. 198 (2007), no. 4, 117–134 (Russian, with Russian summary); English transl., Sb. Math. 198 (2007), no. 3-4, 559–574. MR 2352363, DOI 10.1070/SM2007v198n04ABEH003849
- Miles Reid, Chapters on algebraic surfaces, Complex algebraic geometry (Park City, UT, 1993) IAS/Park City Math. Ser., vol. 3, Amer. Math. Soc., Providence, RI, 1997, pp. 3–159. MR 1442522, DOI 10.1090/pcms/003/02
- B. Saint-Donat, Projective models of $K-3$ surfaces, Amer. J. Math. 96 (1974), 602–639. MR 364263, DOI 10.2307/2373709
- The Stacks Project Authors, Stacks Project, 2018. http://stacks.math.columbia.edu
- Jan Stevens, Rolling factors deformations and extensions of canonical curves, Doc. Math. 6 (2001), 185–226. MR 1849195
- Jonathan M. Wahl, The Jacobian algebra of a graded Gorenstein singularity, Duke Math. J. 55 (1987), no. 4, 843–871. MR 916123, DOI 10.1215/S0012-7094-87-05540-2
Additional Information
- Burt Totaro
- Affiliation: Department of Mathematics, University of California Los Angeles, Box 951555, Los Angeles, California 90095-1555
- MR Author ID: 272212
- Email: totaro@math.ucla.edu
- Received by editor(s): June 1, 2019
- Received by editor(s) in revised form: September 8, 2019, and September 10, 2019
- Published electronically: January 28, 2020
- Additional Notes: This work was supported by National Science Foundation grant DMS-1701237, and by grant DMS-1440140 while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2019 semester.
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 3609-3626
- MSC (2010): Primary 14F17; Secondary 14J26, 14J28
- DOI: https://doi.org/10.1090/tran/8045
- MathSciNet review: 4082249
Dedicated: For William Fulton on his eightieth birthday