Bridgeland stability conditions on threefolds II: An application to Fujita’s conjecture
Authors:
Arend Bayer, Aaron Bertram, Emanuele Macrì and Yukinobu Toda
Journal:
J. Algebraic Geom. 23 (2014), 693-710
DOI:
https://doi.org/10.1090/S1056-3911-2014-00637-8
Published electronically:
January 28, 2014
MathSciNet review:
3263665
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Abstract |
References |
Additional Information
Abstract: We apply a conjectured inequality on third Chern classes of stable two-term complexes on threefolds to Fujita’s conjecture. More precisely, the inequality is shown to imply a Reider-type theorem in dimension three which in turn implies that $K_X + 6L$ is very ample when $L$ is ample, and that $5L$ is very ample when $K_X$ is trivial.
References
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- Arend Bayer, Emanuele Macrì, and Yukinobu Toda. Bridgeland stability conditions on threefolds I: Bogomolov-Gieseker type inequalities, 2011. arXiv:1103.5010.
- Tom Bridgeland, Stability conditions on triangulated categories, Ann. of Math. (2) 166 (2007), no. 2, 317–345. MR 2373143, DOI https://doi.org/10.4007/annals.2007.166.317
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- Takao Fujita. Towards a separation theorem of points by adjoint linear systems on polarized threefolds, 1994. arXiv:alg-geom/9411001.
- Stefan Helmke, On Fujita’s conjecture, Duke Math. J. 88 (1997), no. 2, 201–216. MR 1455517, DOI https://doi.org/10.1215/S0012-7094-97-08807-4
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- Igor Reider, Vector bundles of rank $2$ and linear systems on algebraic surfaces, Ann. of Math. (2) 127 (1988), no. 2, 309–316. MR 932299, DOI https://doi.org/10.2307/2007055
References
- Daniele Arcara and Aaron Bertram, Reider’s theorem and Thaddeus pairs revisited, Grassmannians, moduli spaces and vector bundles, Clay Math. Proc., vol. 14, Amer. Math. Soc., Providence, RI, 2011, pp. 51–68. MR 2807848 (2012f:14027)
- Arend Bayer, Emanuele Macrì, and Yukinobu Toda. Bridgeland stability conditions on threefolds I: Bogomolov-Gieseker type inequalities, 2011. arXiv:1103.5010.
- Tom Bridgeland, Stability conditions on triangulated categories, Ann. of Math. (2) 166 (2007), no. 2, 317–345. MR 2373143 (2009c:14026), DOI https://doi.org/10.4007/annals.2007.166.317
- Lawrence Ein and Robert Lazarsfeld, Global generation of pluricanonical and adjoint linear series on smooth projective threefolds, J. Amer. Math. Soc. 6 (1993), no. 4, 875–903. MR 1207013 (94c:14016), DOI https://doi.org/10.2307/2152744
- Takao Fujita. Remarks on Ein-Lazarsfeld criterion of spannedness of adjoint bundles of polarized threefolds, 1993. arXiv:alg-geom/9311013.
- Takao Fujita. Towards a separation theorem of points by adjoint linear systems on polarized threefolds, 1994. arXiv:alg-geom/9411001.
- Stefan Helmke, On Fujita’s conjecture, Duke Math. J. 88 (1997), no. 2, 201–216. MR 1455517 (99e:14003), DOI https://doi.org/10.1215/S0012-7094-97-08807-4
- Dieter Happel, Idun Reiten, and Sverre O. Smalø, Tilting in abelian categories and quasitilted algebras, Mem. Amer. Math. Soc. 120 (1996), no. 575, viii+ 88. MR 1327209 (97j:16009)
- Yujiro Kawamata, On Fujita’s freeness conjecture for $3$-folds and $4$-folds, Math. Ann. 308 (1997), no. 3, 491–505. MR 1457742 (99c:14008), DOI https://doi.org/10.1007/s002080050085
- Robert Lazarsfeld, Positivity in algebraic geometry. II: Positivity for vector bundles, and multiplier ideals, 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. MR 2095472 (2005k:14001b)
- Emanuele Macrì. A generalized Bogomolov-Gieseker inequality for the three-dimensional projective space, 2012. arXiv:1207.4980.
- Igor Reider, Vector bundles of rank $2$ and linear systems on algebraic surfaces, Ann. of Math. (2) 127 (1988), no. 2, 309–316. MR 932299 (89e:14038), DOI https://doi.org/10.2307/2007055
Additional Information
Arend Bayer
Affiliation:
Department of Mathematics, University of Connecticut U-3009, 196 Auditorium Road, Storrs, Connecticut 06269-3009
MR Author ID:
728427
Email:
bayer@math.uconn.edu
Aaron Bertram
Affiliation:
Department of Mathematics, University of Utah, 155 S 1400 E, Salt Lake City, Utah 84112
MR Author ID:
246391
Email:
bertram@math.utah.edu
Emanuele Macrì
Affiliation:
Mathematical Institute, University of Bonn, Endenicher Allee 60, D-53115 Bonn, Germany; and Department of Mathematics, University of Utah, 155 S 1400 E, Salt Lake City, Utah 84112
Address at time of publication:
Department of Mathematics, The Ohio State University, 231 W 18th Avenue, Columbus, Ohio 43210
Email:
macri.6@math.osu.edu
Yukinobu Toda
Affiliation:
Institute for the Physics and Mathematics of the Universe, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan
Email:
yukinobu.toda@ipmu.jp
Received by editor(s):
June 23, 2011
Published electronically:
January 28, 2014
Additional Notes:
The first author was partially supported by NSF grant DMS-0801356/DMS-1001056
The second author was partially supported by NSF grant DMS-0901128
The third author was partially supported by NSF grant DMS-1001482/DMS-1160466, Hausdorff Center for Mathematics, Bonn, and by SFB/TR 45
The fourth author was supported by World Premier International Research Center Initiative (WPI initiative), MEXT, Japan, and Grant-in AId for Scientific Research grant (22684002), partly (S-19104002), from the Ministry of Education, Culture, Sports, Science and Technology, Japan
Article copyright:
© Copyright 2014
University Press, Inc.
