Algebraic hyperbolicity for surfaces in toric threefolds
Authors:
Christian Haase and Nathan Ilten
Journal:
J. Algebraic Geom. 30 (2021), 573-602
DOI:
https://doi.org/10.1090/jag/770
Published electronically:
January 14, 2021
MathSciNet review:
4283552
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Abstract |
References |
Additional Information
Abstract: Adapting focal loci techniques used by Chiantini and Lopez, we provide lower bounds on the genera of curves contained in very general surfaces in Gorenstein toric threefolds. We illustrate the utility of these bounds by obtaining results on algebraic hyperbolicity of very general surfaces in toric threefolds.
References
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- Christian Haase and Jan Hofmann, Convex-normal (pairs of) polytopes, Canad. Math. Bull. 60 (2017), no. 3, 510–521. MR 3679726, DOI 10.4153/CMB-2016-057-0
- Christian Haase, Andreas Paffenholz, Lindsay C. Piechnik, and Francisco Santos, Existence of unimodular triangulations-positive results, Mem. Amer. Math. Soc. 270 (2021).
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
- Robin Hartshorne, Deformation theory, Graduate Texts in Mathematics, vol. 257, Springer, New York, 2010. MR 2583634, DOI 10.1007/978-1-4419-1596-2
- Shigeru Iitaka, Algebraic geometry, North-Holland Mathematical Library, vol. 24, Springer-Verlag, New York-Berlin, 1982. An introduction to birational geometry of algebraic varieties. MR 637060, DOI 10.1007/978-1-4613-8119-8
- Ji Yong Liu, Leslie E. Trotter Jr., and Günter M. Ziegler, On the height of the minimal Hilbert basis, Results Math. 23 (1993), no. 3-4, 374–376. MR 1215222, DOI 10.1007/BF03322309
- G. Kempf and D. Laksov, The determinantal formula of Schubert calculus, Acta Math. 132 (1974), 153–162. MR 338006, DOI 10.1007/BF02392111
- I. R. Porteous, Simple singularities of maps, Proceedings of Liverpool Singularities Symposium, I (1969/70), Springer, Berlin, 1971, pp. 286–307. Lecture Notes in Math., Vol. 192. MR 0293646
- Bernd Sturmfels, Gröbner bases and convex polytopes, University Lecture Series, vol. 8, American Mathematical Society, Providence, RI, 1996. MR 1363949, DOI 10.1090/ulect/008
- Geng Xu, Subvarieties of general hypersurfaces in projective space, J. Differential Geom. 39 (1994), no. 1, 139–172. MR 1258918
References
- Robert Brody, Compact manifolds and hyperbolicity, Trans. Amer. Math. Soc. 235 (1978), 213–219. MR 470252, DOI 10.2307/1998216
- Damian Brotbek, On the hyperbolicity of general hypersurfaces, Publ. Math. Inst. Hautes Études Sci. 126 (2017), 1–34. MR 3735863, DOI 10.1007/s10240-017-0090-3
- Winfried Bruns, Joseph Gubeladze, and Ngô Viêt Trung, Normal polytopes, triangulations, and Koszul algebras, J. Reine Angew. Math. 485 (1997), 123–160. MR 1442191
- Ugo Bruzzo and Antonella Grassi, Picard group of hypersurfaces in toric 3-folds, Internat. J. Math. 23 (2012), no. 2, 1250028, 14. MR 2890472, DOI 10.1142/S0129167X12500280
- Ugo Bruzzo and Antonella Grassi, The Noether-Lefschetz locus of surfaces in toric threefolds, Commun. Contemp. Math. 20 (2018), no. 5, 1750070, 20. MR 3833909, DOI 10.1142/S0219199717500705
- Luca Chiantini and Ciro Ciliberto, A few remarks on the lifting problem, Journées de Géométrie Algébrique d’Orsay (Orsay, 1992), Astérisque 218 (1993), 95–109. MR 1265310
- Luca Chiantini and Angelo Felice Lopez, Focal loci of families and the genus of curves on surfaces, Proc. Amer. Math. Soc. 127 (1999), no. 12, 3451–3459. MR 1676295, DOI 10.1090/S0002-9939-99-05407-6
- Izzet Coskun and Eric Riedl, Algebraic hyperbolicity of the very general quintic surface in $\mathbb {P}^3$, Adv. Math. 350 (2019), 1314–1323. MR 3949983, DOI 10.1016/j.aim.2019.04.062
- Izzet Coskun and Eric Riedl, Algebraic Hyperbolicity of Very General Surfaces, arXiv:1912:07689 (2019).
- David A. Cox, John B. Little, and Henry K. Schenck, Toric varieties, Graduate Studies in Mathematics, vol. 124, American Mathematical Society, Providence, RI, 2011. MR 2810322, DOI 10.1090/gsm/124
- V. I. Danilov and A. G. Khovanskiĭ, Newton polyhedra and an algorithm for calculating Hodge-Deligne numbers, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 5, 925–945 (Russian). MR 873655
- Jean-Pierre Demailly, Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials, Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 285–360. MR 1492539, DOI 10.1090/pspum/062.2/1492539
- Persi Diaconis and Bernd Sturmfels, Algebraic algorithms for sampling from conditional distributions, Ann. Statist. 26 (1998), no. 1, 363–397. MR 1608156, DOI 10.1214/aos/1030563990
- Steven Diaz and Joe Harris, Ideals associated to deformations of singular plane curves, Trans. Amer. Math. Soc. 309 (1988), no. 2, 433–468. MR 961600, DOI 10.2307/2000919
- Lawrence Ein, Subvarieties of generic complete intersections, Invent. Math. 94 (1988), no. 1, 163–169., DOI 10.1007/BF01394349
- Günter Ewald and Uwe Wessels, On the ampleness of invertible sheaves in complete projective toric varieties, Results Math. 19 (1991), no. 3-4, 275–278. MR 1100674, DOI 10.1007/BF03323286
- William Fulton, Introduction to toric varieties, The William H. Roever Lectures in Geometry, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. MR 1234037, DOI 10.1515/9781400882526
- A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I, Inst. Hautes Études Sci. Publ. Math. 20 (1964), 259 (French). MR 173675
- A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Inst. Hautes Études Sci. Publ. Math. 28 (1966), 255. MR 217086
- A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Inst. Hautes Études Sci. Publ. Math. 32 (1967), 361 (French). MR 238860
- Christian Haase and Jan Hofmann, Convex-normal (pairs of) polytopes, Canad. Math. Bull. 60 (2017), no. 3, 510–521. MR 3679726, DOI 10.4153/CMB-2016-057-0
- Christian Haase, Andreas Paffenholz, Lindsay C. Piechnik, and Francisco Santos, Existence of unimodular triangulations-positive results, Mem. Amer. Math. Soc. 270 (2021).
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
- Robin Hartshorne, Deformation theory, Graduate Texts in Mathematics, vol. 257, Springer, New York, 2010. MR 2583634, DOI 10.1007/978-1-4419-1596-2
- Shigeru Iitaka, Algebraic geometry: An introduction to birational geometry of algebraic varieties, Graduate Texts in Mathematics, vol. 76, North-Holland Mathematical Library, 24, Springer-Verlag, New York-Berlin, 1982. MR 637060
- Ji Yong Liu, Leslie E. Trotter Jr., and Günter M. Ziegler, On the height of the minimal Hilbert basis, Results Math. 23 (1993), no. 3-4, 374–376. MR 1215222, DOI 10.1007/BF03322309
- G. Kempf and D. Laksov, The determinantal formula of Schubert calculus, Acta Math. 132 (1974), 153–162. MR 338006, DOI 10.1007/BF02392111
- I. R. Porteous, Simple singularities of maps, Proceedings of Liverpool Singularities Symposium, I (1969/70), Lecture Notes in Math., Vol. 192, Springer, Berlin, 1971, pp. 286–307. MR 0293646
- Bernd Sturmfels, Gröbner bases and convex polytopes, University Lecture Series, vol. 8, American Mathematical Society, Providence, RI, 1996. MR 1363949
- Geng Xu, Subvarieties of general hypersurfaces in projective space, J. Differential Geom. 39 (1994), no. 1, 139–172. MR 1258918
Additional Information
Christian Haase
Affiliation:
Institut für Mathematik, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany
MR Author ID:
661101
ORCID:
0000-0003-4078-0913
Email:
haase@math.fu-berlin.de
Nathan Ilten
Affiliation:
Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, British Columbia V5A 1S6, Canada
MR Author ID:
864815
Email:
nilten@sfu.ca
Received by editor(s):
May 9, 2019
Received by editor(s) in revised form:
March 5, 2020
Published electronically:
January 14, 2021
Additional Notes:
The work of the first author was partially supported by the grant HA 4383/8-1 of the German Research Foundation DFG. The work of the second author was partially supported by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund
Article copyright:
© Copyright 2021
University Press, Inc.