Algebraic hyperbolicity for surfaces in toric threefolds

Haase, Christian; Ilten, Nathan

doi:10.1090/jag/770

Most recent issue | All issues | Recently published articles

Algebraic hyperbolicity for surfaces in toric threefolds

Authors: Christian Haase and Nathan Ilten
Journal: J. Algebraic Geom.
DOI: https://doi.org/10.1090/jag/770
Published electronically: January 14, 2021
Full-text PDF

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.

[B1] Robert Brody, Compact manifolds and hyperbolicity, Trans. Amer. Math. Soc. 235 (1978), 213–219. MR 470252, https://doi.org/10.1090/S0002-9947-1978-0470252-3
[B2] Damian Brotbek, On the hyperbolicity of general hypersurfaces, Publ. Math. Inst. Hautes Études Sci. 126 (2017), 1–34. MR 3735863, https://doi.org/10.1007/s10240-017-0090-3
[BGNVT] Winfried Bruns, Joseph Gubeladze, and Ngô Viêt Trung, Normal polytopes, triangulations, and Koszul algebras, J. Reine Angew. Math. 485 (1997), 123–160. MR 1442191
[BG1] Ugo Bruzzo and Antonella Grassi, Picard group of hypersurfaces in toric 3-folds, Internat. J. Math. 23 (2012), no. 2, 1250028, 14. MR 2890472, https://doi.org/10.1142/S0129167X12500280
[BG2] 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, https://doi.org/10.1142/S0219199717500705
[CC] Luca Chiantini and Ciro Ciliberto, A few remarks on the lifting problem, Astérisque 218 (1993), 95–109. Journées de Géométrie Algébrique d’Orsay (Orsay, 1992). MR 1265310
[CL] 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, https://doi.org/10.1090/S0002-9939-99-05407-6
[CR1] Izzet Coskun and Eric Riedl, Algebraic hyperbolicity of the very general quintic surface in ℙ³, Adv. Math. 350 (2019), 1314–1323. MR 3949983, https://doi.org/10.1016/j.aim.2019.04.062
[CR2] Izzet Coskun and Eric Riedl, Algebraic Hyperbolicity of Very General Surfaces, arXiv:1912:07689 (2019).
[CLS] 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
[DK] 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
[D] 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, https://doi.org/10.1090/pspum/062.2/1492539
[DS] Persi Diaconis and Bernd Sturmfels, Algebraic algorithms for sampling from conditional distributions, Ann. Statist. 26 (1998), no. 1, 363–397. MR 1608156, https://doi.org/10.1214/aos/1030563990
[DH] 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, https://doi.org/10.1090/S0002-9947-1988-0961600-2
[E] Lawrence Ein, Subvarieties of generic complete intersections, Invent. Math. 94 (1988), no. 1, 163-169., https://doi.org/10.1007/BF01394349
[EW] 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, https://doi.org/10.1007/BF03323286
[F] 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
[G1] 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
[G2] 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
[G3] 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
[HH] Christian Haase and Jan Hofmann, Convex-normal (pairs of) polytopes, Canad. Math. Bull. 60 (2017), no. 3, 510–521. MR 3679726, https://doi.org/10.4153/CMB-2016-057-0
[HPPS] Christian Haase, Andreas Paffenholz, Lindsay C. Piechnik, and Francisco Santos, Existence of unimodular triangulations-positive results, Mem. Amer. Math. Soc. 270 (2021).
[H1] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
[H2] Robin Hartshorne, Deformation theory, Graduate Texts in Mathematics, vol. 257, Springer, New York, 2010. MR 2583634
[I] Shigeru Iitaka, Algebraic geometry, Graduate Texts in Mathematics, vol. 76, Springer-Verlag, New York-Berlin, 1982. An introduction to birational geometry of algebraic varieties; North-Holland Mathematical Library, 24. MR 637060
[LTZ] 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, https://doi.org/10.1007/BF03322309
[KL] G. Kempf and D. Laksov, The determinantal formula of Schubert calculus, Acta Math. 132 (1974), 153–162. MR 338006, https://doi.org/10.1007/BF02392111
[P] 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
[S] Bernd Sturmfels, Gröbner bases and convex polytopes, University Lecture Series, vol. 8, American Mathematical Society, Providence, RI, 1996. MR 1363949
[X] 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
Email: haase@math.fu-berlin.de

Nathan Ilten
Affiliation: Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, British Columbia V5A 1S6, Canada
Email: nilten@sfu.ca

DOI: https://doi.org/10.1090/jag/770
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