Quasismooth hypersurfaces in toric varieties

Artebani, Michela; Comparin, Paola; Guilbot, Robin

doi:10.1090/proc/14515

Quasismooth hypersurfaces in toric varieties
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by Michela Artebani, Paola Comparin and Robin Guilbot PDF

Proc. Amer. Math. Soc. 147 (2019), 4565-4579 Request permission

Abstract:

We provide a combinatorial characterization of monomial linear systems on toric varieties whose general member is quasismooth. This is given both in terms of the Newton polytope and in terms of the matrix of exponents of a monomial basis.

References

Michela Artebani, Paola Comparin, and Robin Guilbot, Families of Calabi-Yau hypersurfaces in $\Bbb {Q}$-Fano toric varieties, J. Math. Pures Appl. (9) 106 (2016), no. 2, 319–341 (English, with English and French summaries). MR 3515305, DOI 10.1016/j.matpur.2016.02.012
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
Per Berglund and Tristan Hübsch, A generalized construction of mirror manifolds, Nuclear Phys. B 393 (1993), no. 1-2, 377–391. MR 1214325, DOI 10.1016/0550-3213(93)90250-S
Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
Jean-François Boutot, Singularités rationnelles et quotients par les groupes réductifs, Invent. Math. 88 (1987), no. 1, 65–68 (French). MR 877006, DOI 10.1007/BF01405091
Weronika Buczyńska, Toryczne przestrzenie rzutowe, Magister thesis (2002), text in Polish, English transl.: arXiv:0805.1211, 2008.
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, de Rham complex on toroidal variety, Algebraic geometry (Chicago, IL, 1989) Lecture Notes in Math., vol. 1479, Springer, Berlin, 1991, pp. 26–38. MR 1181204, DOI 10.1007/BFb0086261
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
Claus Hertling and Ralf Kurbel, On the classification of quasihomogeneous singularities, J. Singul. 4 (2012), 131–153. MR 3044491, DOI 10.5427/jsing.2012.4h
A. R. Iano-Fletcher, Working with weighted complete intersections, Explicit birational geometry of 3-folds, London Math. Soc. Lecture Note Ser., vol. 281, Cambridge Univ. Press, Cambridge, 2000, pp. 101–173. MR 1798982
A. G. Khovanskiĭ, Newton polyhedra (resolution of singularities), Current problems in mathematics, Vol. 22, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1983, pp. 207–239 (Russian). MR 735444
A. G. Kouchnirenko, Polyèdres de Newton et nombres de Milnor, Invent. Math. 32 (1976), no. 1, 1–31 (French). MR 419433, DOI 10.1007/BF01389769
Anatoli G. Kouchnirenko, Criteria for the existence of a non-degenerate quasihomogeneous function with given weights (in Russian), Usp. Mat. Nauk 32 (1977), no. 3, 169–170.
Marc Krawitz, FJRW rings and Landau-Ginzburg mirror symmetry, ProQuest LLC, Ann Arbor, MI, 2010. Thesis (Ph.D.)–University of Michigan. MR 2801653
Maximilian Kreuzer and Harald Skarke, On the classification of quasihomogeneous functions, Comm. Math. Phys. 150 (1992), no. 1, 137–147. MR 1188500, DOI 10.1007/BF02096569