Density of quasismooth hypersurfaces in simplicial toric varieties
Journal de Théorie des Nombres de Bordeaux, Tome 29 (2017) no. 1, pp. 261-288.

Cet article a pour objet la densité des hypersurfaces dans une variété torique projective, normale et simpliciale sur un corps fini ayant une intersection quasi-lisse avec un sous-schéma quasi-lisse fixé. Le résultat géneralise la formule trouvée par B. Poonen pour des variétés projectives lisses. Comme application, nous analysons en outre la densité des hypersurfaces dont le nombre des singularités et la longueur du schéma singulier sont bornés.

This paper investigates the density of hypersurfaces in a projective normal simplicial toric variety over a finite field having a quasismooth intersection with a given quasismooth subscheme. The result generalizes the formula found by B. Poonen for smooth projective varieties. As an application, we further analyze the density of hypersurfaces with bounds on their number of singularities and on the length of their singular schemes.

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DOI : https://doi.org/10.5802/jtnb.979
Classification : 14J70,  11G25,  14G15,  14M25
Mots clés : Bertini theorems, finite fields, toric varieties
@article{JTNB_2017__29_1_261_0,
     author = {Niels Lindner},
     title = {Density of quasismooth hypersurfaces in simplicial toric varieties},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {261--288},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {29},
     number = {1},
     year = {2017},
     doi = {10.5802/jtnb.979},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.979/}
}
Niels Lindner. Density of quasismooth hypersurfaces in simplicial toric varieties. Journal de Théorie des Nombres de Bordeaux, Tome 29 (2017) no. 1, pp. 261-288. doi : 10.5802/jtnb.979. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.979/

[1] Victor V. Batyrev; David A. Cox On the Hodge structure of projective hypersurfaces in toric varieties, Duke Math. J., Volume 75 (1994) no. 2, pp. 293-338 | Article

[2] Yousra Boubakri; Gert-Martin Greuel; Thomas Markwig Invariants of hypersurface singularities in positive characteristic, Rev. Mat. Complut., Volume 25 (2012) no. 1, pp. 61-85 | Article

[3] Ivan Cheltsov Factorial threefold hypersurfaces, J. Algebraic Geom., Volume 19 (2010) no. 4, pp. 781-791 | Article

[4] David A. Cox The homogeneous coordinate ring of a toric variety, J. Algebraic Geom., Volume 4 (1995) no. 1, pp. 17-50 (erratum J. Algebraic Geom. 23, no. 2, 393-398 (2014))

[5] David A. Cox; John B. Little; Henry K. Schenck Toric varieties, Graduate Studies in Mathematics, Volume 124, American Mathematical Society, Providence, 2011, xxiv+841 pages

[6] David Eisenbud; Melvin Hochster A Nullstellensatz with nilpotents and Zariski’s Main Lemma on holomorphic functions, J. Algebra, Volume 58 (1979), pp. 157-161 | Article

[7] Daniel Erman; Melanie Matchett Wood Semiample Bertini theorems over finite fields, Duke Math. J, Volume 164 (2015) no. 1, pp. 1-38 | Article

[8] Robin Hartshorne Algebraic geometry, Graduate Texts in Mathematics, Volume 52, Springer, 1977, xvi+496 pages

[9] Serge Lang; André Weil Number of points of varieties in finite fields, Amer. J. Math, Volume 76 (1954), pp. 819-827 | Article

[10] D. Mumford Varieties defined by quadratic equations, Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969) (1970), pp. 29-100

[11] Bjorn Poonen Bertini theorems over finite fields, Ann. Math., Volume 160 (2005) no. 3, pp. 1099-1127 | Article

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