Degrés d’homogénéité de l’ensemble des intersections complètes singulières  [ Homogeneity degrees of the set of singular complete intersections ]
Annales de l'Institut Fourier, Volume 62 (2012) no. 3, p. 1189-1214
A classical result of Boole shows that, in characteristic 0, the set of singular degree d hypersurfaces in N is a divisor of degree (N+1)(d-1) N in the projective space of all hypersurfaces. We give here analogous formulae for complete intersections in N of arbitrary codimension and degrees, in any characteristic.
Un résultat classique de Boole montre que, sur un corps de caractéristique 0, l’ensemble des hypersurfaces singulières de degré d dans N est un diviseur de degré (N+1)(d-1) N de l’espace projectif de toutes les hypersurfaces. On obtient ici des formules analogues pour des intersections complètes de codimension et de degrés quelconques dans N , en toute caractéristique.
DOI : https://doi.org/10.5802/aif.2720
Classification:  14M10,  14N05,  14M25
Keywords: Complete intersections, projective duality, toric varieties, finite characteristic
@article{AIF_2012__62_3_1189_0,
     author = {Benoist, Olivier},
     title = {Degr\'es d'homog\'en\'eit\'e de l'ensemble des intersections compl\`etes singuli\`eres},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {62},
     number = {3},
     year = {2012},
     pages = {1189-1214},
     doi = {10.5802/aif.2720},
     mrnumber = {3013820},
     zbl = {1254.14061},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_2012__62_3_1189_0}
}
Benoist, Olivier. Degrés d’homogénéité de l’ensemble des intersections complètes singulières. Annales de l'Institut Fourier, Volume 62 (2012) no. 3, pp. 1189-1214. doi : 10.5802/aif.2720. http://www.numdam.org/item/AIF_2012__62_3_1189_0/

[1] Brion, Michel; Vergne, Michèle An equivariant Riemann-Roch theorem for complete, simplicial toric varieties, J. Reine Angew. Math., Tome 482 (1997), pp. 67-92 | MR 1427657 | Zbl 0862.14006

[2] Deligne, P.; Katz, N. Groupes de monodromie en géométrie algébrique. II, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Vol. 340 (1973) (Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II)) | MR 354657

[3] GelʼFand, I. M.; Kapranov, M. M.; Zelevinsky, A. V. Discriminants, resultants, and multidimensional determinants, Birkhäuser Boston Inc., Boston, MA, Mathematics : Theory & Applications (1994) | Article | MR 1264417 | Zbl 0827.14036

[4] Kleiman, Steven L. Tangency and duality, Proceedings of the 1984 Vancouver conference in algebraic geometry, Amer. Math. Soc., Providence, RI (CMS Conf. Proc.) Tome 6 (1986), pp. 163-225 | MR 846021 | Zbl 0601.14046

[5] Miller, Ezra; Sturmfels, Bernd Combinatorial commutative algebra, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 227 (2005) | MR 2110098

[6] Oda, Tadao Convex bodies and algebraic geometry, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Tome 15 (1988) (An introduction to the theory of toric varieties, Translated from the Japanese) | MR 922894 | Zbl 0628.52002

[7] Prasolov, Victor V. Polynomials, Springer-Verlag, Berlin, Algorithms and Computation in Mathematics, Tome 11 (2004) (Translated from the 2001 Russian second edition by Dimitry Leites) | MR 2082772

[8] Wallace, Andrew H. Tangency and duality over arbitrary fields, Proc. London Math. Soc. (3), Tome 6 (1956), pp. 321-342 | Article | MR 80354 | Zbl 0072.16002