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Abstract:

Let V ⊂ double-struck Pn(double-struck F¯q) be a complete intersection defined over a finite field double-struck Fq of dimension r and singular locus of dimension at most s, and let π :V → double-struck Ps+1 (double-struck F¯q) be a generic linear mapping. We obtain an effective version of the Bertini smoothness theorem concerning π, namely an explicit upper bound of the degree of a proper Zariski closed subset of double-struck Ps+1(double-struck F¯q) which contains all the points defining singular fibers of π. For this purpose we make use of the concept of polar variety associated with the set of exceptional points of π. As a consequence, we obtain results of existence of smooth rational points of V, that is, conditions on q which imply that V has a smooth double-struck Fq-rational point. Finally, for s = r - 2 and s = r - 3 we estimate the number of double-struck Fq-rational points and smooth double-struck Fq-rational points of V. © 2014 Elsevier Inc. All rights reserved.

Registro:

Documento: Artículo
Título:Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field
Autor:Cafure, A.; Matera, G.; Privitelli, M.
Filiación:Ciclo Básico Común, Universidad de Buenos Aires, Ciudad Universitaria, Buenos Aires, 1428, Argentina
Instituto del Desarrollo Humano, Universidad Nacional de General Sarmiento, J.M. Gutiérrez 1150, Los Polvorines, Buenos Aires, B1613GSX, Argentina
National Council of Science and Technology (CONICET), Argentina
Instituto de Ciencias, Universidad Nacional de General Sarmiento, J.M. Gutiérrez 1150, Los Polvorines, Buenos Aires, B1613GSX, Argentina
Palabras clave:Bertini smoothness theorem; Deligne estimate; Hooley-Katz estimate; Multihomogeneous Bézout theorem; Polar varieties; Rational points; Singular locus; Varieties over finite fields; Bertini; Deligne estimate; Finite fields; Hooley-Katz estimate; Polar varieties; Rational points; Singular locus
Año:2015
Volumen:31
Página de inicio:42
Página de fin:83
DOI: http://dx.doi.org/10.1016/j.ffa.2014.09.002
Título revista:Finite Fields and their Applications
Título revista abreviado:Finite Fields Appl.
ISSN:10715797
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10715797_v31_n_p42_Cafure

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Citas:

---------- APA ----------
Cafure, A., Matera, G. & Privitelli, M. (2015) . Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field. Finite Fields and their Applications, 31, 42-83.
http://dx.doi.org/10.1016/j.ffa.2014.09.002
---------- CHICAGO ----------
Cafure, A., Matera, G., Privitelli, M. "Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field" . Finite Fields and their Applications 31 (2015) : 42-83.
http://dx.doi.org/10.1016/j.ffa.2014.09.002
---------- MLA ----------
Cafure, A., Matera, G., Privitelli, M. "Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field" . Finite Fields and their Applications, vol. 31, 2015, pp. 42-83.
http://dx.doi.org/10.1016/j.ffa.2014.09.002
---------- VANCOUVER ----------
Cafure, A., Matera, G., Privitelli, M. Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field. Finite Fields Appl. 2015;31:42-83.
http://dx.doi.org/10.1016/j.ffa.2014.09.002