Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field
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 Berti...
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2015
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| Acceso en línea: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10715797_v31_n_p42_Cafure http://hdl.handle.net/20.500.12110/paper_10715797_v31_n_p42_Cafure |
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paper:paper_10715797_v31_n_p42_Cafure2023-06-08T16:04:45Z Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field 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 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. 2015 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10715797_v31_n_p42_Cafure http://hdl.handle.net/20.500.12110/paper_10715797_v31_n_p42_Cafure |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-134 |
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
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 |
| spellingShingle |
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 Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field |
| topic_facet |
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 |
| description |
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. |
| title |
Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field |
| title_short |
Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field |
| title_full |
Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field |
| title_fullStr |
Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field |
| title_full_unstemmed |
Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field |
| title_sort |
polar varieties, bertini's theorems and number of points of singular complete intersections over a finite field |
| publishDate |
2015 |
| url |
https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10715797_v31_n_p42_Cafure http://hdl.handle.net/20.500.12110/paper_10715797_v31_n_p42_Cafure |
| _version_ |
1768542851911647232 |