Continuité des racines d’après Rabinoff

Emeryck Marie

doi:10.5802/crmath.439

Algebraic geometry

Continuité des racines d’après Rabinoff
[Continuity of roots after Rabinoff]

Emeryck Marie ¹

¹ Technische Universität Chemnitz, Fakultät für Mathematik, Reichenhainer Straße 39, 09126 Chemnitz, Germany

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 685-696.

Abstract
Résumé

The content of this paper is a generalization of a theorem by Joseph Rabinoff: if $𝒫$ is a finite family of pointed and rational polyhedra in $N_{ℝ}$ such that there exists a fan in $N_{ℝ}$ that contains all the recession cones of the polyhedra of $𝒫$ , if $k$ is a complete non-archimedean field, if $S$ is a connected and regular $k$ -analytic space (in the sense of Berkovich) and $Y$ is a closed $k$ -analytic subset of $U_{𝒫} \times_{k} S$ which is relative complete intersection and contained in the relative interior of $U_{𝒫} \times_{k} S$ over $S$ , then the quasifiniteness of $π : Y \to S$ implies its flatness and finiteness; moreover, all the finite fibers of $π$ have the same length. This namely gives a analytic justification to the concept of stable intersection used in the theory of tropical intersection.

Ce papier propose une généralisation d’un théorème de Joseph Rabinoff : si $𝒫$ est une famille finie de polyèdres rationnels pointés dans $N_{ℝ}$ telle qu’il existe un éventail dans $N_{ℝ}$ contenant tous les cônes de récession des polyèdres de $𝒫$ , si $k$ est un corps non archimédien complet, si $S$ est un espace $k$ -analytique (au sens de Berkovich) régulier connexe et $Y$ un fermé $k$ -analytique de $U_{𝒫} \times_{k} S$ de complète intersection relative et contenu dans l’intérieur relatif de $U_{𝒫} \times_{k} S$ au-dessus de $S$ , alors la quasi-finitude du morphisme $π : Y \to S$ implique sa platitude et sa finitude. De plus, toutes les fibres finies de $π$ ont la même longueur. Cela fournit notamment une justification analytique au concept d’intersection stable utilisé en théorie de l’intersection tropicale.

Received: 2021-11-24
Revised: 2022-08-01
Accepted: 2022-10-19
Published online: 2023-03-02

DOI: 10.5802/crmath.439

Author's affiliations:

Emeryck Marie ¹

¹ Technische Universität Chemnitz, Fakultät für Mathematik, Reichenhainer Straße 39, 09126 Chemnitz, Germany

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{CRMATH_2023__361_G3_685_0,
     author = {Emeryck Marie},
     title = {Continuit\'e des racines d{\textquoteright}apr\`es {Rabinoff}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {685--696},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.439},
     language = {fr},
}

TY  - JOUR
AU  - Emeryck Marie
TI  - Continuité des racines d’après Rabinoff
JO  - Comptes Rendus. Mathématique
PY  - 2023
SP  - 685
EP  - 696
VL  - 361
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.439
LA  - fr
ID  - CRMATH_2023__361_G3_685_0
ER  -

%0 Journal Article
%A Emeryck Marie
%T Continuité des racines d’après Rabinoff
%J Comptes Rendus. Mathématique
%D 2023
%P 685-696
%V 361
%I Académie des sciences, Paris
%R 10.5802/crmath.439
%G fr
%F CRMATH_2023__361_G3_685_0

Emeryck Marie. Continuité des racines d’après Rabinoff. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 685-696. doi : 10.5802/crmath.439. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.439/

References
Cited by

[1] Vladimir G. Berkovich Spectral theory and analytic geometry over non-archimedean fields, Mathematical Surveys and Monographs, 33, American Mathematical Society, 1990 | MR | Zbl

[2] Vladimir G. Berkovich Étale cohomology for non-archimedean analytic spaces, Publ. Math., Inst. Hautes Étud. Sci., Volume 78 (1993), pp. 5-161 | DOI | Numdam | MR | Zbl

[3] Vladimir I. Danilov The geometry of toric varieties, Russ. Math. Surv., Volume 33 (1978), pp. 97-154 | DOI | MR | Zbl

[4] Antoine Ducros Réduction en famille d’espaces affinoïdes (prépublication, https://webusers.imj-prg.fr/~antoine.ducros/red-aff.pdf)

[5] Antoine Ducros Variation de la dimension relative en géométrie analytique $p$ -adique, Compos. Math., Volume 143 (2007) no. 6, pp. 1511-1532 | DOI | MR | Zbl

[6] Antoine Ducros Les espaces de Berkovich sont excellents, Ann. Inst. Fourier, Volume 59 (2009) no. 4, pp. 1407-1516 | Numdam | MR | Zbl

[7] Antoine Ducros Families of Berkovich spaces, Astérisque, 400, Société Mathématique de France, 2018

[8] William Fulton Introduction to toric varieties, Annals of Mathematics Studies, Princeton University Press, 1993 no. 131 | DOI

[9] Takeshi Kajiwara Tropical toric geometry, Toric topology (Contemporary Mathematics), Volume 460, American Mathematical Society, 2008, pp. 197-207 | DOI | MR | Zbl

[10] Hideyuki Matsumura Commutative ring theory, Cambridge Studies in Advanced Mathematics, 8, Cambridge University Press, 1987 | DOI

[11] Brian Osserman; Sam Payne Lifting tropical intersections, Doc. Math., Volume 18 (2013), pp. 121-176 | DOI | MR | Zbl

[12] Sam Payne Analytification is the limit of all tropicalizations, Math. Res. Lett., Volume 16 (2009) no. 2-3, pp. 543-556 | DOI | MR | Zbl

[13] Joseph Rabinoff Tropical analytic geometry, Newton polygons and tropical interesections, Adv. Math., Volume 229 (2012) no. 6, pp. 3192-3255 | DOI | MR | Zbl

[14] Jürgen Richter-Gebert; Bernd Sturmfels; Thorsten Theobald First steps in tropical geometry, Idempotent mathematics and mathematical physics (Contemporary Mathematics), Volume 377, American Mathematical Society, 2005, pp. 289-317 | DOI | MR | Zbl

[15] Michael Temkin On local properties of non-archimedean spaces I, Ann. Math., Volume 318 (2000) no. 3, pp. 585-607 | DOI | MR

Cited by Sources:

Comments - Policy