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Raphaël Achet

Teaching Assistant

Field of research:
Algebraic Geometry

Institut Fourier, UMR 5582 du CNRS
Université Grenoble Alpes
CS 40700
38058 Grenoble cedex 09

E-mail :

I defended my thesis in September 2017. I am currently a teaching assistant at Grenoble Alpes University.
Picard Group of unipotent groups over a field
Ph.D. advisor
Date of defence
25 September 2017
Let k be any field. We study the Picard group of the (smooth connected) unipotent k-algebraic groups.

As every unipotent algebraic group is an iterated extension of forms of the additive group, we will study the Picard group of the forms of the additive group. In fact we study the Picard group of forms of the additive group and the affine line simultaneously using a geometric method. We obtain an explicit upper bound on the torsion of the Picard group of the forms of the affine line and their regular completion, and a sufficient condition for the Picard group of a form of the affine line to be nontrivial. We also give examples of nontrivial forms of the affine line with trivial Picard groups.

In general, a unipotent k-algebraic group is a form of the affine n-space. In order to study the Picard group of a form X of the affine n-space with a geometric method, we define a "restricted" Picard functor; we show that if X admits a regular completion then the "restricted" Picard functor is representable by a unipotent k-algebraic group (smooth, not necessarily connected). With this "restricted" Picard functor and geometric arguments we show that the Picard group of a unirational form of the affine n-space is finite. Moreover we generalise a result of B. Totaro: if k is separably closed and if the Picard group of a unipotent k-algebraic group is nontrivial then it admits a nontrivial extension by the multiplicative group.

Picard group of the forms of the affine line and of the additive group, Journal of Pure and Applied Algebra Volume 221, Issue 11, November 2017, Pages 2838-2860 (Science Direct link).
My dissertation with Bertrand RÉMY on unipotents algebraic groups (in French).
My CV in pdf.
Notes of a lecture by Guido Pezzini entitled "Geometry and combinatorics of spherical varieties".
Oral presentations
Newcastle University, Workshop on pseudo-reductive groups and related topics.
Rennes University, seminar algebra and geometry.
Lille University, seminar algebraic geometry.
Bourgogne University, "Mini-Workshop Algebraic Geometry".
ESI Vienne, Winter school "Geometry and Representation Theory" (5min presentation).
Institut Fourier, seminar algebra and geometry.
ENS, seminar rational varieties.
Teaching Assistant at Grenoble Alpes university.
Ph.D. at the Fourier Institute under the supervision of Michel Brion.
Studies in mathematics at E.N.S. Lyon, speciality in algebraic geometry and number theory.
Research stages: unipotent algebraic with Bertrand RÉMY (2014), Algebraic number theory with Vincent PILLONI (2012), and L-function with Pierre CHAROLLOIS (2011).
“Agrégation” of mathematics with rank 56 (competitive exam for teaching, 2013).
Preparatory classes (competitive school) at "Lycée Joffre", Montpellier.
Teaching and Popularisation.
Lecture Mat102, Mathematics for Science and Engineering at Grenoble Alpes university.
Lecture Mat103, mathematics for biology at Grenoble Alpes university.
Lecture Mat307, Parametric curves and differential equations for physics at Grenoble Alpes university.
Lecture Mat11a, mathematics for biology at Joseph-Fourier university.
Popularisation of mathematics at high school "Stendhal", Grenoble (France) via math à modeler.
Lecture Mat11a, mathematics for biology at Joseph-Fourier university.
Oral exam in mathematics at the competitive school "Jean Perrin", Lyon (France).
Tutoring for students with academic difficulty at high school "Robert Doisneau" in Lyon (France) via the association trait d'union.
Computer science
Latex, OCaml, Giac/Xcas, Maple, HTML, Magma, Singular.
French (mother language), English (TOEIC 955), Spanish (beginner), Chinese (beginner)
chaos, a popularisation movies on chaos theory and dynamical systems.
dimension, a popularisation movies on geometry and dimensions.
image de math, a website for popularisation of mathematics (in French).
Last update: 27 September 2017.