0000000000004061

AUTHOR

Jérémy Blanc

showing 4 related works from this author

Algebraic models of the Euclidean plane

2018

We introduce a new invariant, the real (logarithmic)-Kodaira dimension, that allows to distinguish smooth real algebraic surfaces up to birational diffeomorphism. As an application, we construct infinite families of smooth rational real algebraic surfaces with trivial homology groups, whose real loci are diffeomorphic to $\mathbb{R}^2$, but which are pairwise not birationally diffeomorphic. There are thus infinitely many non-trivial models of the euclidean plane, contrary to the compact case.

Mathematics - Differential GeometryPure mathematicsaffine complexificationLogarithmReal algebraic model01 natural sciencesMathematics - Algebraic GeometryMathematics::Algebraic Geometry0103 physical sciencesEuclidean geometryAlgebraic surfaceaffine surfaceFOS: Mathematics0101 mathematicsInvariant (mathematics)Algebraic numberMathematics::Symplectic GeometryAlgebraic Geometry (math.AG)MathematicsAlgebra and Number Theory010102 general mathematics[MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG]q-homology planesbirational diffeomorphismDifferential Geometry (math.DG)[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG]rational fibrationPairwise comparison010307 mathematical physicsGeometry and TopologyDiffeomorphism[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG]14R05 14R25 14E05 14P25 14J26[MATH.MATH-DG] Mathematics [math]/Differential Geometry [math.DG]Singular homology
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Automorphisms of $mathbb{A}^{1}$-fibered affine surfaces

2011

We develop technics of birational geometry to study automorphisms of affine surfaces admitting many distinct rational fibrations, with a particular focus on the interactions between automorphisms and these fibrations. In particular, we associate to each surface S of this type a graph encoding equivalence classes of rational fibrations from which it is possible to decide for instance if the automorphism group of S is generated by automorphisms preserving these fibrations.

Surface (mathematics)Graph encodingPure mathematicsApplied MathematicsGeneral MathematicsFibered knotBirational geometryType (model theory)AutomorphismMathematics::Algebraic TopologyMathematics::Group TheoryMathematics::Algebraic GeometryAffine transformationddc:510Focus (optics)Mathematics::Symplectic GeometryMathematics
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Affine Surfaces With a Huge Group of Automorphisms

2013

We describe a family of rational affine surfaces S with huge groups of automorphisms in the following sense: the normal subgroup Aut(S)alg of Aut(S) generated by all algebraic subgroups of Aut(S) is not generated by any countable family of such subgroups, and the quotient Aut(S)/Aut(S)alg cointains a free group over an uncountable set of generators.

Normal subgrouprational fibrationsautomorphismsGroup (mathematics)General Mathematics010102 general mathematicsAutomorphism01 natural sciences[ MATH.MATH-AG ] Mathematics [math]/Algebraic Geometry [math.AG]CombinatoricsMathematics::LogicMathematics - Algebraic GeometryMathematics::Group Theory0103 physical sciencesFree groupCountable setUncountable set[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG]010307 mathematical physics0101 mathematicsAlgebraic number14R25 14R20 14R05 14E05affine surfacesQuotientMathematicsInternational Mathematics Research Notices
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Algebraic models of the real affine plane

2017

We introduce a new invariant, the real (logarithmic)-Kodaira dimension, that allows to distinguish smooth real algebraic surfaces up to birational diffeomorphism. As an application, we construct infinite families of smooth rational real algebraic surfaces with trivial homology groups, whose real loci are diffeomorphic to $\mathbb{R}^2$, but which are pairwise not birationally diffeomorphic. There are thus infinitely many non-trivial models of the real affine plane, contrary to the compact case.

birational diffeomorphismaffine complexificationMathematics::Algebraic Geometry14R05 14R25 14E05 14P25 14J26.affine surface[MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG]rational fibrationReal algebraic model[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG]Mathematics::Symplectic Geometry[ MATH.MATH-AG ] Mathematics [math]/Algebraic Geometry [math.AG]
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