Search results for " 14E05"

showing 4 items of 4 documents

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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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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Torsion of a finite base locus

2018

We interpret geometrically the torsion of the symmetric algebra of the ideal sheaf of a zero-dimensional scheme Z defined by $n+1$ equations in an $n$-dimensional variety. This leads us to generalise a formula of A.Dimca and S.Papadima in positive characteristic for a rational transformation with finite base locus. Among other applications, we construct an explicit example of a homaloidal curve of degree $5$ in characteristic $3$, answering negatively a question of A.V.D\'oria, S.H.Hassanzadeh and A.Simis.

[ MATH ] Mathematics [math][MATH.MATH-AC] Mathematics [math]/Commutative Algebra [math.AC]Mathematics - Algebraic Geometry13D02 14E05 14B05 14H20[MATH.MATH-AC]Mathematics [math]/Commutative Algebra [math.AC][MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG][MATH] Mathematics [math][MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG][MATH]Mathematics [math]Mathematics - Commutative Algebra[ MATH.MATH-AG ] Mathematics [math]/Algebraic Geometry [math.AG][ MATH.MATH-AC ] Mathematics [math]/Commutative Algebra [math.AC]
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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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