6533b7d0fe1ef96bd125a5c7

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Algebraic models of the Euclidean plane

Jérémy BlancAdrien Dubouloz

subject

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

description

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.

yearjournalcountryeditionlanguage
2018-12-05
https://dx.doi.org/10.48550/arxiv.1708.08058