Rational curves and rational singularities
Copyright policy )Rational curves and rational singularities
We study rational curves on algebraic varieties, especially on normal affine varieties endowed with a $\C^*$-action. For varieties with an isolated singularity, we show that the presence of sufficiently many rational curves outside the singular point strongly affects the character of the singularity. This provides an explanation of classical results due to H.A. Schwartz and G.H. Halphen on polynomial solutions of the generalized Fermat equation.
24 pages; uses Paul Taylor's diagrams.tex, see e.g. ftp://ftp.dante.de/tex-archive/macros/generic/diagrams/taylor/V3,88.tex
rational curves, rational singularities, Group actions on varieties or schemes (quotients), 14L30, quotient singularities, Commutative Algebra (math.AC), Higher degree equations; Fermat's equation, [MATH.MATH-AC] Mathematics [math]/Commutative Algebra [math.AC], Mathematics - Algebraic Geometry, C^*-action, C_+-actions, affine surface, FOS: Mathematics, Complex Variables (math.CV), 14J17; 14L30; 13H10, affine algebraic surfaces, quotient singularity, Algebraic Geometry (math.AG), Singularities of surfaces or higher-dimensional varieties, bepress|Physical Sciences and Mathematics|Mathematics, Mathematics - Complex Variables, 13H10, [MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG], Mathematics - Commutative Algebra, quasihomogeneous varieties, Group actions on affine varieties, rational singularity, graded algebra, [MATH.MATH-CV] Mathematics [math]/Complex Variables [math.CV], rational curve, polynomial curve, bepress|Physical Sciences and Mathematics|Mathematics|Algebraic Geometry, Rational and unirational varieties, 14J17
rational curves, rational singularities, Group actions on varieties or schemes (quotients), 14L30, quotient singularities, Commutative Algebra (math.AC), Higher degree equations; Fermat's equation, [MATH.MATH-AC] Mathematics [math]/Commutative Algebra [math.AC], Mathematics - Algebraic Geometry, C^*-action, C_+-actions, affine surface, FOS: Mathematics, Complex Variables (math.CV), 14J17; 14L30; 13H10, affine algebraic surfaces, quotient singularity, Algebraic Geometry (math.AG), Singularities of surfaces or higher-dimensional varieties, bepress|Physical Sciences and Mathematics|Mathematics, Mathematics - Complex Variables, 13H10, [MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG], Mathematics - Commutative Algebra, quasihomogeneous varieties, Group actions on affine varieties, rational singularity, graded algebra, [MATH.MATH-CV] Mathematics [math]/Complex Variables [math.CV], rational curve, polynomial curve, bepress|Physical Sciences and Mathematics|Mathematics|Algebraic Geometry, Rational and unirational varieties, 14J17
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