Displaying similar documents to “Rational 1- and 2-cuspidal plane curves.”

Trivial points on towers of curves

Xavier Xarles (2013)

Journal de Théorie des Nombres de Bordeaux

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In order to study the behavior of the points in a tower of curves, we introduce and study trivial points on towers of curves, and we discuss their finiteness over number fields. We relate the problem of proving that the only rational points are the trivial ones at some level of the tower, to the unboundeness of the gonality of the curves in the tower, which we show under some hypothesis.

A Note on the Rational Cuspidal Curves

Piotr Nayar, Barbara Pilat (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

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In this short note we give an elementary combinatorial argument, showing that the conjecture of J. Fernández de Bobadilla, I. Luengo-Velasco, A. Melle-Hernández and A. Némethi [Proc. London Math. Soc. 92 (2006), 99-138, Conjecture 1] follows from Theorem 5.4 of Brodzik and Livingston [arXiv:1304.1062] in the case of rational cuspidal curves with two critical points.

Counting rational points near planar curves

Ayla Gafni (2014)

Acta Arithmetica

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We find an asymptotic formula for the number of rational points near planar curves. More precisely, if f:ℝ → ℝ is a sufficiently smooth function defined on the interval [η,ξ], then the number of rational points with denominator no larger than Q that lie within a δ-neighborhood of the graph of f is shown to be asymptotically equivalent to (ξ-η)δQ².

Enumerative geometry of divisorial families of rational curves

Ziv Ran (2004)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We compute the number of irreducible rational curves of given degree with 1 tacnode in 2 or 1 node in 3 meeting an appropriate generic collection of points and lines. As a byproduct, we also compute the number of rational plane curves of degree d passing through 3 d - 2 given points and tangent to a given line. The method is ‘classical’, free of Quantum Cohomology.