12,845 research outputs found

    Topology of 2D and 3D Rational Curves

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    In this paper we present algorithms for computing the topology of planar and space rational curves defined by a parametrization. The algorithms given here work directly with the parametrization of the curve, and do not require to compute or use the implicit equation of the curve (in the case of planar curves) or of any projection (in the case of space curves). Moreover, these algorithms have been implemented in Maple; the examples considered and the timings obtained show good performance skills.Comment: 26 pages, 19 figure

    Computing the topology of a planar or space hyperelliptic curve

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    We present algorithms to compute the topology of 2D and 3D hyperelliptic curves. The algorithms are based on the fact that 2D and 3D hyperelliptic curves can be seen as the image of a planar curve (the Weierstrass form of the curve), whose topology is easy to compute, under a birational mapping of the plane or the space. We report on a {\tt Maple} implementation of these algorithms, and present several examples. Complexity and certification issues are also discussed.Comment: 34 pages, lot of figure

    Nearly free divisors and rational cuspidal curves

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    We define a class of plane curves which are close to the free divisors and such that conjecturally it contains the class of rational cuspidal curves. Using a recent result by U. Walther we show that any unicuspidal rational curve with a unique Puiseux pair is either free or belongs to this class.Comment: v3: title modified and the topological results strengthened. In particular, any rational cuspidal curve whose degree is either even or a prime power, or which has an abelian group, is a nearly free diviso

    Logarithmic asymptotics of the genus zero Gromov-Witten invariants of the blown up plane

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    We study the growth of the genus zero Gromov-Witten invariants GW_{nD} of the projective plane P^2_k blown up at k points (where D is a class in the second homology group of P^2_k). We prove that, under some natural restrictions on D, the sequence log GW_{nD} is equivalent to lambda n log n, where lambda = D.c_1(P^2_k).Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper14.abs.htm
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