@MISC{Finch_0.1.rational, author = {Steven Finch}, title = {0.1. Rational Plane Curves Passing Through Points. In the following, we}, year = {} }

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Abstract

Given a complex projective variety V (as defined in [1]), we wish to count the curves in V that satisfy certain prescribed conditions. Let fC denote complex pro-jective -dimensional space. In our first example, V = fC2, the complex projective plane; in the second and third, V is a general hypersurface in fC of degree 2 − 3. Call such V a cubic twofold when = 3 and a quintic threefold when = 4. Our interest is in rational curves, which include all lines (degree 1), conics (degree 2) and singular cubics (degree 3). No elliptic curves are rational. The word “rational ” here refers to the affine parametrization of the curve — a ratio of polynomials — and the curve is of degree if the polynomials are of degree at most . For instance, the circle 2 + 2 = 1 is represented as