Enumerative Geometry

Given a complex projective variety V (as defined in [1]), we wish to count the curves in V that satisfy certain prescribed conditions. Let f C denote complex projective -dimensional space. In our first example, V = f C2, the complex projective plane; in the second and third, V is a general hypersurface in f C 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  +  = 1 is represented as