Introduction to curve counting

Abstract: This talk is intended as a very gentle introduction to the classical subject of enumerative geometry concerned with problems of counting algebraic curves with prescribed properties. In the realm of classical planimetry, we know that there exists a single circle passing through a collection of three points, provided that these points are generic. Here “generic” simply means that the points are not collinear, i.e. don’t belong to the same line but could refer to some other open condition in a different context; the number of points is just right for a problem to be well-stated — there are infinitely many circles passing through any two points and a collection of four points lying on a circle is special. A less trivial example, which will be considered in detail, is the question “How many rational cubic curves on the plane pass through a generic collection of eight points?”. “Rational” here means that a curve has a parametrization by rational functions, and “cubic” refers to a curve defined as a zero locus of a degree three polynomial. The number of points is again chosen just right so that one may expect a non-trivial answer. In the case of real algebraic geometry, the number of cubics may vary depending on the position of the eight generic points, and a priori it is even not clear if such curves exist for all configurations. On the other hand, if we state the same problem over complex numbers the answer becomes definite and independent of the position of the generic points. By an elegant, yet quite elementary, topological reasoning we will deduce that this answer is 12. Adapting the same argument in the real case, we will see that real rational cubics should be counted with signs and the result of this new count becomes a definite -8, proving in particular that for any generic configuration of eight points on the real plane, there exist at least eight rational real cubics passing through them.