Counting Curves on Surfaces: a Guide to New Techniques and Results

1.1. Abstract and summary. A series of recent results solving classical enumerative problems for curves on rational surfaces is described. Impulse to the subject came from recent ideas from quantum field theory leading to the definition of quantum cohomology. As a by-product, formulas enumerating rational curves on certain varieties were derived from the properties of certain generating functions representing the free energy of certain topological field theories. A mathematically acceptable construction of quantum cohomology came soon afterwards ([RT] and [KM]-[K]). Different proofs of some of these formulas were provided later (in [CH1] and [CH2]) using different methods that could be generalized to cases (such as Hirzebruch surfaces) for which the quantum cohomology theory did not give enumerative results. For higher genera, the connection between enumerative geometry and quantum cohomology or quantum field theory is still largely conjectural. On the other hand, a recursive formula enumerating plane curves of any genus has been recently proved using purely algebro-geometric techniques ([CH3]). Moreover a generating function exists together with a differential equation implying such a recursion ([G]). The enumerative problem is precisely stated in the introduction. The second chapter contains a short description of the relation with Quantum Cohomology and Kontsevich’s formula for plane rational curves. A discussion of enumerative problems for Hirzebruch surfaces concludes this part, which is entirely dedicated to rational curves. The general case of plane curves of any genus is described in the third chapter, focussing on the results of [CH3]. The main recursive formula of that paper is explained, together with an outline of the proof and a description of the generating function found in [G]. At the end, there is a discussion of generalizations of the procedure of [CH3] to other varieties. Chapters 2 and 3 are completely independent from each other. In the fourth and last chapter various enumerative techniques are applied to the concrete example of counting rational plane cubics through 8 general points.