Parameter spaces for curves on surfaces and enumeration of rational curves
Let S be a smooth, minimal rational surface. The geometry of the Severi variety parametrising irreducible, rational curves in a given linear system on S is studied. The results obtained are applied to enumerative geometry, in combination with ideas from Quantum Cohomology. Formulas enumerating rational curves are found, some of which generalised Kontsevich‘s formula for plane curves.
Key Words: rational surface; rational curve; enumerative geometry; Severi variety; cross-ratio method..