Parameter spaces for curves on surfaces and enumeration of rational curves

摘要

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 generalise Kontsevich's formula for plane curves. [References: 13]