The Severi Problem for Rational Curves on Del Pezzo Surfaces

Download or read book The Severi Problem for Rational Curves on Del Pezzo Surfaces written by Damiano Testa and published by . This book was released on 2005 with total page 142 pages. Available in PDF, EPUB and Kindle. Book excerpt: Let X be a smooth projective surface and choose a curve C on X. Let VC be the set of all irreducible divisors on X linearly equivalent to C whose normalization is a rational curve. The Severi problem for rational curves on X with divisor class [C] consists of studying the irreducibility of the spaces VC as C varies among all curves on X. In this thesis, we prove that all the spaces VC are irreducible in the case where X is a del Pezzo surface of degree at least two. If the degree of X is one, then we prove the same result only for a general X, with the exception of V-KX, where KX is the canonical divisor of X. It is well known that for general del Pezzo surface of degree one, V-KX consists of twelve points, and thus cannot be irreducible.