Existence of Curves with Prescribed Topological Singularities
Abstract
Throughout this paper we study the existence of irreducible curves C on smooth projective surfaces S with singular points of prescribed topological types S_1,...,S_r. There are necessary conditions for the existence of the type \sum_{i=1}^r \mu(S_i) < aC^2+bC.K+c+1 for some fixed divisor K on S and suitable coefficients a, b and c, and the main sufficient condition that we find is of the same type, saying it is asymptotically optimal. Even for the case where S is the projective plane, ten years ago general results of this quality have not been known. An important ingredient for the proof is a vanishing theorem for invertible sheaves on the blown up S of the form O_{S'}(\pi^*D\sum_{i=1}^r m_iE_i), deduced from the KawamataVieweg Vanishing Theorem. Its proof covers the first part of the paper, while the middle part is devoted to the existence theorems. In the last part we investigate our conditions on ruled surfaces, products of elliptic curves, surfaces in projective 3space, and K3surfaces.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 April 2001
 arXiv:
 arXiv:math/0104062
 Bibcode:
 2001math......4062K
 Keywords:

 Mathematics  Algebraic Geometry;
 14H10;
 14H15 (Primary) 14J26;
 14J27;
 14J28;
 14J70 (Secondary)
 EPrint:
 58 pages