## Families of Curves with Prescribed Singularities

• The study of families of curves with prescribed singularities has a long tradition. Its foundations were laid by Plücker, Severi, Segre, and Zariski at the beginning of the 20th century. Leading to interesting results with applications in singularity theory and in the topology of complex algebraic curves and surfaces it has attained the continuous attraction of algebraic geometers since then. Throughout this thesis we examine the varieties V(D,S1,...,Sr) of irreducible reduced curves in a fixed linear system |D| on a smooth projective surface S over the complex numbers having precisely r singular points of types S1,...,Sr. We are mainly interested in the following three questions: 1) Is V(D,S1,...,Sr) non-empty? 2) Is V(D,S1,...,Sr) T-smooth, that is smooth of the expected dimension? 3) Is V(D,S1,...Sr) irreducible? We would like to answer the questions in such a way that we present numerical conditions depending on invariants of the divisor D and of the singularity types S1,...,Sr, which ensure a positive answer. The main conditions which we derive will be of the type inv(S1)+...+inv(Sr) < aD^2+bD.K+c, where inv is some invariant of singularity types, a, b and c are some constants, and K is some fixed divisor. The case that S is the projective plane has been very well studied by many authors, and on other surfaces some results for curves with nodes and cusps have been derived in the past. We, however, consider arbitrary singularity types, and the results which we derive apply to large classes of surfaces, including surfaces in projective three-space, K3-surfaces, products of curves and geometrically ruled surfaces.
• Families of Curves with Prescribed Singularities

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Author: Thomas Keilen urn:nbn:de:bsz:386-kluedo-13214 Gert-Martin Greuel Doctoral Thesis English 2001 2001 Technische Universität Kaiserslautern Technische Universität Kaiserslautern 2002/10/30 2001/12/05 equisingular families ; projective surfaces; singularities Kurvenschar; Projektive Fläche; Singularität; Verschwindungsatz Fachbereich Mathematik 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik 14-XX ALGEBRAIC GEOMETRY / 14Hxx Curves / 14H10 Families, moduli (algebraic) 14-XX ALGEBRAIC GEOMETRY / 14Hxx Curves / 14H15 Families, moduli (analytic) [See also 30F10, 32Gxx] 14-XX ALGEBRAIC GEOMETRY / 14Hxx Curves / 14H20 Singularities, local rings [See also 13Hxx, 14B05] 14-XX ALGEBRAIC GEOMETRY / 14Jxx Surfaces and higher-dimensional varieties (For analytic theory, see 32Jxx) / 14J26 Rational and ruled surfaces 14-XX ALGEBRAIC GEOMETRY / 14Jxx Surfaces and higher-dimensional varieties (For analytic theory, see 32Jxx) / 14J27 Elliptic surfaces 14-XX ALGEBRAIC GEOMETRY / 14Jxx Surfaces and higher-dimensional varieties (For analytic theory, see 32Jxx) / 14J28 K3 surfaces and Enriques surfaces 14-XX ALGEBRAIC GEOMETRY / 14Jxx Surfaces and higher-dimensional varieties (For analytic theory, see 32Jxx) / 14J70 Hypersurfaces Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011

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