## 14H20 Singularities, local rings [See also 13Hxx, 14B05]

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#### Schlagworte

- Kurvenschar (1)
- Projektive Fläche (1)
- Singularität (1)
- Verschwindungsatz (1)
- algebroid curve (1)
- canonical ideal (1)
- curve singularity (1)
- duality (1)
- endomorphism ring (1)
- equisingular families (1)

Using valuation theory we associate to a one-dimensional equidimensional semilocal Cohen-Macaulay ring \(R\) its semigroup of values, and to a fractional ideal of \(R\) we associate its value semigroup ideal. For a class of curve singularities (here called admissible rings) including algebroid curves the semigroups of values, respectively the value semigroup ideals, satisfy combinatorial properties defining good semigroups, respectively good semigroup ideals. Notably, the class of good semigroups strictly contains the class of value semigroups of admissible rings. On good semigroups we establish combinatorial versions of algebraic concepts on admissible rings which are compatible with their prototypes under taking values. Primarily we examine duality and quasihomogeneity.
We give a definition for canonical semigroup ideals of good semigroups which characterizes canonical fractional ideals of an admissible ring in terms of their value semigroup ideals. Moreover, a canonical semigroup ideal induces a duality on the set of good semigroup ideals of a good semigroup. This duality is compatible with the Cohen-Macaulay duality on fractional ideals under taking values.
The properties of the semigroup of values of a quasihomogeneous curve singularity lead to a notion of quasihomogeneity on good semigroups which is compatible with its algebraic prototype. We give a combinatorial criterion which allows to construct from a quasihomogeneous semigroup \(S\) a quasihomogeneous curve singularity having \(S\) as semigroup of values.
As an application we use the semigroup of values to compute endomorphism rings of maximal ideals of algebroid curves. This yields an explicit description of the intermediate rings in an algorithmic normalization of plane central arrangements of smooth curves based on a criterion by Grauert and Remmert. Applying this result to hyperplane arrangements we determine the number of steps needed to compute the normalization of a the arrangement in terms of its Möbius function.

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.