TAILIEUCHUNG - Tyurina components and rational cycles for rational singularities

In this paper, we give a geometric proof of Pinkham’s theorem on the positive cycles supported on the exceptional divisor of a rational singularity. In order to do this, we give several properties of the Tyurina components of the exceptional divisor and of the points of blowing-up surface of a rational singularity. | Turk J Math 23 (1999) , 361 – 374. ¨ ITAK ˙ c TUB TYURINA COMPONENTS AND RATIONAL CYCLES FOR RATIONAL SINGULARITIES Meral Tosun Abstract In this paper, we give a geometric proof of Pinkham’s theorem on the positive cycles supported on the exceptional divisor of a rational singularity. In order to do this, we give several properties of the Tyurina components of the exceptional divisor and of the points of blowing-up surface of a rational singularity. Introduction An isolated singularity of a complex surface S is rational if the stalk at the singularity of the coherent sheave R1 π∗ OX is equal to zero where π : X → S is a resolution of S at the singularity. The numerical characterization of a rational singularity, given by M. Artin in [1] (see theorem below), permits us to study these singularities by the exceptional divisor of a resolution of the singularity (see [15] or [17]). Here we are interested in the positive cycles supported on the exceptional divisor of a resolution divisor of a rational singularity. In [1] and [11], it has been shown that these cycles correspond to some special functions on S. In section 4, we use this correspondance to prove Pinkham’s theorem given in [12] (see theorem below). We start our paper by introducing some notations. In section 3, following the general case of a theorem of M. Artin [1] (see theorem below), we give a proof on the nature of the exceptional divisor of a resoution of a rational singularity (see corollary below). After giving some properties on the blowing up surface of a rational singularity, we finish the section by giving a bound on the non-Tyurina components of the exceptional divisor 361 TOSUN of a resolution of a rational singularity. Recall Let (S, ξ) be a germ of a normal analytic surface embedded in C N . Denote by S a sufficiently small representative of the germ (S, ξ). A resolution of S is a complex analytic surface X and a proper holomorphic map π : X → S such that its .

• Toán học
• Tyurina components
• Rational cycles
• Rational singularities
• Exceptional divisor
• Several properties
• Blowing up surface