By I. Burban, Yu. Drozd, G.-M Greuel (auth.), Ciro Ciliberto, Friedrich Hirzebruch, Rick Miranda, Mina Teicher (eds.)

An up to date record at the present prestige of significant study subject matters in algebraic geometry and its functions, corresponding to computational algebra and geometry, singularity conception algorithms, numerical suggestions of polynomial platforms, coding concept, verbal exchange networks, and desktop imaginative and prescient. Contributions on extra basic elements of algebraic geometry contain expositions concerning counting issues on kinds over finite fields, Mori thought, linear structures, Abelian kinds, vector bundles on singular curves, degenerations of surfaces, and replicate symmetry of Calabi-Yau manifolds.

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**Example text**

Mathematics Subject Classification (2000): 14J26, 14C20. 1. Introduction Consider the linear system of plane curves of a given degree d having at 11 specified, but general, points of the plane, singularities of multiplicities say m I, ... , m n . The classical multivariate polynomial interpolation problem consists in evaluting the dimension of the linear system in question. , up to a scalar, the coefficients of their equation, we see that the expected dimension of the linear system we are dealing with is the maximum between d(d;3) - I;l=1 mi(nt l ) and -I, the latter being the dimension of the empty system.

Up to a scalar, the coefficients of their equation, we see that the expected dimension of the linear system we are dealing with is the maximum between d(d;3) - I;l=1 mi(nt l ) and -I, the latter being the dimension of the empty system. A system for which the dimension is bigger than the expected one is called c\pecial, and it is a long-standing problem to classify all special linear systems of the above type. e. a rational curve which, on the blow up ofthe plane at the multiple base points, ends up having self-intersection -1.

Moreover E . (C hE) = 0 and hi (S, Os( C)) :::: (h; I) (Ciliberto and Miranda, 1998). This motivates the above definition, inasmuch as a (-I )-special system is indeed special. a(p711 , ... £ on S. a(p~1 , ... ,p':/") referring to the self-intersection of the curves in the proper transform system £ on S, etc. The following conjecture is essentially due to Harbourne (1986) and Hirschowitz ( 1989) and is related to an earlier conjecture by Gimigliano (1987). We will formulate it as follows. 1. a(p';lt, ...