By Tracy Kompelien
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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, ...