Hilbert functions of fat point subschemes of the plane: the two-fold way (Q2895430)

The aim of the paper is to describe the Hilbert function (H.f. in the following) for schemes of fat points in the projective plane having the same multiplicities. Note that if \(X\subseteq {\mathbb P}^2\) is a scheme of \(n\) reduced points in the plane, the m-fat points scheme \(mX\) is defined by the homogeneous ideal \((I_X)^{(m)}\) (the saturation of \(I_X^m\)). Two approaches are used in the paper to attack the problem of classifying all the possible H. f. of schemes \(mX\) for given \(n, m\): the first way tries to find costraints which such functions must satisfy and then, for each numerical function a prori ``allowed'', seeks to find a scheme which has such function as its H. f. The second way is to consider the surface \(Y\) obtained by a blow up \(Y \rightarrow {\mathbb P}^2\) at \(X\), and then to determine the cohomology of the divisors of type \(dH-mE\), where \(E\) is the exceptional divisor on \(Y\) and \(H\) is the strict transform of a generic line (knowing \(h^0(Y,mE)\) amounts to knowing the Hilbert function of \(mX\) in degree \(d\)).NEWLINENEWLINEThe first approach allows the authors to determine all the possible H.f. of \(2X\) for \(n=9\) (the first open case for \(m=2\)). The second approach is used to study the case of \(n\geq 9\) points (every \(m\)) when \(X\) is a set of smooth points of an irreducible cubic curve. Other results for points on a cubic are obtained via the first appoach, dropping the hypothesis that the points are smooth for the curve.NEWLINENEWLINEFor the entire collection see [Zbl 1237.13005].