Maximal rank for schemes of small multiplicity by Évain’s differential Horace method

Abstract

The Hilbert function of the union of n n general e e -fold points in the plane is maximal if n ≥ 4 e 2 n\ge 4e^2 or n n is a square. The Hilbert function of a union of A A , D D , E E singularity schemes in general position is maximal in every degree > 28 >28 . The proofs use a computation of limits of families of linear systems whose special members acquire base divisors, an interesting problem in itself.