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The Hilbert function of the union of n general e-fold points in the plane is maximal if n ≥ 4e² or n is a square. The Hilbert function of a union of A, D, E singularity schemes in general position is maximal in every degree > 28. The proofs use computation of limits of families of linear systems whose special members acquire base divisors, an interesting problem in itself

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