  Given a smooth family F/Y of geometrically irreducible surfaces, we study
sequences of arbitrarily near T-points of F/Y; they generalize the traditional
sequences of infinitely near points of a single smooth surface. We distinguish
a special sort of these new sequences, the strict sequences. To each strict
sequence, we associate an ordered unweighted Enriques diagram. We prove that
the various sequences with a fixed diagram form a functor, and we represent it
by a smooth Y-scheme.
  We equip this Y-scheme with a free action of the automorphism group of the
diagram. We equip the diagram with weights, take the subgroup of those
automorphisms preserving the weights, and form the corresponding quotient
scheme. Our main theorem constructs a canonical universally injective map \Psi
from this quotient scheme to the Hilbert scheme of F/Y; further, this map is an
embedding in characteristic 0. However, in every positive characteristic, we
give an example, in Appendix B, where the map is purely inseparable.
