Enriques diagrams, arbitrarily near points, and Hilbert schemes

TL;DR

This paper introduces a new framework for studying sequences of near points on surfaces using Enriques diagrams, establishing a functorial scheme representation and connecting it to Hilbert schemes, with characteristic-dependent properties.

Contribution

It generalizes sequences of infinitely near points to arbitrary families, associates Enriques diagrams, and constructs a scheme representing these sequences linked to Hilbert schemes.

Findings

Constructs a functorial scheme for sequences with fixed diagrams.

Establishes a canonical map to the Hilbert scheme, which is an embedding in characteristic 0.

Shows the map can be inseparable in positive characteristic.

Abstract

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.