# Enriques diagrams, arbitrarily near points, and Hilbert schemes

### Steven Kleiman

Massachusetts Institute of Technology, Cambridge, USA### Ragni Piene

University of Oslo, Norway### Ilya Tyomkin

Ben Gurion University of the Negev, Beer Sheva, Israel

## 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 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.

## Cite this article

Steven Kleiman, Ragni Piene, Ilya Tyomkin, Enriques diagrams, arbitrarily near points, and Hilbert schemes. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 22 (2011), no. 4, pp. 411–451

DOI 10.4171/RLM/608