Enriques diagrams, arbitrarily near points, and Hilbert schemes
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.
Keywords
Cite
@article{arxiv.0905.2169,
title = {Enriques diagrams, arbitrarily near points, and Hilbert schemes},
author = {Steven Kleiman and Ragni Piene and Ilya Tyomkin},
journal= {arXiv preprint arXiv:0905.2169},
year = {2011}
}
Comments
36 pages. Final version for Rendiconti Lincei. Notable changes: (1) Introduction enhanced at beginning. (2) "Arbitrarily near" replaces "infinitely near" as the T-point need not lie entirely within the exceptional divisor. (3) Prop.5.9 now applies to any diagram with every vertex a root. (4) Appendix B gives examples of diagrams with a nonroot, yet \Psi is an embedding in every characteristic