Enriques diagrams, arbitrarily near points, and Hilbert schemes
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Kleiman_Enriques Diagrams.pdf
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Author(s)
Kleiman, Steven L.
Piene, Ragni
Tyomkin, Ilya
Date Issued
September 2011
Journal
Rendiconti Lincei. Matematica e Applicazioni
Publisher
European Mathematical Society
Citation
Kleiman, Steven, Ragni Piene, and Ilya Tyomkin. “Enriques Diagrams, Arbitrarily Near Points, and Hilbert Schemes.” Rendiconti Lincei - Matematica e Applicazioni 22.4 (2011): 411–451. Web.
Version
Author's final manuscript
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike 3.0
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DOI of Published Version
https://doi.org/10.4171/rlm/608
