Enriques diagrams, arbitrarily near points, and Hilbert schemes

S. Kleiman, R. Piene, I. Tyomkin

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


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.

Original languageEnglish
Pages (from-to)411-451
Number of pages41
JournalAtti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica E Applicazioni
Issue number4
StatePublished - 1 Dec 2011


  • Arbitrarily near points
  • Enriques diagrams
  • Hilbert schemes
  • Property of exchange

ASJC Scopus subject areas

  • General Mathematics


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