Existence of curves with prescribed topological singularities

Thomas Keilen, Ilya Tyomkin

Research output: Contribution to journalArticlepeer-review

12 Scopus citations


Throughout this paper we study the existence of irreducible curves C on smooth projective surfaces ∑ with singular points of prescribed topological types S1,...,Sr. There are necessary conditions for the existence of the type ∑1=1r μ(Si) ≤ αC2+βC.K+γ for some fixed divisor K on ∑ and suitable coefficients α, β and γ, and the main sufficient condition that we find is of the same type, saying it is asymptotically proper. Ten years ago general results of this quality were not known even for the case ∑ = ℙC2. An important ingredient for the proof is a vanishing theorem for invertible sheaves on the blown up ∑ of the form O(π* D-∑i=1r miEi), deduced from the Kawamata-Vieweg Vanishing Theorem. Its proof covers the first part of the paper, while the middle part is devoted to the existence theorems. In the last part we investigate our conditions on ruled surfaces, products of elliptic curves, surfaces in ℙC3, and K3-surfaces.

Original languageEnglish
Pages (from-to)1837-1860
Number of pages24
JournalTransactions of the American Mathematical Society
Issue number5
StatePublished - 1 Jan 2002
Externally publishedYes


  • Algebraic geometry
  • Singularity theory

ASJC Scopus subject areas

  • Mathematics (all)
  • Applied Mathematics


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