Family blowup formula, admissible graphs and the enumeration of singular curves. I.
type: Item
instance of: scholarly article
title: Family blowup formula, admissible graphs and the enumeration of singular curves. I.
zbMATH Open document ID: 1036.14014
author: Ai-Ko Liu
published in: Journal of Differential Geometry
publication date: 2002-08-15T00:00:00Z
review text: The author discusses the scheme of enumerating the singular holomorphic curves in a linear system on an algebraic surface. This problem is classically known as the Severi degree. Recursive formulas for the Severi degrees have been recently derived by \textit{Z. Ran} [Invent. Math. 97, 447--465 (1989; Zbl 0702.14040)] and later by \textit{J. Harris} and \textit{L. Caporaso} [Invent. Math. 131, 345--392 (1998; Zbl 0934.14040)]. In this direction the author presents the following main results. Theorem 1.1: Let \(M\) be an algebraic surface. Let \(L\) be a sufficiently very ample line bundle on \(M\). Let the number \(n_L (\delta)\) denote the number of \(\delta \)-nodes nodal curves in the linear system \(| L| \). Then the number can be expressed as an universal polynomial in terms of \(L.L\), \(L.c_1(M)\), \(c_1(M)^2\) and \(c_2(M)\). Theorem 1.2: Let \(M\) be a Kähler surface and \(L\) be a sufficiently very ample linear divisor on \(M\). Let \(n_L( \Gamma, L - \Sigma m_i E_i )\) denote the ``number of singular curves'' in a generic \( \Sigma (m_i^2 + m_i -4)/2 \)-dimensional linear subsystem of \(| L| \) with a fixed topological type of plane curve singularities specified by \(\Gamma \) and \(m_i\), etc. The virtual number can be expressed as an universal polynomial in \(L.L\), \(L.c_1(M)\), \(c_1(M)^2\) and \(c_2(M)\). It depends on \(\Gamma\), the admissible graph, and \(m_i\), the multiplicities of the singularities.
reviewed by: Isidro Nieto Baños
Mathematics Subject Classification ID: 14H20
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14N10
zbMATH DE Number: 1782671
zbMATH Keywords: enumerative problems
zbMATH Keywords: singularities
zbMATH Keywords: blowup formula
zbMATH Keywords: linear system
zbMATH Keywords: algebraic surface
MaRDI profile type: Publication
Wikidata QID: Q115202928
full work available at URL: https://doi.org/10.4310/jdg/1090347696
OpenAlex ID: W1640023898
DOI: 10.4310/JDG/1090347696
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description: scientific article; zbMATH DE number 1782671