Family blowup formula, admissible graphs and the enumeration of singular curves. I. (Q1609805)

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.