Universal polynomials for singular curves on surfaces (Q2874607)

Given a collection of analytic or topological types of isolated plane curve singularity \(\underline{\alpha} = (\alpha_1,\ldots,\alpha_l)\) and a polarized smooth complex projective surface \((S,L)\). The authors prove the existence of a universal polynomial \(T_{\underline{\alpha}}\), depending only on \(\underline{\alpha}\), such that \( T_{\underline{\alpha}}(c_1(L)^2,c_1(L)c_1(K_S), c_1(S)^2, c_2(S))\) equals the number of curves with singularity type \(\underline{\alpha}\) in a general sub-linear system of \(|L|\) of correct codimension, provided \(L\) is sufficiently ample. This result generalizes Göttsche's universality conjecture proven independently by \textit{Y.-J. Tzeng} [J. Differ. Geom. 90, No. 3, 439--472 (2012; Zbl 1253.14054)] and \textit{M. Kool} et al. [Geom. Topol. 15, No. 1, 397--406 (2011; Zbl 1210.14011)].NEWLINENEWLINEA generalization to higher dimensional cases is also discussed in the article. Current results concerning the computation of \(d_{\underline{\alpha}}(S,L)\) are listed after the proof of the existence theorem. Despite the existence of the universal polynomial, exact values of \(d_{\underline{\alpha}}(S,L)\) are only known in a few cases, not even in the case of the projective plane for most singularity types.NEWLINENEWLINEAt the end, the authors show that the strata (resp. the closure of a stratum) of the Severi variety given by analytic singularity type of a sufficiently ample linear system on a smooth complex projective surface are smooth (resp. is irreducible).