Computing Severi degrees with long-edge graphs (Q487061)

The number of (possibly reducible) algebraic curves in the complex projective plane of degree \(d\) with \(\delta\) nodes passing through the appropriate number of general points is denoted \(N^{d,\delta}\) and is referred to as a Severi degree. The Severi variety is the Hilbert scheme parameterizing degree \(d\) plane curves with \(\delta\) nodes, and of course it has a natural embedding in the projective space of all degree \(d\) plane curves; as one would hope, passing through generic points imposes generic linear constraints and the degree of the Severi variety in this projective embedding is indeed the finite number of curves mentioned above, hence the term Severi degree. This enumerative problem dates back to the 19th century and was the subject of intense research for a period then, as well as sporadically in the 20th century until the last few decades where interest -- and progress -- has grown tremendously. Note, for instance, that when the degree is sufficiently large relative to the number of nodes then all such curves are irreducible and hence the Severi degree is in fact a Gromov-Witten invariant, i.e., and intersection number on a Kontsevich moduli space of stable maps. The introduction of the present paper nicely summarizes the situation and is recommended to the reader. In brief, it was conjectured and fairly recently proven that \(N^{d,\delta}\) is polynomial in \(d\), for \(d \gg 0\) and fixed \(\delta\). One can consider the generating series \[ \mathcal{N}(d) = \sum_{\delta \geq 0} N^{d,\delta}x^\delta \] and its formal logarithm \[ \mathcal{Q}(d) = \sum_{\delta \geq 1}Q^{d,\delta}x^\delta. \] The Göttsche-Yau-Zaslow formula (proved by \textit{Y.-J. Tzeng} [J. Differ. Geom. 90, No. 3, 439--472 (2012; Zbl 1253.14054)] using some technical machinery) shows that \(Q^{d,\delta}\) is quadratic in \(d \gg 0\) for fixed \(\delta\). The present paper provides an elementary, combinatorial proof of this fact, and discusses a relation to tropical enumerative geometry as well.