Abstract
Based on results by Brugallé and Mikhalkin, Fomin and Mikhalkin give formulas for computing classical Severi degrees Nd,δ using long-edge graphs. In 2012, Block, Colley and Kennedy considered the logarithmic versionof a special function associated to long-edge graphs which appeared in Fomin-Mikhalkin’s formula, and conjecturedit to be linear. They have since proved their conjecture. At the same time, motivated by their conjecture, we considera special multivariate function associated to long-edge graphs that generalizes their function. The main result of thispaper is that the multivariate function we define is always linear.The first application of our linearity result is that by applying it to classical Severi degrees, we recover quadraticity of Qd,δ and a bound δ for the threshold of polynomiality ofNd,δ.Next, in joint work with Osserman, we apply thelinearity result to a special family of toric surfaces and obtain universal polynomial results having connections to the Göttsche-Yau-Zaslow formula. As a result, we provide combinatorial formulas for the two unidentified power series B1(q) and B2(q) appearing in the Göttsche-Yau-Zaslow formula.The proof of our linearity result is completely combinatorial. We defineτ-graphs which generalize long-edge graphs,and a closely related family of combinatorial objects we call (τ,n)-words. By introducing height functions and aconcept of irreducibility, we describe ways to decompose certain families of (τ,n)-words into irreducible words,which leads to the desired results.
Highlights
Background on Severi degreesThe classical Severi degree, denoted by N d,δ, is the degree of the Severi variety
In 1994, Di Francesco and Itzykson [6] conjectured that for fixed δ, the Severi degree N d,δ is given by a node polynomial Nδ(d) for sufficiently large d
We prove a certain function associated to long-edge graphs is linear
Summary
The classical Severi degree, denoted by N d,δ, is the degree of the Severi variety. It counts the number of curves of degree d with δ nodes passing through d(d+3) 2. In 1994, Di Francesco and Itzykson [6] conjectured that for fixed δ, the Severi degree N d,δ is given by a node polynomial Nδ(d) for sufficiently large d. It is natural for us to consider the generating function for classical Severi degrees and its formal logarithm: N (d) := 1 +. We prove a certain function associated to long-edge graphs is linear Using this result, we provide another proof of Corollary 1.2 as well as a combinatorial way of computing the power series A1(t) and A2(t) of Proposition 1.1. In joint work with Osserman [12], the linearity result enables us to obtain a universal polynomiality property of Severi degrees on families of toric surfaces. We give a brief introduction to the objects in our results, and state the main theorem of the paper
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