Refined curve counting with tropical geometry (Q2787622)

This a breakthrough work, which, in fact, opens a new direction in the tropical enumerative geometry as well as in the classical enumerative geometry of curves. According to \textit{G. Mikhalkin}'s correspondence theorem [J. Am. Math. Soc. 18, No. 2, 313--377 (2005; Zbl 1092.14068)] the number of irreducible plane curves of degree \(d\) and genus \(g\) passing through \(3d-1+g\) generic points (so-called Severi degree) equals the number of irreducible plane trivalent tropical curves of the same degree and genus passing through \(3d-1+g\) generic point in \({\mathbb R}^2\) and counted with Mikhalkin's multiplicity, which is equal to the product of some positive integer weights of trivalent vertices (Mikhalkin's weights). The authors suggest to refine Mikhalkin's formula by replacing each vertex weight \(n\) with the quantum number \([n]_y=\frac{y^{n/2}-y^{-n/2}}{y^{1/2}-y^{-1/2}}\). The refined count of the above tropical curves yields a symmetric Laurent polynomial, which does not depend on the choice of the point constraint as shown by \textit{I. Itenberg} [Int. Math. Res. Not. 2013, No. 23, 5289--5320 (2013; Zbl 1329.14114)], and remarkably specializes to the classical Severi degree as \(y=1\) and specializes to the tropical Welschinger invariant (equal for \(g=0\) to the Welschinger invariant of the plane) as \(y=-1\). The authors show that the refined Severi degrees for fixed \(\delta=\frac{(d-1)(d-2)}{2}-g\) and large \(d\) are polynomials in \(d\) and in \(y\), which, moreover, coincide with the Göttsche-Schende refinements of Severi degrees [\textit{L. Göttsche} and \textit{V. Shende}, Geom. Topol. 18, No. 4, 2245--2307 (2014; Zbl 1310.14012)] under some additional restrictions. The authors show that this theory smoothly extends to other toric surfaces and link it with the enumeration of floor diagrams.