Polynomiality properties of tropical refined invariants

Abstract

Tropical refined invariants of toric surfaces constitute a fascinating interpolation between real and complex enumerative geometries via tropical geometry. They were originally introduced by Block and Göttsche, and further extended by Göttsche and Schroeter in the case of rational curves.In this paper, we study the polynomial behavior of coefficients of these tropical refined invariants. We prove that coefficients of small codegree are polynomials in the Newton polygon of the curves under enumeration, when one fixes the genus of the latter. This provides a somehow surprising resurgence, in some sort of dual setting, of the so-called node polynomials and Göttsche conjecture. Our methods are entirely combinatorial, hence our results may suggest phenomenons in complex enumerative geometry that have not been studied yet.In the particular case of rational curves, we extend our polynomiality results by including the extra parameter s recording the number of ψ classes. Contrary to the polynomiality with respect to ∆, the one with respect to s may be expected from considerations on Welschinger invariants in real enumerative geometry. This pleads in particular in favor of a geometric definition of Göttsche-Schroeter invariants.

Tropical curves in abelian surfaces III: pearl diagrams and multiple cover formulas

“…The result from [10] is similar to the conjecture from L. Göttsche [16] (proved in [35] and [22]) asserting that the number of curves with a prescribed number of nodes δ in a linear system L on a surface S passing through a given number of points is a polynomial. The difference is now that the genus is fixed rather than the number of nodes.…”

“…Then, we prove an analog of the result of E. Brugallé and A. Puentes [10]. They prove a polynomiality statement for the refined invariants of toric surfaces for which we have a floor diagram algorithm.…”

“…The obtained polynomials are known as node polynomials. Such an universality property for polynomials from [10] remains out of reach due to a lack of definition of the refined invariants in a general setting. Expanding the range of examples could contribute to checking such a universality.…”

“…Coefficients of fixed codegree. In [10], E. Brugallé and A. Puentes prove the following regularity result for the refined invariants of specific toric surfaces. For simplicity, we state it for the projective plane.…”

Floor diagrams and enumerative invariants of line bundles over an elliptic curve

“…Recently, E. Brugallé and A. J. Puentes [13] proved a polynomiality statement for the coefficients of a fixed codegree in the families of refined invariance of toric surfaces. Such a behavior could also be studied for the hereby introduced refined invariants.…”