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Wall crossings for double Hurwitz numbers. (English) Zbl 1231.14023
This paper continues the very active investigation of the enumerative geometry of ramified covers of the projective line. Specifically, the authors study double Hurwitz numbers \(H_g(\underline{x})\), which by definition count covers of \(\mathbb{CP}^1\) by algebraic curves of genus \(g\) with ramification profile over 0 and \(\infty\) specified by the vector \(\underline{x}\) and with simple ramification at the remaining number of points prescribed by the Riemann-Hurwitz formula. Here \(\underline{x}\) has both positive and negative entries, and they view the positive (resp. negative) ones as indicating the monodromy over 0 (resp. \(\infty\)). The point is to fix the genus but let \(\underline{x}\) vary and thus view the double Hurwitz numbers as a function defined on the locus \(|\underline{x}|=0\). The paper is a beautiful extension/application of the authors’ previous work [“Tropical Hurwitz numbers”, J. Algebr. Comb. 32, 241–265 (2010; Zbl 1218.14058)] where a combinatorial framework was developed for describing these numbers. The authors study their earlier formula and use it to answer several interesting questions about this double Hurwitz number function. Although their proofs are very much in the combinatorial realm, much of the motivation for the theorems they prove comes from geometry, so let us briefly recall this context while at the same time describing in more detail the main theorems of the paper.
The traditional Hurwitz numbers, counting covers with all ramification simple except possibly at a single point, have had an exciting role in modern mathematics, touching on many areas and stimulating much research. One particularly remarkable instance is the ELSV formula, which gives a precise description of each Hurwitz number for fixed genus and ramification profile in terms of an integral over \(\overline{\mathcal{M}}_{g,n}\) of a certain formal power series involving tautological classes. Of course, here “integral” means the algebro-geometric formulation in intersection theory of capping with the fundamental class \([\overline{\mathcal{M}}_{g,n}]\). This formula serves as an intriguing bridge between the enumerative geometry of Hurwitz theory and the intersection theory of the (Deligne-Mumford) moduli space of curves. It implies, in particular, a polynomality condition on the Hurwitz numbers. For double Hurwitz numbers, the enumerative geometry is expected to be related to the intersection theory of some compactified universal Picard variety Pic\(_{g,n}\) parameterizing marked curves with a line bundle. This is described in the paper of I.P. Goulden, D. M. Jackson and R. Vakil [“Towards the geometry of double Hurwitz numbers”, Adv. Math. 198, No. 1, 43–92 (2005; Zbl 1086.14022)] which serves as the main background for the present paper. In particular, GJV shows that the double Hurwitz numbers are piece-wise polynomial, and as they mention, understanding the behavior of these functions can potentially help predict the appropriate compactification of the universal Picard variety to consider.
The first main result of the present paper is a new proof of GJV’s piece-wise polynomality result concerning double Hurwitz numbers. In the authors’ earlier above-cited work, they describe a formula for double Hurwitz numbers as a sum of some combinatorial data over a finite collection of graphs. Roughly, they consider directed trivalent genus \(g\) graphs with leaves labeled by the ramification entries \(x_i\) and a natural number associated to each internal edge such that these weights satisfy a balancing condition. They call these monodromy graphs. Their formula then reads \[ H_g(\underline{x}) = \sum_\Gamma \frac{\varphi_\Gamma}{|Aut(\Gamma)|} \] where the sum is over all monodromy graphs \(\Gamma\) and \(\varphi_\Gamma\) is the product of the weights of all internal edges. In genus zero the edge weights are uniquely determined by \(\underline{x}\), whereas in positive genus this is no longer the case, so much of the paper is devoted to studying the space of possible edge-weightings.
In addition to the piece-wise polynomality result, which as mentioned had already been proven using entirely different methods in GJV, the present paper goes on to use their monodromy graph formula and results about the space of edge-weightings to prove several other impressive results: i) they characterize the walls of polynomality, ii) perhaps most importantly, they produce a wall-crossing formula, and iii) they show that the double Hurwitz number function is either even or odd.
The authors nicely summarize the combinatoric tools underlying their analysis of monodromy graphs and begin the paper with an illustrative example, which this reviewer recommends reading. In short, the basic idea is that for a fixed directed graph and vector \(\underline{x}\), the space of possible internal edge weightings can be viewed as the lattice points of certain bounded polytopes. This enables them to translate the Hurwitz theory questions that interest them into questions about lattice points of polytopes, for which there are ready-made tools available. The translation is not as trivial as this sounds, but this is the basic idea underpinning their paper.

MSC:
14H30 Coverings of curves, fundamental group
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14N20 Configurations and arrangements of linear subspaces
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
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