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We describe and investigate a connection between the topology of isolated singularities of plane curves and the mutation equivalence, in the sense of cluster algebra theory, of the quivers associated with their morsifications.
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The stratum $mathcal{H}(a,-b_{1},dots,-b_{p})$ of meromorphic $1$-forms with a zero of order $a$ and poles of orders $b_{1},dots,b_{p}$ on the Riemann sphere has a map, the isoresidual fibration, defined by assigning to any differential its residues at the poles. We show that above the complement of a hyperplane arrangement, the resonance arrangement, the isoresidual fibration is an unramified cover of degree $frac{a!}{(a+2-p)!}$. Moreover, the monodromy of the fibration is computed for strata with at most three poles and a system of generators and relations is given for all strata. These results are obtained by associating to special differentials of the strata a tree, and by studying the relationship between the geometric properties of the differentials and the combinatorial properties of these trees.
We express the Masur-Veech volume and the area Siegel-Veech constant of the moduli space of meromorphic quadratic differential with simple poles as polynomials in the intersection numbers of psi-classes supported on the boundary cycles of the Deligne-Mumford compactification of the moduli space of curves. Our formulae are derived from lattice point count involving the Kontsevich volume polynomials that also appear in Mirzakhanis recursion for the Weil-Petersson volumes of the moduli space of bordered hyperbolic Riemann surfaces. A similar formula for the Masur-Veech volume (though without explicit evaluation) was obtained earlier by Mirzakhani through completely different approach. We prove further result: up to an explicit normalization factor depending only on the genus and on the number of cusps, the density of the orbit of any simple closed multicurve computed by Mirzakhani coincides with the density of square-tiled surfaces having horizontal cylinder decomposition associated to the simple closed multicurve. We study the resulting densities in more detail in the special case when there are no cusps. In particular, we compute explicitly the asymptotic frequencies of separating and non-separating simple closed geodesics on a closed hyperbolic surface of genus g for all small genera g and we show that in large genera the separating closed geodesics are exponentially less frequent. We conclude with detailed conjectural description of combinatorial geometry of a random simple closed multicurve on a surface of large genus and of a random square-tiled surface of large genus. This description is conditional to the conjectural asymptotic formula for the Masur-Veech volume in large genera and to the conjectural uniform asymptotic formula for certain sums of intersection numbers of psi-classes in large genera.
We express the Masur-Veech volume and the area Siegel-Veech constant of the moduli space $mathcal{Q}_{g,n}$ of genus $g$ meromorphic quadratic differentials with $n$ simple poles as polynomials in the intersection numbers of $psi$-classes with explicit rational coefficients. The formulae obtained in this article result from lattice point counts involving the Kontsevich volume polynomials that also appear in Mirzakhanis recursion for the Weil-Petersson volumes of the moduli spaces of bordered hyperbolic surfaces with geodesic boundaries. A similar formula for the Masur-Veech volume (though without explicit evaluation) was obtained earlier by Mirzakhani via completely different approach. Furthermore, we prove that the density of the mapping class group orbit of any simple closed multicurve $gamma$ inside the ambient set of integral measured laminations computed by Mirzakhani coincides with the density of square-tiled surfaces having horizontal cylinder decomposition associated to $gamma$ among all square-tiled surfaces in $mathcal{Q}_{g,n}$. We study the resulting densities (or, equivalently, volume contributions) in more detail in the special case $n=0$. In particular, we compute the asymptotic frequencies of separating and non-separating simple closed geodesics on a closed hyperbolic surface of genus $g$ for small $g$ and we show that for large genera the separating closed geodesics are $sqrt{frac{2}{3pi g}}cdotfrac{1}{4^g}$ times less frequent.
It is known that the coordinate ring of the Grassmannian has a cluster structure, which is induced from the combinatorial structure of a plabic graph. A plabic graph is a certain bipartite graph described on the disk, and there is a family of plabic graphs giving a cluster structure of the same Grassmannian. Such plabic graphs are related by the operation called square move which can be considered as the mutation in cluster theory. By using a plabic graph, we also obtain the Newton-Okounkov polytope which gives a toric degeneration of the Grassmannian. The purposes of this article is to survey these phenomena and observe the behavior of Newton-Okounkov polytopes under the operation called the combinatorial mutation of polytopes. In particular, we reinterpret some operations defined for Newton-Okounkov polytopes using the combinatorial mutation.
Let $S$ be a connected closed oriented surface of genus $g$. Given a triangulation (resp. quadrangulation) of $S$, define the index of each of its vertices to be the number of edges originating from this vertex minus $6$ (resp. minus $4$). Call the set of integers recording the non-zero indices the profile of the triangulation (resp. quadrangulation). If $kappa$ is a profile for triangulations (resp. quadrangulations) of $S$, for any $min mathbb{Z}_{>0}$, denote by $mathscr{T}(kappa,m)$ (resp. $mathscr{Q}(kappa,m)$) the set of (equivalence classes of) triangulations (resp. quadrangulations) with profile $kappa$ which contain at most $m$ triangles (resp. squares). In this paper, we will show that if $kappa$ is a profile for triangulations (resp. for quadrangulations) of $S$ such that none of the indices in $kappa$ is divisible by $6$ (resp. by $4$), then $mathscr{T}(kappa,m)sim c_3(kappa)m^{2g+|kappa|-2}$ (resp. $mathscr{Q}(kappa,m) sim c_4(kappa)m^{2g+|kappa|-2}$), where $c_3(kappa) in mathbb{Q}cdot(sqrt{3}pi)^{2g+|kappa|-2}$ and $c_4(kappa)in mathbb{Q}cdotpi^{2g+|kappa|-2}$. The key ingredient of the proof is a result of J. Kollar on the link between the curvature of the Hogde metric on vector subbundles of a variation of Hodge structure over algebraic varieties, and Chern classes of their extensions. By the same method, we also obtain the rationality (up to some power of $pi$) of the Masur-Veech volume of arithmetic affine submanifolds of translation surfaces that are transverse to the kernel foliation.
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