A. We derive explicit formulae for the generating series of mixed Grothendieck dessins d'enfant/monotone/simple Hurwitz numbers, via the semi-in nite wedge formalism. This reveals the strong piecewise polynomiality in the sense of Goulden-Jackson-Vakil, generalising a result of Johnson, and provides a new explicit proof of the piecewise polynomiality of the mixed case. Moreover, we derive wall-crossing formulae for the mixed case. These statements specialise to any of the three types of Hurwitz numbers, and to the mixed case of any pair. double case is still an active topic of research. A tropical approach for the monotone case is developed in [DK ] and in [Hah ].Grothendieck dessins d'enfant or strictly monotone Hurwitz numbers. Dessins d'enfant have been introduced by Grothendieck in [Gro ]. Their enumeration counts Hurwitz coverings of genus and degree d over the Riemann sphere, with two rami cations µ and ν over 0 and ∞, and a single further rami cation over 1, whose length is determined by the Riemann-Hurwitz formula.The CEO recursion for the r = 2 orbifold case (i.e. ν = (2) d /2 ) was proved in [Nor ; DMSS ] and is known as enumeration of ribbon graphs. In [DM ] the CEO recursion was conjectured for the general r -orbifold case (i.e. ν = (r ) d /r ), which was then proved in [DOPS ] by combinatorial methods. Moreover, CEO recursion was proved in [KZ ] for the case of two consecutive intermediate rami cations of xed lengths instead of one, together with a proof of the KP integrability and the Virasoro constraints for the same case. ELSV formulae for simple, orbifold or double cases are still not known. The connection between strictly monotone numbers and dessins d'enfant counting is explained in the following.Mixed cases. It is natural to interpolate several Hurwitz enumerative problems, by allowing di erent conditions on di erent blocks of intermediate rami cations. In fact hypergeometric tau functions for the D Toda integrable hierarchy have been proved to have several explicit combinatorial interpretations [HO ]-one of them is in terms of mixed double strictly monotone/weakly monotone Hurwitz numbers, another one involves a mixed case of combinatorial problems, in which the part relative to the strictly monotone rami cations can be interpreted in terms of Grothendieck dessins d'enfant. This implies indirectly that the enumeration of Grothendieck dessins and strictly monotone numbers coincide. A direct proof of this fact through the Jucys correspondence [Juc ] is derived in [ALS ].A combinatorial study of the mixed double monotone-simple case can be found in [GGN ], in which piecewise polynomiality is proved. A tropical interpretation providing an algorithm to compute the chamber polynomials and wall-crossing formulae via Erhart theory is developed in [Hah ]. Further developments on CEO topological recursion for general Hurwitz enumerative geometric problems appear in [ACEH ]. This study con rms the existence of an ELSV-type formula for mixed Hurwitz enumerative problems.
In [8], it was shown that not all abstract non-hyperelliptic tropical curves of genus 3 can be realized as a tropicalization of a quartic in R 2 . In this paper, we focus on the interior of the maximal cones in the moduli space and classify all curves which can be realized as a faithful tropicalization in a tropical plane. Reflecting the algebro-geometric world, we show that these are all curves but the tropicalizations of realizably hyperelliptic algebraic curves.Our approach is constructive: For a curve which is not the tropicalization of a hyperelliptic algebraic curve, we explicitly construct a realizable model of the tropical plane in R n , and a faithfully tropicalized quartic in it. These constructions rely on modifications resp. tropical refinements. Conversely, we prove that the tropicalizations of hyperelliptic algebraic curves cannot be embedded in such a fashion. For that, we rely on the theory of tropical divisors and embeddings from linear systems [3,21], and recent advances in the realizability of sections of the tropical canonical divisor [30].
Hurwitz numbers count genus g, degree d covers of the complex projective line with fixed branched locus and fixed ramification data. An equivalent description is given by factorisations in the symmetric group. Simple double Hurwitz numbers are a class of Hurwitz-type counts of specific interest. In recent years a related counting problem in the context of random matrix theory was introduced as so-called monotone Hurwitz numbers. These can be viewed as a desymmetrised version of the Hurwitz-problem. A combinatorial interpolation between simple and monotone double Hurwitz numbers was introduced as mixed double Hurwitz numbers and it was proved that these objects are piecewise polynomial in a certain sense. Moreover, the notion of strictly monotone Hurwitz numbers has risen in interest as it is equivalent to a certain Grothendieck dessins d'enfant count. In this paper, we introduce a combinatorial interpolation between simple, monotone and strictly monotone double Hurwitz numbers as triply interpolated Hurwitz numbers. Our aim is twofold: Using a connection between triply interpolated Hurwitz numbers and tropical covers in terms of so-called monodromy graphs, we give algorithms to compute the polynomials for triply interpolated Hurwitz numbers in all genera using Erhart theory. We further use this approach to study the wall-crossing behaviour of triply interpolated Hurwitz numbers in genus 0 in terms of related Hurwitz-type counts. All those results specialise to the extremal cases of simple, monotone and Grothendieck dessins d'enfants Hurwitz numbers.
In this paper, we study linear spaces of matrices defined over discretely valued fields and discuss their dimension and minimal rank drops over the associated residue fields. To this end, we take first steps into the theory of rank-metric codes over discrete valuation rings by means of skew algebras derived from Galois extensions of rings. Additionally, we model projectivizations of rank-metric codes via Mustafin varieties, which we then employ to give sufficient conditions for a decrease in the dimension.
Mustafin varieties are flat degenerations of projective spaces, induced by a set of lattices in a vector space over a non-archimedean field. They were introduced by Mustafin (Math USSR-Sbornik 34(2):187, 1978) in the 70s in order to generalise Mumford's groundbreaking work on the unformisation of curves to higher dimension. These varieties have a rich combinatorial structure as can be seen in pioneering work of Cartwright et al. (Selecta Math 17(4):757-793, 2011). In this paper, we introduce a new approach to Mustafin varieties in terms of images of rational maps, which were studied in Li (IMRN, 2017). Applying tropical intersection theory and tropical convex hull computations, we use this method to give a new combinatorial description of the irreducible components of the special fibers of Mustafin varieties. Finally, we outline a first application of our results in limit linear series theory.
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