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## On Hurwitz--Severi numbers

Cornell University
,
2016.
No. 06935.

Yurii Burman, Shapiro B.

For a point p in a complex projective plane and a triple (g,d,l) of non-negative
integers we define a plane Hurwitz number of the Severi variety
W_{g,d,l} consisting of all reduced irreducible plane curves of
genus g and degree d+l having an l-fold node at p and at
most ordinary nodes as singularities at the other points. In the cases
d+l >= g+2 and d+2l >= g+2 > d+l we express the plane
Hurwitz numbers via appropriate ordinary Hurwitz numbers. The
remaining case d+2l<g+2 is still widely open.

Keywords: числа ГурвицапроекцияHurwitz numbersplane algebraic curveprojectionплоская алгебраическая кривая

Publication based on the results of:

Yu. Burman, Serge Lvovski, Moscow Mathematical Journal 2015 Vol. 15 No. 1 P. 31-48

Suppose that C ⊂ P^2 is a general enough nodal plane curve
of degree > 2, : \hat C → C is its normalization, and π: C′ → P^1 is a finite
morphism simply ramified over the same set of points as a projection
pr_p ◦ν : \hat C → P1, where p ∈ P^2 ...

Added: January 14, 2015

Burman Y. M., Lvovsky S., / Cornell University. Series math "arxiv.org". 2013. No. 1904.

Suppose that C⊂P2 is a general enough smooth plane curve of degree >2 and that π:C→P1 is a finite morphism simply ramified over the same set of points as a projection prp:C→P1, where p∈P2∖C. We prove that the morphism π is equivalent to such a projection if and only if it extends to a finite ...

Added: November 14, 2013

Mironov A., Morozov A., Natanzon S. M., Journal of Knot Theory and Its Ramifications 2014 Vol. 23 No. 6 P. 1-16

The classical Hurwitz numbers of degree n together with the Hurwitz numbers of the seamed surfaces of degree n give rise to the Klein topological field theory. We extend this construction to the Hurwitz numbers of all degrees at once. The corresponding Cardy-Frobenius algebra is induced by arbitrary Young diagrams and arbitrary bipartite graphs. It ...

Added: April 2, 2014

Bychkov B., Функциональный анализ и его приложения 2015 Т. 49 № 2 С. 1-6

The investigation of decompositions of a permutation into a product of permutations
satisfying certain conditions plays a key role in the study of meromorphic functions or, equivalently,
branched coverings of the 2-sphere; it goes back to A. Hurwitz' work in the late nineteenth century.
In 2000 M. Bousquet-Melou and G. Schaeffer obtained an elegant formula for the number ...

Added: July 18, 2015

Mironov A., Morozov A., Natanzon S. M., Journal of Geometry and Physics 2012 Vol. 62 P. 148-155

We establish a correspondence between Young diagrams and differential operators of infinitely many variables. These operators form a commutative associative algebra isomorphic to the algebra of the conjugated classes of finite permutations of the set of natural numbers. The Schur functions form a complete system of common eigenfunctions of these differential operators, and their eigenvalues ...

Added: September 19, 2012

Natanzon S. M., Journal of Physics: Conference Series 2016 Vol. 670 P. 1-6

We describe all formal symmetric solutions of dispersionless 2D Toda hierarchy. This classication we use for solving of two classical problems: 1) The calculation of conformal mapping of an arbitrary simply connected domain to the standard disk; 2) Calculation of 2-Hurwitz numbers of genus 0. ...

Added: February 10, 2016

Dunin-Barkowski P., Kramer R., Popolitov A. et al., Journal of Geometry and Physics 2019 Vol. 137 P. 1-6

We give a new proof of the cut-and-join equation for the monotone Hurwitz numbers, derived first by Goulden, Guay-Paquet, and Novak. The main interest in this particular equation is its close relation to the quadratic loop equation in the theory of spectral curve topological recursion, and we recall this motivation giving a new proof of ...

Added: February 20, 2019

Zabrodin A., Journal of Physics A: Mathematical and Theoretical 2013 Vol. 46 No. 18 P. 185203

We study the integrable structure of the 2D Laplacian growth problem with zero surface tension in an infinite channel with periodic boundary conditions in a transverse direction. Similarly to the Laplacian growth in radial geometry, this problem can be embedded into the 2D Toda lattice hierarchy in the zero dispersion limit. However, the relevant solution ...

Added: April 29, 2013

Alexandrov A., Mironov A., Morozov A. et al., Journal of Physics A: Mathematical and Theoretical 2012 No. 45 P. 1-10

We construct partition functions that are tau-functions of integrable hierarchies. ...

Added: September 19, 2012

Shirokov D., Вестник Самарского государственного технического университета. Серия: Физико-математические науки 2015 Т. 19 № 1 С. 117-135

In this paper we consider expressions in real and complex Clifford algebras, which we call contractions or averaging. We consider contractions of arbitrary Clifford algebra element. Each contraction is a sum of several summands with different basis elements of Clifford algebra. We consider even and odd contractions, contractions on ranks and contractions on quaternion types. ...

Added: October 16, 2015

Bychkov B., Dunin-Barkowski P., Kazaryan M. et al., / Cornell University. Series math "arxiv.org". 2020. No. 2012.14723.

We study the n-point differentials corresponding to Kadomtsev-Petviashvili tau functions of hypergeometric type (also known as Orlov-Scherbin partition functions), with an emphasis on their ℏ2-deformations and expansions.
Under the naturally required analytic assumptions, we prove certain higher loop equations that, in particular, contain the standard linear and quadratic loop equations, and thus imply the blobbed topological recursion. We ...

Added: April 20, 2022

Burman Y. M., Zvonkine D., European Journal of Combinatorics 2010 Vol. 31 No. 1 P. 129-144

Consider factorizations into transpositions of an n-cycle in the symmetric group Sn. To every such factorization we assign a monomial in variables wij that retains the transpositions used, but forgets their order. Summing over all possible factorizations of n-cycles we obtain a polynomial that happens to admit a closed expression. From this expression we deduce ...

Added: November 7, 2012

Natanzon S. M., Orlov A. Y., Theoretical and Mathematical Physics 2020 Vol. 204 No. 3 P. 1166-1194

To obtain a generating function of the most general form for Hurwitz numbers with arbitrary base surfaceand arbitrary ramification profiles, we consider a matrix model constructed according to a graph on anoriented connected surfaceΣwith no boundary. The vertices of this graph, called stars, are small discs,and the graph itself is a clean dessin d’enfants. We ...

Added: September 27, 2020

Kazaryan M., Lando S., Успехи математических наук 2015 Т. 70 № 3 С. 70-106

This paper reviews modern approaches to the construction of formal solutions to integrable hierarchies of mathematical physics whose coefficients are answers to various enumerative problems. The relationship between these approaches and the combinatorics of symmetric groups and their representations is explained. Applications of the results to the construction of efficient computations in problems related to ...

Added: September 21, 2015

Bychkov B., Dunin-Barkowski P., Shadrin S., European Journal of Combinatorics 2020 Vol. 90 P. 103184

In this paper we prove, in a purely combinatorial-algebraic way, a structural quasi-polynomiality property for the Bousquet-Mélou–Schaeffer numbers. Conjecturally, this property should follow from the Chekhov–Eynard–Orantin topological recursion for these numbers (or, to be more precise, the Bouchard–Eynard version of the topological recursion for higher order critical points), which we derive in this paper from ...

Added: September 22, 2020

Alexeevski A., Natanzon S. M., American Mathematical Society Translations 2014 Vol. 234 P. 1-12

In 2001 Ivanov and Kerov associated with the infinite permutation group S∞ certain commutative associative algebra A∞ called the algebra of conjugacy classes of partial elements. A standard basis of A∞ islabeled by Yang diagrams of all orders. Mironov, Morozov, Natanzon, 2012, have proved that the completion of A∞ is isomorphic to the direct product ...

Added: April 2, 2014

Costa A., Sergey Natanzon, Shapiro B., Annales Academiae Scientiarum Fennicae Mathematica 2018 Vol. 43 P. 349-363

In this article, to each generic real meromorphic function (i.e., having only simple branch points in the appropriate sense) we associate a certain combinatorial gadget which we call the park of a function. We show that the park determines the topological type of the generic real meromorphic function and the set of parks produce an stratification ...

Added: March 4, 2018

Natanzon S. M., Zabrodin A., International Mathematics Research Notices 2015 Vol. 2015 No. 8 P. 2082-2110

We explicitly construct the series expansion for a certain class of solutions to the 2D Toda hierarchy in the zero dispersion limit, which we call symmetric solutions. We express the Taylor coefficients through some universal combinatorial constants and find recurrence relations for them. These results are used to obtain new formulas for the genus 0 ...

Added: April 2, 2014

Dunin-Barkowski P., Lewanski D., Popolitov A. et al., Journal of London Mathematical Society 2015 Vol. 92 No. 3 P. 547-565

In this paper, we present an example of a derivation of an ELSV-type formula using the methods of topological recursion. Namely, for orbifold Hurwitz numbers we give a new proof of the spectral curve topological recursion, in the sense of Chekhov, Eynard and Orantin, where the main new step compared to the existing proofs is ...

Added: November 16, 2015

А.Д. Миронов, А.Ю. Морозов, С.М. Натанзон, Теоретическая и математическая физика 2011 Т. 166 № 1 С. 3-27

We define cut-and-join operators in Hurwitz theory for merging two branch points of an arbitrary type. These operators have two alternative descriptions: (1) the GL characters are their eigenfunctions and the symmetric group characters are their eigenvalues; (2) they can be represented as W-type differential operators (in particular, acting on the time variables in the ...

Added: November 24, 2012

Dunin-Barkowski Petr, Kazarian Maxim, Orantin N. et al., Advances in Mathematics 2015 Vol. 279 P. 67-103

In this paper we give a new proof of the ELSV formula. First, we refine an argument of Okounkov and Pandharipande in order to prove (quasi-)polynomiality of Hurwitz numbers without using the ELSV formula (the only way to do that before used the ELSV formula). Then, using this polynomiality we give a new proof of ...

Added: September 24, 2015

Shapiro B., Yurii Burman, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 2019 Vol. XIX No. 1 P. 155-167

For a point p of the complex projective plane and a triple (g,d,l) of non-negative integers we define a Hurwitz--Severi number H(g,d,l) as the number of generic irreducible plane curves of genus g and degree d+l having an l-fold node at p and at most ordinary nodes as singularities at the other points, such that the ...

Added: April 14, 2017

Bychkov B., Функциональный анализ и его приложения 2019 Т. 53 № 1 С. 16-30

В данной работе мы получили новые формулы для степеней стратов пространств Гурвица рода 0, отвечающих функциям с двумя непростыми критическими значениями с предписанными разбиениями кратностей прообразов. При этом один из прообразов имеет произвольную кратность, а другой кратность коразмерности 1. В качестве следствий мы получили новые выражения для серий двойных чисел Гурвица. ...

Added: October 20, 2016

Orlov A. Y., / Mathematical Physics. Series arXiv:1807.11056 " arXiv.org > math-ph > arXiv:1807.11056". 2018. No. 5.

Abstract: We shall consider the product of complex random matrices from the independent complex Ginibre ensembles. The product includes complex matrices
Zi,Z†i,i=1,…,n and 2n sources (complex matrices Ci and C∗i). Any such product can be represented by a chord diagram that encodes the order of the matrices in the product. We introduce the Euler characteristic E∗ of ...

Added: November 12, 2018