Hurwitz Schemes Research Articles

Routh–Hurwitz Stability and Quasiperiodic Attractors in a Fractional‐Order Model for Awareness Programs: Applications to COVID‐19 Pandemic

This work explores Routh–Hurwitz stability and complex dynamics in models for awareness programs to mitigate the spread of epidemics. Here, the investigated models are the integer‐order model for awareness programs and their corresponding fractional form. A non‐negative solution is shown to exist inside the globally attracting set (GAS) of the fractional model. It is also shown that the diseasefree steady state is locally asymptotically stable (LAS) given that R0 is less than one, where R0 is the basic reproduction number. However, as R0 > 1, an endemic steady state is created whose stability analysis is studied according to the extended fractional Routh–Hurwitz scheme, as the order lies in the interval (0,2]. Furthermore, the proposed awareness program models are numerically simulated based on the predictor‐corrector algorithm and some clinical data of the COVID‐19 pandemic in KSA. Besides, the model’s basic reproduction number in KSA is calculated using the selected data (R0 = 1.977828168). In conclusion, the findings indicate the effectiveness of fractional‐order calculus to simulate, predict, and control the spread of epidemiological diseases.

An algebraic lifting invariant of Ellenberg, Venkatesh, and Westerland

We define and prove basic properties of a lifting invariant of curves over an algebraically closed field k with a map to the projective line $${\mathbb {P}}^1_{k}$$ that was introduced by Ellenberg, Venkatesh, and Westerland. In other work of Liu, Zureick-Brown, and the author, this lifting invariant, and its connection to components of Hurwitz schemes, is applied to help understand the average number of unramified extensions of function fields with a given Galois group.

On the gonality of stable curves

AbstractIn this paper we use admissible covers to investigate the gonality of a stable curve C over . If C is irreducible, we compare its gonality to that of its normalization. If C is reducible, we compare its gonality to that of its irreducible components. In both cases we obtain lower and upper bounds. Furthermore, we show that four admissible covers constructed give rise to generically injective maps between Hurwitz schemes. We show that the closures of the images of three of these maps are components of the boundary of the target Hurwitz schemes, and the closure of the image of the remaining map is a component of a certain codimension‐1 subscheme of the boundary of the target Hurwitz scheme.

The unirationality of the Hurwitz schemes $\mathcal H_{10, 8}$ and $\mathcal H_{13, 7}$

We show that the Hurwitz scheme \mathcal{H}_{g,d} parametrizing d -sheeted simply branched covers of the projective line by smooth curves of genus g , up to isomorphism, is unirational for (g,d)=(10,8) and (13,7) . The unirationality is settled by using liaison constructions in \mathbb{P}^1 \times \mathbb{P}^2 and \mathbb{P}^6 respectively, and through the explicit computation of single examples over a finite field.

Hurwitz spaces and liftings to the Valentiner group

We study the components of the Hurwitz scheme of ramified coverings of P1 with monodromy given by the alternating group A6 and elements in the conjugacy class of product of two disjoint cycles. In order to detect the connected components of the Hurwitz scheme, inspired by the case of the spin structures studied by Fried for the 3-cycles, we use as invariant the lifting to the Valentiner group, a triple covering of A6. We prove that the Hurwitz scheme has two irreducible components when the genus of the covering is greater than zero, in accordance with the asymptotic solution found by Bogomolov and Kulikov.

Tropicalizing the space of admissible covers

We study the relationship between tropical and classical Hurwitz moduli spaces. Following recent work of Abramovich, Caporaso and Payne, we outline a tropicalization for the moduli space of generalized Hurwitz covers of an arbitrary genus curve. Our approach is to appeal to the geometry of admissible covers, which compactify the Hurwitz scheme. We study the relationship between a combinatorial moduli space of tropical admissible covers and the skeleton of the Berkovich analytification of the classical space of admissible covers. We use techniques from non-archimedean geometry to show that the tropical and classical tautological maps are compatible via tropicalization, and that the degree of the classical branch map can be recovered from the tropical side. As a consequence, we obtain a proof, at the level of moduli spaces, of the equality of classical and tropical Hurwitz numbers.

Connected components of Hurwitz schemes and Malle's conjecture

Let Z(X) be the number of degree-n extensions of Fq(t) with some specified Galois group and with discriminant bounded by X. The problem of computing the asymptotics for Z(X) can be related to a problem of counting Fq-rational points on certain Hurwitz spaces. Ellenberg and Venkatesh used this idea to develop a heuristic for the asymptotic behavior of Z0(X), the number of – geometrically connected – extensions, and showed that this agrees with the conjectures of Malle for function fields. We extend Ellenberg–Venkatesh's argument to handle the more complicated case of covers of P1 which may not be geometrically connected, and show that the resulting heuristic suggests a natural modification to Malle's conjecture which avoids the counterexamples, due to Klüners, to the original conjecture.

Coverings with prescribed ramification and Virasoro groups

In this article we study coverings with prescribed ramification from the point of view of the Sato Grassmannian and of the algebro-geometric theory of solitons. We show that the moduli space of such coverings, which is a Hurwitz scheme, is a subscheme of the Grassmannian. We give its equations and show that there is a Virasoro group that uniformizes it. We also characterize when a curve is a covering in terms of bilinear identities.

Equations of Hurwitz schemes in the infinite Grassmannian

AbstractThe main result proved in the paper is the computation of the explicit equations defining the Hurwitz schemes of coverings with punctures as subschemes of the Sato infinite Grassmannian. As an application, we characterize the existence of certain linear series on a smooth curve in terms of soliton equations. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Generating functions for Hurwitz–Hodge integrals

We describe explicit generating functions for a large class of Hurwitz–Hodge integrals. These are integrals of tautological classes on moduli spaces of admissible covers, a (stackily) smooth compactification of the Hurwitz schemes. Admissible covers and their tautological classes are interesting mathematical objects on their own, but recently they have proved to be a useful tool for the study of the tautological ring of the moduli space of curves, and the orbifold Gromov–Witten theory of DM stacks.

On the moduli of surfaces admitting genus two fibrations over elliptic curves

In this paper, we study the structure, deformations and the moduli spaces of complex projective surfaces admitting genus two fibrations over elliptic curves. We observe that a surface admitting a smooth fibration as above is elliptic, and we employ results on the moduli of polarized elliptic surfaces to construct moduli spaces of these smooth fibrations. In the case of nonsmooth fibrations, we relate the moduli spaces to the Hurwitz schemes \({\mathcal{H}}(1,X(d), n)\) of morphisms of degree n from elliptic curves to the modular curve X(d), d ≥ 3. Ultimately, we show that the moduli spaces in the nonsmooth case are fiber spaces over the affine line \({\mathbb{A}}^1\) with fibers determined by the components of \({\mathcal{H}}(1,X(d), n)\).

Irreducibility of Hurwitz spaces of coverings with one special fiber

Let Y be a smooth, projective complex curve of genus g ⩾ 1. Let d be an integer ⩾ 3, let e = { e 1, e 2,..., e r } be a partition of d and let | e | = Σ i=1 r ( e i − 1). In this paper we study the Hurwitz spaces which parametrize coverings of degree d of Y branched in n points of which n − 1 are points of simple ramification and one is a special point whose local monodromy has cyclic type e and furthermore the coverings have full monodromy group S d . We prove the irreducibility of these Hurwitz spaces when n − 1 + | e | ⩾ 2 d, thus generalizing a result of Graber, Harris and Starr [A note on Hurwitz schemes of covers of a positive genus curve, Preprint, math. AG/0205056].

A characterization of bielliptic curves and applications to their moduli spaces

We deal with the covers of degree 4 naturally associated to a bielliptic curve of genus g≥6, giving a proof of the unirationality of the moduli space ? g be of such curves, of the rationality of the Hurwitz scheme ℌ be 4, g of bielliptic curves of even genus g, whereas, when g is odd, we construct a finite map ℂ2 g -2→? g be and compute its degree.

Uniqueness of the Riemann minimal examples

We prove that a properly embedded minimal surface in R 3 of genus zero with infinite symmetry group is a plane, a catenoid, a helicoid or a Riemann minimal example. We introduce the language of Hurwitz schemes to understand the underlying moduli space of surfaces in our setting.

HITCHIN SYSTEMS AND KP EQUATIONS

International Journal of MathematicsVol. 07, No. 02, pp. 227-244 (1996) No AccessHITCHIN SYSTEMS AND KP EQUATIONSYINGCHEN LI and MOTOHICO MULASEYINGCHEN LIDepartment of Mathematics, University of California, Davis, CA 95616, USA and MOTOHICO MULASEDepartment of Mathematics, University of California, Davis, CA 95616, USAhttps://doi.org/10.1142/S0129167X9600013XCited by:7 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail FiguresReferencesRelatedDetailsCited By 7Tangency and a ruled surface associated with a Hitchin systemTaejung Kim1 May 2014 | Journal of Geometry and Physics, Vol. 79Differential Algebras with Banach-Algebra Coefficients I: from C*-Algebras to the K-Theory of the Spectral CurveMaurice J. Dupré, James F. Glazebrook and Emma Previato17 February 2012 | Complex Analysis and Operator Theory, Vol. 7, No. 4EQUATIONS OF THE MODULI OF HIGGS PAIRS AND INFINITE GRASSMANNIAND. HERNÁNDEZ-SERRANO, J. M. MUÑOZ PORRAS, and F. J. PLAZA MARTÍN2 May 2012 | International Journal of Mathematics, Vol. 20, No. 08Equations of Hurwitz schemes in the infinite GrassmannianJosé M. Muñoz Porras and Francisco J. Plaza Martín1 Jul 2008 | Mathematische Nachrichten, Vol. 281, No. 7Flows of Calogero–Moser SystemsDavid Ben-Zvi and Thomas Nevins1 January 2007 | International Mathematics Research Notices, Vol. 2007HMichiel Hazewinkel1 Jan 2000Секущие абелевых многообразий, тэта-функции и солитонные уравненияИскандер Асанович Тайманов and Iskander Asanovich Taimanov1 Jan 1997 | Успехи математических наук, Vol. 52, No. 1 Recommended Vol. 07, No. 02 Metrics History Received 9 October 1995 PDF download

The Geometry of the Compactification of the Hurwitz Scheme

The Geometry of the Compactification of the Hurwitz Scheme

On Row Relaxation Methods for Large Constrained Least Squares Problems

This paper addresses the question of how to construct a row relaxation method for solving large unstructured linear least squares problems, with or without linear constraints. The proposed approach combines the Herman–Lent–Hurwitz scheme for solving regularized least squares problems with the Lent–Censor–Hildreth method for solving linear constraints. However, numerical experiments show that the Herman–Lent–Hurwitz scheme has difficulty reaching a least squares solution. This difficulty is resolved by applying the Riley–Golub iterative improvement process.

On the existence of components of the Hilbert scheme with the expected number of moduli

In the study of the geometry of families of algebraic curves one common tool is the comparison between the extrinsic geometry, represented by properties of their projective embeddings and of the Hilbert scheme and the intrinsic geometry, represented by abstract properties and the moduli space. This comparison is often best represented by the natural map π : Hn,g,r →Mg from the Hilbert scheme Hn,g,r of smooth curves in IP r of degree n and genus g (or the Hurwitz scheme for r = 1, the Severi variety for r = 2) to the moduli space of smooth curves of genus g. The behaviour of π is very strongly influenced by the nature of its fibers, the BrillNoether varieties. Now, whileMg is an irreducible variety, the Hilbert scheme Hn,g,r is in general reducible and has in fact many components for r ≥ 3. This difference has inspired many authors in the search of what could be a good component of the Hilbert scheme. One possible answer to such a question is to apply the above principle of comparison ([Ser],

Hurwitz Schemes and Irreducibility of Moduli of Algebraic Curves

Hurwitz Schemes and Irreducibility of Moduli of Algebraic Curves