Curves Of Genus Research Articles

Belyi injectivity for outer representations on certain quotients of étale fundamental groups of hyperbolic curves of genus zero

In the present paper, we study certain quotients of the étale fundamental group of a hyperbolic curve over a field. We prove that the action of the outer automorphism group of a certain quotient of the étale fundamental group of a hyperbolic curve over an algebraically closed field on its conjugacy classes of open subgroups is faithful. Also, we prove that, if k is either a number field or a p-adic local field, then the outer Galois representation associated to a certain quotient of the geometric fundamental group of ℙk1\{0,1,∞} is injective.

The Hecke system of harmonic Maass functions and applications to modular curves of higher genera

In monstrous moonshine, the replication formula and the Hecke operator played a central role. We generalize the replication formula and the Hecke operator to higher genus modular curves, with an eye toward extending moonshine to these cases. Specifically, we extend the definitions of replicates and a Hecke operator to harmonic Maass functions on modular curves of higher genera and obtain uniform proofs for numerous arithmetic properties of Fourier coefficients of modular functions of arbitrary level, which have been proved only for special cases of curves of genus zero or small prime levels.

Asymptotic properties of conditional U-statistics using delta sequences

Stute (1991) introduced a class of so-called conditional U-statistics, which may be viewed as a generalization of the Nadaraya-Watson estimates of a regression function. Stute proved their strong pointwise consistency to: r ( k ) ( φ , t ) : = E [ φ ( Y 1 , … , Y k ) | ( X 1 , … , X k ) = t ] , for t ∈ R d k . This article deals with a quite general non parametric statistical curve estimation setting including the Stute estimator as a particular case. The class of “delta sequence estimators” is defined and treated here. This class includes also the orthogonal series and histogram methods. The theoretical results concerning the exponential inequalities and the asymptotic normality, established in this article, are (or will be) key tools for many further developments in functional estimation. As a by-product of our proofs, we state consistency results for the delta sequences conditional U-statistics estimator, under the random censoring. Potential applications include discrimination problems, metric learning and multipartite ranking, Kendall rank correlation coefficient, generalized U-statistics, and set indexed conditional U-statistics.

Cyclic étale coverings of generic curves and ordinariness of dormant opers

The ordinariness of elliptic curves is essential in proving various expected properties of elliptic curves in positive characteristic and can be extended to algebraic curves of arbitrary genus. The present paper deals with another kind of extension, i.e., ordinariness of dormant sl2-opers, or more generally, dormant sln-opers. We prove that, for dormant sl2-opers on elliptic curves, this notion is essentially equivalent to the classical ordinariness. Moreover, the main result of the present paper asserts that the pull-back of an ordinary dormant sln-oper on a general curve by a cyclic covering is ordinary whenever the order of its Galois group is prime to the characteristic of the base field. This result may be regarded as an analogue of a result by S. Nakajima for ordinary algebraic curves.

General Reference and Design S–N Curves Obtained for 1.2709 Tool Steel

At present, due to advanced fatigue calculation models, it is becoming more crucial to find a reliable source for design S-N curves, especially in the case of new 3D-printed materials. Such obtained steel components are becoming very popular and are often used for important parts of dynamically loaded structures. One of the commonly used printing steels is EN 1.2709 tool steel, which has good strength properties and high abrasion resistance, and can be hardened. The research shows, however, that its fatigue strength may differ depending on the printing method, and may be characterized by a wide scatter of the fatigue life. This paper presents selected S-N curves for EN 1.2709 steel after printing with the selective laser melting method. The characteristics are compared, and conclusions are presented regarding the resistance of this material to fatigue loading, especially in the tension-compression state. A combined general mean reference and design fatigue curve is presented, which incorporates our own experimental results as well as those from the literature for the tension-compression loading state. The design curve may be implemented in the finite element method by engineers and scientists in order to calculate the fatigue life.

Stable Cohomology of the Moduli Space of Trigonal Curves

Abstract We prove that the rational cohomology $H^{i}(\mathcal {T}_{g};\mathbf {Q})$ of the moduli space of trigonal curves of genus $g$ is independent of $g$ in degree $i<\lfloor g/4\rfloor .$ This makes possible to define the stable cohomology ring as $H^{\bullet }(\mathcal {T}_{g};\mathbf {Q})$ for a sufficiently large $g.$ We also compute the stable cohomology ring, which turns out to be isomorphic to the tautological ring. This is done by studying the embedding of trigonal curves in Hirzebruch surfaces and using Gorinov–Vassiliev’s method.

Separation of tripeptides in binary mixtures using ion-exchange membrane adsorber

The adsorption of three tripeptides in an ion-exchange membrane adsorber was analyzed in single and binary solutions, with the aim of evaluating the capability of the membrane adsorber to separate triglycine (GGG) from two other tripeptides: glycine-histidine-glycine (GHG) and glycine-tyrosine-glycine (GTG). The equilibrium adsorption of single peptide solutions followed the Langmuir isotherm and GTG showed the highest adsorption affinity. The dynamic adsorption was fitted with a generalized model, which was defined using dimensionless parameters and based on the continuity equation. In general, the calculated and experimental breakthrough curves were correlated with high agreement. It was found that the axial dispersion coefficient was independent of the peptide molecule and that it increased with flow rate. The competitive adsorption between peptides in binary solutions was analyzed using the extended and modified Langmuir equations. The adsorption equilibrium data were satisfactorily fitted with the modified Langmuir isotherm for GGG/GHG solutions, while the extended Langmuir isotherm was a better fit to the data for GGG/GTG solutions. The experimental breakthrough curves of the two peptide binary mixtures were simulated using the parameters calculated from the competitive isotherms and the parameters obtained from the breakthrough curves of the single peptide solutions. The separation of GGG from the GGG/GHG mixtures was possible. The GGG recovery was higher than 35% and the GGG molar fraction in the outlet stream was higher than 0.994.

Elliptic Dimers on Minimal Graphs and Genus 1 Harnack Curves

This paper provides a comprehensive study of the dimer model on infinite minimal graphs with Fock’s elliptic weights (Fock, Inverse spectral problem for GK integrable system. arXiv e-prints arXiv:1503.00289 , 2015). Specific instances of such models were studied in Boutillier et al. (Invent Math 208(1):109–189, 2017), Boutillier et al. (Probability theory and related fields, 2018) and de Tilière (Electron J Probab 26:1–86, 2021); we now handle the general genus 1 case, thus proving a non-trivial extension of the genus 0 results of Kenyon (Invent Math 150(2):409–439, 2002) and Kenyon and Okounkov (Duke Math J 131(3):499–524, 2006) on isoradial critical models. We give an explicit local expression for a two-parameter family of inverses of the Kasteleyn operator with no periodicity assumption on the underlying graph. When the minimal graph satisfies a natural condition, we construct a family of dimer Gibbs measures from these inverses, and describe the phase diagram of the model by deriving asymptotics of correlations in each phase. In the $$\mathbb {Z}^2$$ -periodic case, this gives an alternative description of the full set of ergodic Gibbs measures constructed in Kenyon et al. (Ann Math 163(3):1019–1056, 2006). We also establish a correspondence between elliptic dimer models on periodic minimal graphs and Harnack curves of genus 1. Finally, we show that a bipartite dimer model is invariant under the shrinking/expanding of 2-valent vertices and spider moves if and only if the associated Kasteleyn coefficients are antisymmetric and satisfy Fay’s trisecant identity.

Relative Richardson varieties

AbstractA Richardson variety in a flag variety is an intersection of two Schubert varieties defined by transverse flags. We define and study relative Richardson varieties, which are defined over a base scheme with a vector bundle and two flags. To do so, we generalise transversality of flags to a relative notion, versality, that allows the flags to be non-transverse over some fibers. Relative Richardson varieties share many of the geometric properties of Richardson varieties. We generalise several geometric and cohomological facts about Richardson varieties to relative Richardson varieties. We also prove that the local geometry of a relative Richardson variety is governed, in a precise sense, by the two intersecting Schubert varieties, giving a generalisation, in the flag variety case, of a theorem of Knutson–Woo–Yong; we also generalise this result to intersections of arbitrarily many relative Schubert varieties. We give an application to Brill–Noether varieties on elliptic curves, and a conjectural generalisation to higher genus curves.

Universal asymptotic correlation functions for point group C_{6v} and an observation for triangular lattice Q-state Potts model.

We investigate a universal curve in asymptotic correlation functions of off-critical systems that possess C_{6v} symmetry following the argument for C_{4v} symmetry in our previous paper [Phys. Rev. E 102, 032141 (2020)2470-004510.1103/PhysRevE.102.032141]. Unlike the C_{4v} case, a minimal asymptotic form exists, which contains only two free parameters: the normalization constant and the modulus of the universal curve. We perform large-scale Monte Carlo simulations of the triangular lattice Q-state Potts model above the transition temperature. For Q=1, 2, 3, and 4, we successfully obtain numerical evidence that the minimal form gives the leading asymptotic behavior. We also discuss the possibility that the corrections to the minimal form are expressed using this form as a building block. From the minimal form with optimized parameters, we derive the equilibrium crystal shape of the honeycomb lattice Potts model, which is given by an algebraic curve of genus 1 and is universal among models with C_{6v}. Although the curve differs from those obtained in the C_{4v} case, the latter curves also have genus 1. We indicate that the birational equivalence concept can play an important role in comparing asymptotic forms for different point group symmetries, for example, C_{6v} and C_{4v}.

The dimension of the moduli space of pointed algebraic curves of low genus

We explicitly compute the moduli space pointed algebraic curves with a given numerical semigroup as Weierstrass semigroup for many cases of genus at most seven and when possible describe the structure as cone over an explicitly given variety. We determine the dimension of the moduli space for all semigroups of genus seven.

Nonabelian Hodge theory for stacks and a stacky P=W conjecture

We introduce a version of the P=W conjecture relating the Borel–Moore homology of the stack of representations of the fundamental group of a genus g Riemann surface with the Borel–Moore homology of the stack of degree zero semistable Higgs bundles on a smooth projective complex curve of genus g. In order to state the conjecture we propose a construction of a canonical isomorphism between these Borel–Moore homology groups. We relate the stacky P=W conjecture with the original P=W conjecture concerning the cohomology of smooth moduli spaces, and the PI=WI conjecture concerning the intersection cohomology groups of singular moduli spaces. In genus zero and one, we prove the conjectures that we introduce in this paper.

On the Approximate Polar Curves of Foliations

We present a decomposition theorem of the generic polar curves of a generalized curve foliation with only one separatrix and the Hamiltonian foliations defined by the approximate roots of the generatrix. This is a generalization to foliations of the decomposition theorem of approximate Jacobians given by García Barroso and Gwoździewicz for plane branches.

Degenerations of bundle moduli

Over a family X of complete curves of genus g, which gives the degeneration of a smooth curve into one with nodal singularities, we build a moduli space which is the moduli space of holomorphic SL(n,C) bundles over the generic smooth curve Xt in the family, and is a moduli space of bundles equipped with extra structure at the nodes for the nodal curves in the family. This moduli space is a quotient by (C⁎)s of a moduli space on the desingularization. Taking a “maximal” degeneration of the curve into a nodal curve built from the glueing of three-pointed spheres, we obtain a degeneration of the moduli space of bundles into a (C⁎)(3g−3)(n−1)-quotient of a (2g−2)-th power of a space associated to the three-pointed sphere. Via the Narasimhan-Seshadri theorem, the moduli space of SL(n,C) bundles on the smooth curve is a space of representations of the fundamental group into SU(n) (the “symplectic picture”). We obtain the degenerations also in this symplectic context, in a way that is compatible with the holomorphic degeneration, so that our limit space is also a (S1)(3g−3)(n−1) symplectic quotient of a (2g−2)-th power of a space associated to the three-pointed sphere.

Slope equality of non-hyperelliptic Eisenbud–Harris special fibrations of genus 4

AbstractThe Horikawa index and the local signature are introduced for relatively minimal fibered surfaces whose general fiber is a non-hyperelliptic curve of genus 4 with unique trigonal structure.

The CM class number one problem for curves of genus 2

Gauss’s class number one problem, solved by Heegner, Baker, and Stark, asked for all imaginary quadratic fields for which the ideal class group is trivial. An application of this solution gives all elliptic curves that can be defined over the rationals and have a large endomorphism ring (CM). Analogously, to get all CM curves of genus two defined over the smallest number fields, we need to find all quartic CM fields for which the CM class group (a quotient of the ideal class group) is trivial. We solve this CM class number one problem. We prove that the list given in Bouyer–Streng [LMS J Comput Math 18(1):507–538, 2015, Tables 1a, 1b, 2b, and 2c] of maximal CM curves of genus two defined over the reflex field is complete. We also prove that there are exactly 21 simple CM curves of genus two over $$\mathbb {C}$$ that can be defined over $$\mathbb {Q}$$ .

Generalized Tevelev degrees of [formula omitted

Let (C,p1,…,pn) be a general curve. We consider the problem of enumerating covers of the projective line by C subject to incidence conditions at the marked points. These counts have been obtained by the first named author with Pandharipande and Schmitt via intersection theory on Hurwitz spaces and by the second named author with Farkas via limit linear series. In this paper, we build on these two approaches to generalize these counts to the situation where the covers are constrained to have arbitrary ramification profiles: that is, additional ramification conditions are imposed at the marked points, and some collections of marked points are constrained to have equal image.

A Laplacian to Compute Intersection Numbers on overline{{{mathcal {M}}}}_{g,n} and Correlation Functions in NCQFT

Let F_g(t) be the generating function of intersection numbers of psi -classes on the moduli spaces overline{{{mathcal {M}}}}_{g,n} of stable complex curves of genus g. As by-product of a complete solution of all non-planar correlation functions of the renormalised Phi ^3-matrical QFT model, we explicitly construct a Laplacian Delta _t on a space of formal parameters t_i which satisfies exp (sum _{gge 2} N^{2-2g}F_g(t))=exp ((-Delta _t+F_2(t))/N^2)1 as formal power series in 1/N^2. The result is achieved via Dyson-Schwinger equations from noncommutative quantum field theory combined with residue techniques from topological recursion. The genus-g correlation functions of the Phi ^3-matricial QFT model are obtained by repeated application of another differential operator to F_g(t) and taking for t_i the renormalised moments of a measure constructed from the covariance of the model.

Quadratic Chabauty for Atkin–Lehner quotients of modular curves of prime level and genus 4, 5, 6

Quadratic Chabauty for Atkin–Lehner quotients of modular curves of prime level and genus 4, 5, 6

Curves of stationary acceleration according to alternative frame

This paper investigates curves of stationary acceleration by using alternative frame which includes the principal normal vector, the derivative of principal normal vector and the Darboux vector. Mentioned curves are studied by the way of rigid body motions, that is to say a point in the moving body follows the curve and the alternative frame in the moving body stays aligned with the members of frame. It is determined that in which condition this special motion becomes to stationary acceleration motion. The matrix representations of a constant vector related to velocity vector of the motion which is used to characterize stationary acceleration is obtained by means of alternative frame curvatures. Some examinations are developed with some solutions of differential equations. The main result is attained as: general helix curves with linear curvature and torsion functions are curves of stationary acceleration which are curves in the rigid body motions group SE(3) correlated with robotics. The paths designed as stationary acceleration curves can lead the way to control the end-effectors of robots. Finally, some explanatory examples are imputed.