Curves Of Genus Research Articles

Hyperelliptic Covers of Different Degree for Elliptic Curves

In elliptic curve cryptography (ECC) and hyperelliptic curve cryptography (HECC), the size of cipher-text space defined by the cardinality of Jacobian is a significant factor to measure the security level. Counting problems on Jacobians of elliptic curve can be solved in polynomial time by Schoof–Elkies–Atkin (SEA) algorithm. However, counting problems on Jacobians of hyperelliptic curves are solved less satisfactorily than those on elliptic curves. So, we consider the construction of the cover map from the hyperelliptic curves to the elliptic curves to convert point counting problems on hyperelliptic curves to those on elliptic curves. We can also use the cover map as a kind of cover attacks. Given an elliptic curve over an extension field of degree n , one might try to use the cover attack to reduce the discrete logarithm problem (DLP) in the group of rational points of the elliptic curve to DLPs in the Jacobian of a curve of genus g ≥ n over the base field. An algorithm has been proposed for finding genus 3 hyperelliptic covers as a cover attack for elliptic curves with cofactor 2. Our algorithms are about the cover map from hyperelliptic curves of genus 2 to elliptic curves of prime order. As an application, an example of an elliptic curve whose order is a 256-bit prime vulnerable to our algorithms is given.

Edge States for Generalized Iwatsuka Models: Magnetic Fields Having a Fast Transition Across a Curve

In this paper, we study the localization and propagation properties of the edge states associated with a class of magnetic Laplacians in mathbb {R}^2. We assume that the intensity of the magnetic field has a fast transition along a regular and compact curve Gamma . Our main results extend to a general regular curve the study of the localized eigenfunction obtained when Gamma is a straight line (i.e. Iwatsuka models). Furthermore, we include in our analysis the case of magnetic fields that slowly change along the curve Gamma and we obtain a rigorous and explicit characterization of the asymptotic mass distribution of the edge state along Gamma .

On the maximality of hyperelliptic Howe curves of genus 3

In this paper, we study a Howe curve $C$ in positive characteristic $p \geq 3$ which is of genus 3 and is hyperelliptic. We will show that if $C$ is superspecial, then its standard form is maximal or minimal over $\mathbf{F}_{p^2}$ without taking its $\mathbf{F}_{p^2}$-form.

An exact threshold for separator bifurcation

Reconnection involving magnetic separators is known to lead to the spontaneous generation of new separator pairs. In this work, we explore the bifurcation process for a system composed of a pair of null points with a joining separator. We begin with a simplified analytical model to derive the basic principles of bifurcation in this system and then consider models with more general separator curve geometry and generic localised null structure. We demonstrate that the maximum pairwise linking (net-winding) of the separator and the local fan plane always approaches a multiple of 0.25 just before bifurcation. Additionally, we show the integrated twisting along the separator (the field strength normalised parallel current) can be used to determine when this limit will definitely lead to bifurcation. We present step-by-step algorithms to assess how close such systems are to bifurcation.

The extended coset leader weight enumerator of a twisted cubic code

The extended coset leader weight enumerator of the generalized Reed–Solomon [q+1,q-3,5]_q code is computed. In this computation methods in finite geometry, combinatorics and algebraic geometry are used. For this we need the classification of the points, lines and planes in the projective three space under projectivities that leave the twisted cubic invariant. A line in three space determines a rational function of degree at most three and vice versa. Furthermore, the double point scheme of a rational function is studied. The pencil of a true passant of the twisted cubic, not in an osculation plane gives a curve of genus one as double point scheme. With the Hasse–Weil bound on {mathbb F}_q-rational points we show that there is a 3-plane containing the passant.

On the square peg problem and its relatives

Toeplitz’s Square Peg Problem asks whether every continuous simple closed curve in the plane contains the four vertices of a square. It has been proved for various classes of sufficiently smooth curves, some of which are dense, none of which are open. In this paper we prove it for several open classes of curves, one of which is also dense. This can be interpreted in saying that the Square Peg Problem is solved for generic curves. The latter class contains all previously known classes for which the Square Peg Problem has been proved in the affirmative. We also prove results about rectangles inscribed in immersed curves. Finally, we show that the problem of finding a regular octahedron on metric 2-spheres has a “topological counter-example”, that is, a certain test map with boundary condition exists.

Equivariant approach to weighted projective curves

We investigate group actions on the category of coherent sheaves over weighted projective lines. We show that the equivariant category with respect to certain finite group action is equivalent to the category of coherent sheaves over a weighted projective curve, where the genus of the underlying smooth projective curve and the weight data are given by explicit formulas. Moreover, we obtain a trichotomy result for all such equivariant relations.This equivariant approach provides an effective way to investigate smooth projective curves via weighted projective lines. As an application, we give a description for the category of coherent sheaves over a hyperelliptic curve of arbitrary genus. Links to smooth projective curves of genus 2 and also to Arnold's exceptional unimodal singularities are also established.

Testing Site Amplification Curves in Hybrid Broadband Ground Motion Simulations of M6.0, 24 August 2016 Amatrice Earthquake, Italy

This research focuses on predicting and assessing earthquake impact due to future scenarios regarding the ground motion seismic hazard by accounting mainly for site effect in the Central Apennines. To this end, we produced synthetic broadband seismograms by adopting a hybrid simulation technique for the Mw6.0 Amatrice earthquake, Central Italy, on 24 August 2016, accounting for site conditions by means of amplification curves, computed with different approaches. Simulations were validated by comparing with data recorded at 57 strong-motion stations, the majority installed in urban areas. This station sample was selected among stations recording the Amatrice earthquake within an epicentral distance of 150 km and potentially prone to experience site amplification effects because of lying in particular site conditions (sedimentary basins, topographic irregularities, and fault zones). The evaluation of amplification curves best suited to describe local effects is of great importance because many towns and villages in central Italy are built in very different geomorphological conditions, from valleys and sedimentary basins to topographies. In order to well reproduce observed ground motions, we accounted for the site amplification effect by testing various generic and empirical amplification curves such as horizontal-to-vertical spectral ratios (calculated from Fourier spectra using both earthquake, HVSR, and ambient noise, HVNSR, recordings) and those derived from the generalized inversion technique (GIT). The site amplifications emanated from GIT improve the match between observed and simulated data, especially in the case of stations installed in sedimentary basins, where the empirical amplification curve effectively reproduces spectral peaks. On the contrary, the worst performances are for the spectral ratios between components, even compared to the generic site amplification, although the latter ignores the strong bedrock/soil seismic impedance contrasts. At sites on topography, we did not observe any systematic behavior, the use of empirical curves ameliorating the fit only in a small percentage of cases. These results may provide a valuable framework for developing ground motion models for earthquake seismic hazard assessment and risk mitigation, especially in urban areas located in the seismically active central Italy region.

Automorphisms of tropical Hassett spaces

Given an integer g \geq 0 and a weight vector w \in \mathbb{Q}^n \cap (0, 1]^n satisfying 2g - 2 +\sum w_i > 0 , let \Delta_{g, w} denote the moduli space of n -marked, w -stable tropical curves of genus g and volume one. We calculate the automorphism group \operatorname{Aut}(\Delta_{g, w}) for g \geq 1 and arbitrary w , and we calculate the group \operatorname{Aut}(\Delta_{0, w}) when w is heavy/light. In both of these cases, we show that \operatorname{Aut}(\Delta_{g, w}) \cong \operatorname{Aut}(K_w) , where K_w is the abstract simplicial complex on \{1, \ldots, n\} whose faces are subsets with w -weight at most 1. We show that these groups are precisely the finite direct products of symmetric groups. The space \Delta_{g, w} may also be identified with the dual complex of the divisor of singular curves in the algebraic Hassett space \overline{\mathcal{M}}_{g, w} . Following the work of Massarenti and Mella (2017) on the biregular automorphism group \operatorname{Aut}(\overline{\mathcal{M}}_{g, w}) , we show that \operatorname{Aut}(\Delta_{g, w}) is naturally identified with the subgroup of automorphisms which preserve the divisor of singular curves.

LINEAR SERIES ON GENERAL CURVES WITH PRESCRIBED INCIDENCE CONDITIONS

AbstractUsing degeneration and Schubert calculus, we consider the problem of computing the number of linear series of given degree d and dimension r on a general curve of genus g satisfying prescribed incidence conditions at n points. We determine these numbers completely for linear series of arbitrary dimension when d is sufficiently large, and for all d when either $r=1$ or $n=r+2$ . Our formulas generalise and give new proofs of recent results of Tevelev and of Cela, Pandharipande and Schmitt.

Comet 21P/Giacobini–Zinner: Narrowband Photometry of the Prototype of Carbon-chain Depleted Comets at Multiple Apparitions

We obtained extensive narrowband photoelectric photometry of Comet 21P/Giacobini–Zinner with observations spanning 33 yr. The original data from 1985 were re-reduced and are presented along with data from three additional apparitions, including 2018/19. The original conclusion regarding Giacobini–Zinner’s chemical composition remains unchanged, with it having a 4–6× depletion in the carbon-chain molecules C2 and C3 and in NH, as compared with both OH and CN. The comet continues to exhibit a large asymmetry in production rates as a function of time and heliocentric distance, with production reaching a peak 3–5 weeks prior to perihelion. All species, including dust, follow the same general production rate curve each apparition, and the carbon-bearing species are always very similar to one another. However, OH and NH each differ in detail from the carbon-bearing species, implying somewhat varied composition between source regions. Longer term, there are only small secular changes among the apparitions before and near perihelion, but larger changes are evident as the comet recedes from the Sun, suggestive of a progressive precession of the rotation axis.

Slow dynamics and time-composition superposition in gels of cellulose nanocrystals.

Cellulose nanocrystals (CNCs) are rodlike biosourced colloidal particles used as key building blocks in a growing number of materials with innovative mechanical or optical properties. While CNCs form stable suspensions at low volume fractions in pure water, they aggregate in the presence of salt and form colloidal gels with time-dependent properties. Here, we study the impact of salt concentration on the slow aging dynamics of CNC gels following the cessation of a high-shear flow that fully fluidizes the sample. We show that the higher the salt content, the faster the recovery of elasticity upon flow cessation. Most remarkably, the elastic modulus G' obeys a time-composition superposition principle: the temporal evolution of G' can be rescaled onto a universal sigmoidal master curve spanning 13 orders of magnitude in time for a wide range of salt concentrations. Such a rescaling is obtained through a time-shift factor that follows a steep power-law decay with increasing salt concentration until it saturates at large salt content. These findings are robust to changes in the type of salt and the CNC content. We further show that both linear and nonlinear rheological properties of CNC gels of various compositions, including, e.g., the frequency-dependence of viscoelastic spectra and the yield strain, can be rescaled based on the sample age along the general master curve. Our results provide strong evidence for universality in the aging dynamics of CNC gels and call for microstructural investigations during recovery as well as theoretical modeling of time-composition superposition in rodlike colloids.

ON THE TEMPERED FUNDAMENTAL GROUPS OF HYPERBOLIC CURVES OF GENUS OVER

AbstractLet p be a prime number. In the present paper, we prove that the moduli of hyperbolic curves of genus $0$ over an algebraic closure of the field of p-adic numbers may be completely determined by their tempered fundamental groups.

Hodge conjecture for the moduli space of semi-stable sheaves over a nodal curve

In this article, we prove the Hodge conjecture for a desingularization of the moduli space of rank 2, semi-stable, torsion-free sheaves with fixed odd degree determinant over a very general irreducible nodal curve of genus at least 2. We also compute the algebraic Poincaré polynomial of the associated cohomology ring.

The local-global principle for integral points on stacky curves

We construct a stacky curve of genus 1 / 2 1/2 (i.e., Euler characteristic 1 1 ) over Z \mathbb {Z} that has an R \mathbb {R} -point and a Z p \mathbb {Z}_p -point for every prime p p but no Z \mathbb {Z} -point. This is best possible: we also prove that any stacky curve of genus less than 1 / 2 1/2 over a ring of S S -integers of a global field satisfies the local-global principle for integral points.

Globally generated vector bundles on and the unirationality of

Chang and Ran proved, in 1984, the unirationality of the moduli spaces of curves of genus 11, 12, and 13. They consider a family U of nonsingular space curves. If U is non-empty, it dominates the moduli space. Considering certain vector bundles related to the curves, one shows that U is unirational. Finally, Chang and Ran prove that U is non-empty by constructing smoothable reducible curves. We prove, instead, the existence of (almost) globally generated vector bundles of the above kind. The dependency locus of (rank − 1) general global sections of such a bundle is a nonsingular curve from U.

Singular intersections of subgroups and character varieties

We prove a global local rigidity result for character varieties of 3-manifolds into \(\mathrm {SL}_2({\mathbb {C}})\). Given a 3-manifold with toric boundary M satisfying some technical hypotheses, we prove that all but a finite number of its Dehn fillings \(M_{p/q}\) are globally locally rigid in the following sense: every irreducible representation \(\rho :\pi _1(M_{p/q})\rightarrow \mathrm {SL}_2({\mathbb {C}})\) is infinitesimally rigid, meaning that \(H^1(M_{p/q},\text {Ad}_\rho )=0\). This question arose from the study of asymptotics problems in topological quantum field theory developed in Charles and Marché (Knot state asymptotics II. Irreducible representations and the Witten conjecture, 2011). The proof relies heavily on recent progress in diophantine geometry and raises new questions of Zilber–Pink type. The main step is to show that a generic curve lying in a plane multiplicative torus intersects transversally almost all subtori of codimension 1. We prove an effective result of this form, based mainly on a height upper bound of Habegger.

Development of analytical seismic fragility functions for the common buildings in Iran

One of the main components for developing regional seismic risk models is the fragility functions of common building types. Due to the differences between the national design codes, construction practices, and construction materials, it is necessary to develop specific fragility functions for the common building types which are constructed in each region. For this reason, the present study is devoted to filling this essential gap for Iran. Therefore, in the first step, the existing building types in the country are classified into 31 categories regarding material, lateral-load-resisting system, height, and code level. Also, by conducting comprehensive studies on all previously performed research in the country, structural and dynamic parameters have been collected for buildings in each class. This information was used to compute a large set of backbone curves for Iranian buildings taxonomy. Then nearly three hundred appropriate fragility functions were generated using non-linear time-history analyses on the generic backbone curves and a large set of ground motion records. Based on the existing empirical data from past earthquakes in the country, the validation of the resulting fragility functions was carried out. The resulted fragility functions can be utilized in seismic risk assessment studies in the country.

Restricted Tangent Bundles for General Free Rational Curves

Abstract Suppose that $X$ is a smooth projective variety and that $C$ is a general member of a family of free rational curves on $X$. We prove several statements showing that the Harder–Narasimhan filtration of $T_{X}|_{C}$ is approximately the same as the restriction of the Harder–Narasimhan filtration of $T_{X}$ with respect to the class of $C$. When $X$ is a Fano variety, we prove that the set of all restricted tangent bundles for general free rational curves is controlled by a finite set of data. We then apply our work to analyze Peyre’s “freeness” formulation of Manin’s Conjecture in the setting of rational curves.

The Sato-Tate conjecture and Nagao's conjecture

Nagao's conjecture relates the rank of an elliptic surface to a limit formula arising from a weighted average of fibral Frobenius traces, and it is further generalized to smooth irreducible projective surfaces. We show that the Sato-Tate conjecture based on the random matrix model implies Nagao's conjecture for surfaces which form families of quadratic twists of fixed elliptic curves or fixed hyperelliptic curves of genus g≥1.