Curves Of Genus Research Articles

The trigonal construction in the ramified case

AbstractTo every double cover ramified in two points of a general trigonal curve of genus , one can associate an étale double cover of a tetragonal curve of genus . We show that the corresponding Prym varieties are canonically isomorphic as principally polarized abelian varieties. This extends Recillas' trigonal construction to covers ramified in two points.

Search of Fractal Space-Filling Curves with Minimal Dilation

We introduce an algorithm for a search of extremal fractal curves in large curve classes. It heavily uses SAT-solvers—heuristic algorithms that find models for CNF boolean formulas. Our algorithm was implemented and applied to the search of fractal surjective curves \(\gamma :[0,1]\rightarrow [0,1]^d\) with minimal dilation $$\begin{aligned} \sup _{t_1<t_2}\frac{\Vert \gamma (t_2)-\gamma (t_1)\Vert ^d}{t_2-t_1}. \end{aligned}$$We report new results of that search in the case of Euclidean norm. We have found a new curve that we call “YE”, a self-similar (monofractal) plane curve of genus \(5\times 5\) with dilation \(5+{43}/{73}=5.5890\ldots \) In dimension 3 we have found facet-gated bifractals (which we call “Spring”) of genus \(2\times 2\times 2\) with dilation \(<17\). In dimension 4 we obtained that there is a curve with dilation \(<62\). Some lower bounds on the dilation for wider classes of cubically decomposable curves are proven.

Linear Complexity of Sequences on Koblitz Curves of Genus 2

Abstract In this paper, we consider the hyperelliptic analogue of the Frobenius endomorphism generator and show that it produces sequences with large linear complexity on the Jacobian of genus 2 curves.

Derived equivalence and fibrations over curves and surfaces

We prove that the bounded derived category of coherent sheaves on a smooth projective complex variety reconstructs the isomorphism classes of fibrations onto smooth projective curves of genus g≥2. Moreover, in dimension at most four, we prove that the same category reconstructs the isomorphism classes of fibrations onto normal projective surfaces with positive holomorphic Euler characteristic and admitting a finite morphism to an abelian variety. Finally, we study the derived invariance of a class of fibrations with minimal base dimension under the condition that all the Hodge numbers of type h0,p(X) are derived invariant.

Autonomous Navigation in Unknown Environments With Sparse Bayesian Kernel-Based Occupancy Mapping

This article focuses on online occupancy mapping and real-time collision checking onboard an autonomous robot navigating in a large unknown environment. Commonly used voxel and octree map representations can be easily maintained in a small environment but have increasing memory requirements as the environment grows. We propose a fundamentally different approach for occupancy mapping, in which the boundary between occupied and free space is viewed as the decision boundary of a machine learning classifier. This work generalizes a kernel perceptron model which maintains a very sparse set of support vectors to represent the environment boundaries efficiently. We develop a probabilistic formulation based on relevance vector machines, handling measurement noise, and probabilistic occupancy classification, supporting autonomous navigation. We provide an online training algorithm, updating the sparse Bayesian map incrementally from streaming range data, and an efficient collision-checking method for general curves, representing potential robot trajectories. The effectiveness of our mapping and collision checking algorithms is evaluated in tasks requiring autonomous robot navigation and active mapping in unknown environments.

Group-theoreticity of numerical invariants and distinguished subgroups of configuration space groups

Let $\Sigma$ be a set of prime numbers which is either of cardinality one or equal to the set of all prime numbers. In this paper, we prove that various objects that arise from the geometry of the configuration space of a hyperbolic curve over an algebraically closed field of characteristic zero may be reconstructed group-theoretically from the pro-$\Sigma$ fundamental group of the configuration space. Let $X$ be a hyperbolic curve of type $(g,r)$ over a field $k$ of characteristic zero. Thus, $X$ is obtained by removing from a proper smooth curve of genus $g$ over $k$ a closed subscheme [i.e., the "divisor of cusps"] of $X$ whose structure morphism to Spec$(k)$ is finite étale of degree $r$; $2g-2+r>0$. Write $X_n$ for the $n$-th configuration space associated to $X$, i.e., the complement of the various diagonal divisors in the fiber product over $k$ of $n$ copies of $X$. Then, when $k$ is algebraically closed, we show that the triple $(n,g,r)$ and the generalized fiber subgroups—i.e., the subgroups that arise from the various natural morphisms $X_n \to X_m$ $[m$ < $n]$, which we refer to as generalized projection morphisms—of the pro-$\Sigma$ fundamental group $\Pi_n$ of $X_n$ may be reconstructed group-theoretically from $\Pi_n$ whenever $n \geq 2$. This result generalizes results obtained previously by the first and third authors and A. Tamagawa to the case of arbitrary hyperbolic curves [i.e., without restrictions on $(g,r)$]. As an application, in the case where $(g,r)= (0,3)$ and $n \geq 2$, we conclude that there exists a direct product decomposition Out$(\Pi_n) = {\mathrm{GT}^{\Sigma}} \times \mathfrak{S}_{n + 3}$ —where we write "Out$(-)$" for the group of outer automorphisms [i.e., without any auxiliary restrictions!] of the profinite group in parentheses and ${\mathrm{GT}^{\Sigma}}$ (respectively, $\mathfrak{S}_{n + 3}$) for the pro-$\Sigma$ Grothendieck-Teichmüller group (respectively, symmetric group on $n + 3$ letters). This direct product decomposition may be applied to obtain a simplified purely group-theoretic equivalent definition—i.e., as the centralizer in ${\rm Out}(\Pi_n)$ of the union of the centers of the open subgroups of ${\rm Out}(\Pi_n)$—of ${\mathrm{GT}^{\Sigma}}$. One of the key notions underlying the theory of the present paper is the notion of a pro-$\Sigma$ log-full subgroup—which may be regarded as a sort of higher-dimensional analogue of the notion of a pro-$\Sigma$ cuspidal inertia subgroup of a surface group—of $\Pi_n$. In the final section of the present paper, we show that, when $X$ and $k$ satisfy certain conditions concerning "weights", the pro-$l$ log-full subgroups may be reconstructed group-theoretically from the natural outer action of the absolute Galois group of $k$ on the geometric pro-$l$ fundamental group of $X_n$.

Triangular modular curves of small genus

Triangular modular curves are a generalization of modular curves that arise from quotients of the upper half-plane by congruence subgroups of hyperbolic triangle groups. These curves also arise naturally as a source of Belyi maps with monodromy text {PGL}_2(mathbb {F}_q) or text {PSL}_2(mathbb {F}_q). We present a computational approach to enumerate Borel-type triangular modular curves of low genus, and we carry out this enumeration for prime level and small genus.

On the tautological ring of Humbert curves

We exhibit a 2-dimensional family of non-hyperelliptic curves of genus 5, called Humbert curves, for which the tautological ring injects into cohomology. In particular, Humbert curves have a multiplicative Chow-K\"unneth decomposition (in the sense of Shen-Vial), and their Ceresa cycle is torsion.

Monodromy of Kodaira fibrations of genus 3

AbstractA Kodaira fibration is a non‐isotrivial fibration from a smooth algebraic surface S to a smooth algebraic curve B such that all fibers are smooth algebraic curves of genus g. Such fibrations arise as complete curves inside the moduli space of genus g algebraic curves. We investigate here the possible connected monodromy groups of a Kodaira fibration in the case and classify which such groups can arise from a Kodaira fibration obtained as a general complete intersection curve inside a subvariety of parametrizing curves whose Jacobians have extra endomorphisms.

Effects of land clearing for agriculture on soil organic carbon stocks in drylands: A meta-analysis.

Agricultural activities have been expanding globally with the pressure to provide food security to the earth's growing population. These agricultural activities have profoundly impacted soil organic carbon (SOC) stocks in global drylands. However, the effects of clearing natural ecosystems for cropland (CNEC) on SOC are uncertain. To improve our understanding of carbon emissions and sequestration under different land uses, it is necessary to characterize the response patterns of SOC stocks to different types of CNEC. We conducted a meta-analysis with mixed-effect model based on 873 paired observations of SOC in croplands and adjacent natural ecosystems from 159 individual studies in global drylands. Our results indicate that CNEC significantly (p < .05) affects SOC stocks, resulting from a combination of natural land clearing, cropland management practices (fertilizer application, crop species, cultivation duration) and the significant negative effects of initial SOC stocks. Increases in SOC stocks (in 1m depth) were found in croplands which previously natural land (deserts and shrublands) had low SOC stocks, and the increases were 278.86% (95% confidence interval, 196.43%-361.29%) and 45.38% (26.53%-62.23%), respectively. In contrast, SOC stocks (in 1m depth) decreased by 24.11% (18.38%-29.85%) and 10.70% (1.80%-19.59%) in clearing forests and grasslands for cropland, respectively. We also established the general response curves of SOC stocks change to increasing cultivation duration, which is crucial for accurately estimating regional carbon dynamics following CNEC. SOC stocks increased significantly (p < .05) with high long-term fertilizer consumption in cleared grasslands with low initial SOC stocks (about 27.2Mg ha-1 ). The results derived from our meta-analysis could be used for refining the estimation of dryland carbon dynamics and developing SOC sequestration strategies to achieve the removal of CO2 from the atmosphere.

Defect detection in composites by deep learning using solitary waves

This paper proposes a real-time non-destructive evaluation technique to detect defects in laminated composites by deep learning using highly nonlinear solitary waves (HNSWs). HNSW data are collected by conducting experiments using a granular crystal sensor composed of a vertical array of steel beads directly contacting an AS4/PEEK composite plate. Using HNSW data, a deep learning algorithm based on the convolution neural networks (CNN) is trained and tested for the identification of delamination in AS4/PEEK composites. The influence of the number of hidden layers and various CNN parameters is investigated for improved classification accuracy of the deep learning algorithm. A general curve fit is presented in order to facilitate the correct choice of the input pixel and batch size. Moreover, a multiple mode testing scheme, classifying defects using multiple HNSW signals, is introduced to improve the accuracy of the algorithm. The efficiency and accuracy of using three different types of the input signal (i.e., original (without pre-processing) and time-sliced/time-sliced noise-cutting signals (with pre-processing)) are examined for the real-time detection of defects. Mathematical formulations are established to obtain time-sliced and time-sliced noise-cutting signals from the original HNSW signals. It was found that accuracy could be improved by increasing both the number of hidden layers and the input pixel size, reducing the learning rate, and by using a batch normalization process and RELU activation function. For all three input signals, accuracy levels of over 90% were achieved in identifying the existence and location of delamination in AS4/PEEK composites, highlighting the possibility of using the proposed deep learning algorithm for the real-time detection of defects in laminated composites.

Polynomial curves with projections to PH curves

Despite the fact that the orthogonal projection of a spatial Pythagorean hodograph (PH) curve into the plane is not a planar PH curve in general, we can find special cases such that the PH property is preserved when the curve is projected. In Farouki et al. (2021) the authors studied how to generate spatial PH curves with planar PH projections. Their approach and presented results motivated us to continue and extend this investigation. We study geometric conditions under which a spatial curve is projected to a PH curve. For this purpose, we introduced a suitable geometric characterization of the curves with PH property via intersection multiplicity of the associated curves described by the hodograph mapping with the absolute conic. As a consequence we will show that a generic polynomial curve of degree higher than five possesses no parallel projection to a PH curve. On contrary, for a spatial cubic there are finitely many ways how to orthogonally project it to a planar PH cubic. And the same holds for oblique parallel projections of spatial quintics. Hence these cases are examined in more detail.

Topological recursion for Masur–Veech volumes

We study the Masur–Veech volumes M V g , n $MV_{g,n}$ of the principal stratum of the moduli space of quadratic differentials of unit area on curves of genus g $g$ with n $n$ punctures. We show that the volumes M V g , n $MV_{g,n}$ are the constant terms of a family of polynomials in n $n$ variables governed by the topological recursion/Virasoro constraints. This is equivalent to a formula giving these polynomials as a sum over stable graphs, and retrieves a result of [Delecroix, Goujard, Zograf, Zorich, Duke J. Math 170 (2021), no. 12, math.GT/1908.08611] proved by combinatorial arguments. Our method is different: it relies on the geometric recursion and its application to statistics of hyperbolic lengths of multicurves developed in [Andersen, Borot, Orantin, Geometric recursion, math.GT/1711.04729, 2017]. We also obtain an expression of the area Siegel–Veech constants in terms of hyperbolic geometry. The topological recursion allows numerical computations of Masur–Veech volumes, and thus of area Siegel–Veech constants, for low g $g$ and n $n$ , which leads us to propose conjectural formulae for low g $g$ but all n $n$ . We also relate our polynomials to the asymptotic counting of square-tiled surfaces with large boundaries.

On the Rank of Quadratic Equations for Curves of High Degree

Let $${\mathcal {C}} \subset {{\mathbb {P}}}^r$$ be a linearly normal curve of arithmetic genus g and degree d. In Saint-Donat (CR Acad Sci Paris Ser A 274: 324–327, 1972), B. Saint-Donat proved that the homogeneous ideal $$I({\mathcal {C}})$$ of $${\mathcal {C}}$$ is generated by quadratic equations of rank at most 4 whenever $$d \ge 2g+2$$ . Also, in Eisenbud et al. (Amer J Math 110: 513–539, 1988) Eisenbud, Koh and Stillman proved that $$I({\mathcal {C}})$$ admits a determinantal presentation if $$d \ge 4g+2$$ . In this paper, we will show that $$I({\mathcal {C}})$$ can be generated by quadratic equations of rank 3 if either $$g=0,1$$ and $$d \ge 2g+2$$ or else $$g \ge 2$$ and $$d \ge 4g+4$$ .

Moments of quadratic Dirichlet L-functions over function fields

In this paper, we establish the expected order of magnitude of the kth-moment of quadratic Dirichlet L-functions associated to hyperelliptic curves of genus g as well as of prime moduli over a fixed finite field Fq for all real k≥0.

Perverse sheaves on $\mathbf{C}^{2}$ without vanishing cycles at the origin along a general plane curve with singularities

Generalizing MacPherson-Vilonen’s method [2] to arbitrary plane curve singularities, we provide a classification of perverse sheaves on the neighborhood of the origin in the complex plane, which are adapted to a germ of a complex analytic plane curve. We rely on the presentation of the fundamental group of the complement of the curve as obtained by Neto and Silva [5]. The main result is an equivalence of categories between the category of perverse sheaves on $ \mathbf{C}^{2}$ stratified with respect to a singular plane curve and the category of $n$-tuples of finite dimensional vector spaces and linear maps satisfying a finite number of suitable relations. As an application, we classify perverse sheaves with no vanishing cycles at the origin for a special case.

The modularity of elliptic curves over all but finitely many totally real fields of degree 5

We study the finiteness of low degree points on certain modular curves and their Atkin--Lehner quotients, and, as an application, prove the modularity of elliptic curves over all but finitely many totally real fields of degree $5$. On the way, we prove a criterion for the finiteness of rational points of degree $5$ on a curve of large genus over a number field using the results of Abramovich--Harris and Faltings on subvarieties of Jacobians.

A note on semiorthogonal indecomposability of some Cohen-Macaulay varieties

In this short note, we observe that the criterion proven in [12] for semiorthogonal indecomposability of the derived category of smooth DM stacks based on the canonical bundle can be extended to the case of projective varieties with Cohen-Macaulay singularities. As a consequence, all projective curves of positive arithmetic genus have weakly indecomposable bounded derived categories and indecomposable categories of perfect complexes.

SN 2019zrk, a bright SN 2009ip analog with a precursor

We present photometric and spectroscopic observations of the Type IIn supernova SN 2019zrk (also known as ZTF 20aacbyec). The SN shows a &gt; 100 day precursor, with a slow rise, followed by a rapid rise toM ≈ −19.2 in therandgbands. The post-peak light-curve decline is well fit with an exponential decay with a timescale of ∼39 days, but it shows prominent undulations, with an amplitude of ∼1 mag. Both the light curve and spectra are dominated by an interaction with a dense circumstellar medium (CSM), probably from previous mass ejections. The spectra evolve from a scattering-dominated Type IIn spectrum to a spectrum with strong P-Cygni absorptions. The expansion velocity is high, ∼16 000 km s−1, even in the last spectra. The last spectrum ∼110 days after the main eruption reveals no evidence for advanced nucleosynthesis. From analysis of the spectra and light curves, we estimate the mass-loss rate to be ∼4 × 10−2 M⊙yr−1for a CSM velocity of 100 km s−1, and a CSM mass of 1 M⊙. We find strong similarities for both the precursor, general light curve, and spectral evolution with SN 2009ip and similar SNe, although SN 2019zrk displays a brighter peak magnitude. Different scenarios for the nature of the 09ip-class of SNe, based on pulsational pair instability eruptions, wave heating, and mergers, are discussed.

Équation de Pell–Abel et applications

In this paper, we show that there are solutions of degree r of the equation of Pell–Abel on some real hyperelliptic curve of genus g if and only if r>g. This result, which is known to the experts, has consequences, which seem to be unknown to the experts. First, we deduce the existence of a primitive k-differential on an hyperelliptic curve of genus g with a unique zero of order k(2g-2) for every (k,g)≠(2,2). Moreover, we show that there exists a non Weierstrass point of order n modulo a Weierstrass point on a hyperelliptic curve of genus g if and only if n>2g.