Algebraic Curve Of Genus Research Articles

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}.

On algebraic curves with many automorphisms in characteristic p

Let \({\mathcal {X}}\) be an irreducible, non-singular, algebraic curve defined over a field of odd characteristic p. Let g and \(\gamma \) be the genus and p-rank of \({\mathcal {X}}\), respectively. The influence of g and \(\gamma \) on the automorphism group \(Aut({\mathcal {X}})\) of \({\mathcal {X}}\) is well-known in the literature. If \(g \ge 2\) then \(Aut({\mathcal {X}})\) is a finite group, and unless \({\mathcal {X}}\) is the so-called Hermitian curve, its order is upper bounded by a polynomial in g of degree four (Stichtenoth). In 1978 Henn proposed a refinement of Stichtenoth’s bound of degree 3 in g up to few exceptions, all having p-rank zero. In this paper a further refinement of Henn’s result is proposed. First, we prove that if an algebraic curve of genus \(g \ge 2\) has more than \(336g^2\) automorphisms then its automorphism group has exactly two short orbits, one tame and one non-tame, that is, the action of the group is completely known. Finally when \(|Aut({\mathcal {X}})| \ge 900g^2\) sufficient conditions for \({\mathcal {X}}\) to have p-rank zero are provided.

Minimal rational curves on the moduli spaces of symplectic and orthogonal bundles

Let $C$ be an algebraic curve of genus $g$ and $L$ a line bundle over $C$. Let $\mathcal{MS}_C(n,L)$ and $\mathcal{MO}_C(n,L)$ be the moduli spaces of $L$-valued symplectic and orthogonal bundles respectively, over $C$ of rank $n$. We construct rational curves on these moduli spaces which generalize Hecke curves on the moduli space of vector bundles. As a main result, we show that these Hecke type curves have the minimal degree among the rational curves passing through a general point of the moduli spaces. As its byproducts, we show the non-abelian Torelli theorem and compute the automorphism group of moduli spaces.

Bound on the order of the decomposition groups of an algebraic curve in positive characteristic

In this paper, X is an algebraic curve of genus g≥2 defined over an algebraically closed field K of positive characteristic p, G is an automorphism group of X which fixes K element-wise, and, for a point P∈X, GP is the subgroup of G which fixes P. The question “how large GP can be compared to g” has been the subject of several papers. We are concerned with the case where the second ramification group GP(2) of GP is trivial. Under this condition Theorem 3.1 states that if |GP|>12(g−1) then X is either an ordinary hyperelliptic curve, or it has zero p-rank and p≠3. More precisely, up to birational equivalence, there exists a separable p-linearized polynomial L(T)∈K[T] of degree q such that an affine equation of X is L(y)=ax+1/x with a∈K⁎ in the former case, and L(y)=x3+bx with b∈K in the latter case. In 1987 Nakajima proved that if X is an ordinary curve (more generally, the second ramification group of G is trivial for every P∈X), then the order of G does not exceed 84g(g−1). We show that Theorem 3.1 together with some refinements of Nakajima's computations provide a slight improvement in Nakajima's bound from 84g(g−1) to 48(g−1)2.

Аналитические и теоретико-числовые свойства двумерных сигма-функций

Обзор посвящен классическим и современным задачам, связанным с целой функцией $\sigma({\bf u};\lambda)$, которая определяется семейством неособых алгебраических кривых рода 2, где ${\bf u}= (u_1,u_3)$, $\lambda=(\lambda_4, \lambda_6, \lambda_8, \lambda_{10})$. Эта функция является аналогом сигма-функции Вейерштрасса $\sigma({{u}};g_2,g_3)$ семейства эллиптических кривых. Логарифмические производные порядка 2 и выше функции ${\sigma({\mathbf{u}};\lambda)}$ порождают поле гиперэллиптических функций от ${\mathbf{u}} = (u_1,u_3)$ на якобианах кривых с фиксированным значением вектора параметров $\lambda$. Мы рассматриваем три ряда Гурвица $\sigma({\mathbf{u}};\lambda)=\sum_{m,n\ge0}a_{m,n}(\lambda)\frac{u_1^mu_3^n}{m!n!}$, $\sigma({\mathbf{u}};\lambda) = \sum_{k\ge 0}\xi_k(u_1;\lambda)\frac{u_3^k}{k!}$ и $\sigma({\mathbf{u}};\lambda) = \sum_{k\ge 0}\mu_k(u_3;\lambda)\frac{u_1^k}{k!}$. Обзор посвящен теоретико-числовым свойствам функций $a_{m,n}(\lambda)$, $\xi_k(u_1;\lambda)$ и $\mu_k(u_3;\lambda)$. Он включает самые последние результаты, доказательства которых использует тот фундаментальный факт, что функция $\sigma ({\mathbf{u}};\lambda)$ определяется системой четырех уравнений теплопроводности в неголономном репере шестимерного пространства.

Large odd prime power order automorphism groups of algebraic curves in any characteristic

Let X be a (projective, geometrically irreducible, nonsingular) algebraic curve of genus g≥2 defined over an algebraically closed field K of odd characteristic p≥0, and let Aut(X) be the group of all automorphisms of X which fix K element-wise. For any a subgroup G of Aut(X) whose order is a power of an odd prime d other than p, the bound proven by Zomorrodian for Riemann surfaces is |G|≤9(g−1) where the extremal case can only be obtained for d=3 and g≥10. We prove Zomorrodian's result for any K. The essential part of our paper is devoted to extremal 3-Zomorrodian curves X. Two cases are distinguished according as the quotient curve X/Z for a central subgroup Z of Aut(X) of order 3 is either elliptic, or not. For elliptic type extremal 3-Zomorrodian curves X, we completely determine the two possibilities for the abstract structure of G using deeper results on finite 3-groups. We also show infinite families of extremal 3-Zomorrodian curves for both types, of elliptic or non-elliptic. Our paper does not adapt methods from the theory of Riemann surfaces, nevertheless it sheds a new light on the connection between Riemann surfaces and their automorphism groups.

Algebraic curves with many automorphisms

Let X be a (projective, geometrically irreducible, non-singular) algebraic curve of genus g≥2 defined over an algebraically closed field K of odd characteristic p. Let Aut(X) be the group of all automorphisms of X which fix K element-wise. It is known that if |Aut(X)|≥8g3 then the p-rank (equivalently, the Hasse–Witt invariant) of X is zero. This raises the problem of determining the (minimum-value) function f(g) such that whenever |Aut(X)|≥f(g) then X has zero p-rank. For eveng we prove that f(g)≤900g2. The odd genus case appears to be much more difficult although, for any genus g≥2, if Aut(X) has a solvable subgroup G such that |G|>252g2 then X has zero p-rank and G fixes a point of X. Our proofs use the Hurwitz genus formula and the Deuring Shafarevich formula together with a few deep results from Group theory characterizing finite simple groups whose Sylow 2-subgroups have a cyclic subgroup of index 2. We also point out some connections with the Abhyankar conjecture and the Katz–Gabber covers.

Ordinary algebraic curves with many automorphisms in positive characteristic

Let $\mathcal{X}$ be an ordinary (projective, geometrically irreducible, nonsingular) algebraic curve of genus $\mathcal{g}(\mathcal{X}) \ge 2$ defined over an algebraically closed field $\mathbb{K}$ of odd characteristic $p$. Let $Aut(\mathcal{X})$ be the group of all automorphisms of $\mathcal{X}$ which fix $\mathbb{K}$ element-wise. For any solvable subgroup $G$ of $Aut(\mathcal{X})$ we prove that $|G|\leq 34 (\mathcal{g}(\mathcal{X})+1)^{3/2}$. There are known curves attaining this bound up to the constant $34$. For $p$ odd, our result improves the classical Nakajima bound $|G|\leq 84(\mathcal{g}(\mathcal{X})-1)\mathcal{g}(\mathcal{X})$, and, for solvable groups $G$, the Gunby-Smith-Yuan bound $|G|\leq 6(\mathcal{g}(\mathcal{X})^2+12\sqrt{21}\mathcal{g}(\mathcal{X})^{3/2})$ where $\mathcal{g}(\mathcal{X})>cp^2$ for some positive constant $c$.

Symmetry breaking in quantum curves and super Chern-Simons matrix models

It was known that quantum curves and super Chern-Simons matrix models correspond to each other. From the viewpoint of symmetry, the algebraic curve of genus one, called the del Pezzo curve, enjoys symmetry of the exceptional algebra, while the super Chern-Simons matrix model is described by the free energy of topological strings on the del Pezzo background with the symmetry broken. We study the symmetry breaking of the quantum cousin of the algebraic curve and reproduce the results in the super Chern-Simons matrix model.

Three-Webs Defined by Symmetric Functions

We study local differential-geometrical properties of curvilinear k-webs defined by symmetric functions (webs SW(k)). This class of k-webs contains in particular algebraic rectilinear k-webs defined by algebraic curves of genus 0. On a web SW(3), there are three three-parameter families of closed Thomsen configurations. We find equations of a rectilinear web SW(k) in terms of adapted coordinates and prove that the curvature of a symmetric three-web is a skew-symmetric function with respect to adapted coordinates. In conclusion, we formulate some open problems.

Wiman’s and Edge’s sextics attaining Serre’s bound

We find new explicit algebraic curves of genus 6 over the finite fields $$\mathbb {F}_{67^3},\mathbb {F}_{5393}$$ and $$\mathbb {F}_{9173}$$ attaining Serre’s bound. They are Wiman’s and Edge’s sextics. Also we obtain a condition on p for Wiman’s sextic over the finite field $$\mathbb {F}_{p^2}$$ to be maximal, and give new entries of genus 6 in the tables at manYPoints.org by computer search on Edge’s sextics.

On the parametrization of a certain algebraic curve of genus 2

On the parametrization of a certain algebraic curve of genus 2

A characterization of the Artin–Mumford curve

Let M be the Artin–Mumford curve over the finite prime field Fp with p>2. By a result of Valentini and Madan, AutFp(M)≅H with H=(Cp×Cp)⋊Dp−1. We prove that if X is an algebraic curve of genus g=(p−1)2 defined over Fp such that AutFp(X) contains a subgroup isomorphic to H then X is birationally equivalent over Fp to the Artin–Mumford curve M.

On the Parametrization of a Certain Algebraic Curve of Genus 2

On the Parametrization of a Certain Algebraic Curve of Genus 2

On the Gonality of an Algebraic Curve and Its Abelian Automorphism Groups

Let G be an abelian automorphism group of a complex algebraic curve of genus g ≥ 2 with gonality d, and let |G| be its order. We prove that except for the Fermat curve of degree d + 1, and if G is cyclic. Furthermore, if |G| > 2g − 2 + 2d and C admits a unique finite cover f: C → ℙ1 of degree d, then f is a Galois cover.

A criterion for the simplicity of the Lyapunov spectrum of square-tiled surfaces

We present a Galois-theoretical criterion for the simplicity of the Lyapunov spectrum of the Kontsevich-Zorich cocycle over the Teichmueller flow on the $SL_2(R)$-orbit of a square-tiled surface. The simplicity of the Lyapunov spectrum has been proved by A. Avila and M. Viana with respect to the so-called Masur-Veech measures associated to connected components of moduli spaces of translation surfaces, but is not always true for square-tiled surfaces of genus $\geq 3$. We apply our criterion to square-tiled surfaces of genus 3 with one single zero. Conditionally to a conjecture of Delecroix and Leli\`evre, we prove with the aid of Siegel's theorem (on integral points on algebraic curves of genus $>0$) that all but finitely many such square-tiled surfaces have simple Lyapunov spectrum.

Infinite-dimensional prolongation Lie algebras and multicomponent Landau–Lifshitz systems associated with higher genus curves

The Wahlquist–Estabrook prolongation method constructs for some PDEs a Lie algebra that is responsible for Lax pairs and Bäcklund transformations of certain type. We present some general properties of Wahlquist–Estabrook algebras for (1+1)-dimensional evolution PDEs and compute this algebra for the n-component Landau–Lifshitz system of Golubchik and Sokolov for any n≥3. We prove that the resulting algebra is isomorphic to the direct sum of a 2-dimensional abelian Lie algebra and an infinite-dimensional Lie algebra L(n) of certain matrix-valued functions on an algebraic curve of genus 1+(n−3)2n−2. This curve was used by Golubchik, Sokolov, Skrypnyk, and Holod in constructions of Lax pairs. Also, we find a presentation for the algebra L(n) in terms of a finite number of generators and relations. These results help to obtain a partial answer to the problem of classification of multicomponent Landau–Lifshitz systems with respect to Bäcklund transformations.Furthermore, we construct a family of integrable evolution PDEs that are connected with the n-component Landau–Lifshitz system by Miura type transformations parametrized by the above-mentioned curve. Some solutions of these PDEs are described.

On the generalized Chaplygin system

We discuss two polynomial bi-Hamiltonian structures for the generalized integrable Chaplygin system on the sphere S^2 with an additional integral of fourth order in momenta. An explicit procedure to find the variables of separation, the separation relations and the transformation of the corresponding algebraic curves of genus two is considered in detail.

Moduli Spaces of Coherent Systems of Small Slope on Algebraic Curves

Let C be an algebraic curve of genus g ≥ 2. A coherent system on C consists of a pair (E, V), where E is an algebraic vector bundle over C of rank n and degree d and V is a subspace of dimension k of the space of sections of E. The stability of the coherent system depends on a parameter α. We study the geometry of the moduli space of coherent systems for 0 < d ≤ 2n. We show that these spaces are irreducible whenever they are nonempty and obtain necessary and sufficient conditions for nonemptiness.

Double Kodaira fibrations

The existence of a Kodaira fibration, i.e., of a fibration of a compact complex surface $S$ onto a complex curve $B$ which is a differentiable but not a holomorphic bundle, forces the geographical slope $ \nu(S) = c_1^2 (S) / c_2 (S)$ to lie in the interval $(2,3)$. But up to now all the known examples had slope $ \nu(S) \leq 2 + 1/3$. In this paper we consider a special class of surfaces admitting two such Kodaira fibrations, and we can construct many new examples, showing in particular that there are such fibrations attaining the slope $ \nu(S) = 2 + 2/3$. We are able to explicitly describe the moduli space of such class of surfaces, and we show the existence of Kodaira fibrations which yield rigid surfaces. We observe an interesting connection between the problem of the slope of Kodaira fibrations and a 'packing' problem for automorphisms of algebraic curves of genus $\geq 2$.