On the weights of the fixed points of an automorphism of a compact Riemann surface

In this work we study automorphisms of compact Riemann surfaces with more than four fixed points. We obtain a lower bound for the weight of each of these fixed points. The discussion depends on the parities of the order of the automorphism and the number of fixed points. Moreover, we discuss the sharpness of our bounds. Received: 15 February 2005     

Double coverings of hyperelliptic real algebraic curves

We consider double and (possibly) branched coverings π:X→X^′ between real algebraic curves where X is hyperelliptic. We are interested in the topology of such coverings and also in describing them in terms of algebraic equations. In this article we completely solve these two problems. We first analyse the topological features and ramification data of such coverings. Second, for each isomorphism class of these coverings we then describe a representative, with defining polynomial equations for X and for X^′, a formula for the involution that generates the covering transformation group, and a rational formula for the covering projection π:X→X^′.     

Weierstrass weight of the hyperosculating points of generalized Fermat curves

Let $(S,H)$ be a generalized Fermat pair of the type $(k,n)$. If $F?S$ is the set of fixed points of the non-trivial elements of the group H, then F is exactly the set of hyperosculating points of the standard embedding $S?{\mathbb{P}}^n$. We provide an optimal lower bound (this being sharp in a dense open set of the moduli space of the generalized Fermat curves) for the Weierstrass weight of these points.     

Topologically unique maximal elementary abelian group actions on compact oriented surfaces

We determine all finite maximal elementary abelian group actions on compact oriented surfaces of genus σ≥2 which are unique up to topological equivalence. For certain special classes of such actions, we determine group extensions which also define unique actions. In addition, we explore in detail one of the families of such surfaces considered as compact Riemann surfaces and tackle the classical problem of constructing defining equations.     

The gonality of Riemann surfaces under projections by normal coverings

A compact Riemann surface X of genus g≥2 which can be realized as a q-fold, normal covering of a compact Riemann surface of genus p is said to be (q,p)-gonal. In particular the notion of (2,p)-gonality coincides with p-hyperellipticity and (q,0)-gonality coincides with ordinary q-gonality. Here we completely determine the relationship between the gonalities of X and Y for an N-fold normal covering X→Y between compact Riemann surfaces X and Y. As a consequence we obtain classical results due to Maclachlan (1971) [5] and Martens (1977) [6].     

On Singerman symmetries of a class of Belyi Riemann surfaces

By virtue of the Belyi Theorem an algebraic curve can be defined over the algebraic numbers if and only if the corresponding Riemann surface can be uniformized by a subgroup of a Fuchsian triangle group. Such surfaces are known as Belyi surfaces and an important class of them consists of Riemann surfaces having the so-called large group of automorphisms. Necessary and sufficient algebraic conditions for these surfaces to be symmetric were found by Singerman in the middle of the seventies and, by a recent result of Köck and Singerman, the algebraic numbers above can be chosen to be real if and only if the respective surface is symmetric. The aim of this paper is to give, in similar terms, the formulas for the number of ovals of the corresponding symmetries, which we refer to as the Singerman symmetries.     

Uniqueness of orientation-preserving free actions of finite abelian groups on surfaces

Let G be a finite abelian group. We list all the cases where the topological equivalence class of orientation-preserving free G-actions on a closed surface is unique. Moreover, we obtain the classification of topological equivalence classes of orientation-preserving free actions of finite abelian groups of rank 2.     

Imaginary automorphisms on real hyperelliptic curves

A real hyperelliptic curve X is said to be Gaussian if there is an automorphism such that , where [-1] denotes the hyperelliptic involution on X. Gaussian curves arise naturally in several contexts, for example when one studies real Jacobians. In the present paper, we study the properties of Gaussian curves and we describe their moduli spaces.     

On geometrically transitive Hopf algebroids

This paper contributes to the characterization of a certain class of commutative Hopf algebroids. It is shown that a commutative flat Hopf algebroid with a non zero base ring and a nonempty character groupoid is geometrically transitive if and only if any base change morphism is a weak equivalence (in particular, if any extension of the base ring is Landweber exact), if and only if any trivial bundle is a principal bi-bundle, and if and only if any two objects are fpqc locally isomorphic. As a consequence, any two isotropy Hopf algebras of a geometrically transitive Hopf algebroid (as above) are weakly equivalent. Furthermore, the character groupoid is transitive and any two isotropy Hopf algebras are conjugated. Several other characterizations of these Hopf algebroids in relation to transitive groupoids are also given.     

p-Groups acting on Riemann surfaces

Let p be a prime integer and let $r\ge 3$ be an integer so that $p\ge 5r?7$. We show that a closed Riemann surface S of genus $g\ge 2$ has at most one p-group H of conformal automorphisms so that $S/H$ has genus zero and exactly r cone points. This, in particular, asserts that, for $r=3$ and $p\ge 11$, the minimal field of definition of S coincides with that of $(S,H)$. Another application of this fact, for the case that S is pseudo-real, is that $\mathrm{Aut}(S)/H$ must be either trivial or a cyclic group and that r is necessarily even. This generalizes a result due to Bujalance–Costa for the case of pseudo-real cyclic p-gonal Riemann surfaces.     

On the weight of numerical semigroups

We investigate the weights of a family of numerical semigroups by means of even gaps and the Weierstrass property for such a family.     

A generalization of the Weierstrass semigroup

The Weierstrass semigroup H(P) is well known and has been studied. Recently there has been a renewed interest in these semigroups because of applications in coding theory. Generalizations of the Weierstrass semigroup H(P) to n-tuples P₁,…,P_n have been made and studied. We will state and study another possible generalization.     

Local structure of Koszul-Vinberg and of Lie algebroids

In this short note we continue our study of Koszul-Vinberg algebroids which form a subcategory of the category of Lie algebroids, and which appear naturally in the study of affine structures, affine and transversally affine foliations [N. Nguiffo Boyom, R. Wolak, J. Geom. Phys. 42 (2002) 307-317]. We prove a local decomposition theorem for KV-algebroids. Using the notion of KV-algebroids we introduce a new class of singular foliations: affine singular foliations. In the last section we study the holonomy of these foliations and prove a stability theorem.     

Salem numbers defined by Coxeter transformation

A real algebraic integer α>1 is called a Salem number if all its remaining conjugates have modulus at most 1 with at least one having modulus exactly 1. It is known [J.A. de la Peña, Coxeter transformations and the representation theory of algebras, in: V. Dlab et al. (Eds.), Finite Dimensional Algebras and Related Topics, Proceedings of the NATO Advanced Research Workshop on Representations of Algebras and Related Topics, Ottawa, Canada, Kluwer, August 10-18, 1992, pp. 223-253; J.F. McKee, P. Rowlinson, C.J. Smyth, Salem numbers and Pisot numbers from stars, Number theory in progress. in: K. Gy?ry et al. (Eds.), Proc. Int. Conf. Banach Int. Math. Center, Diophantine problems and polynomials, vol. 1, de Gruyter, Berlin, 1999, pp. 309-319; P. Lakatos, On Coxeter polynomials of wild stars, Linear Algebra Appl. 293 (1999) 159-170] that the spectral radii of Coxeter transformation defined by stars, which are neither of Dynkin nor of extended Dynkin type, are Salem numbers. We prove that the spectral radii of the Coxeter transformation of generalized stars are also Salem numbers. A generalized star is a connected graph without multiple edges and loops that has exactly one vertex of degree at least 3.     

A stochastic Hopf bifurcation

Summary Let {x _t :t0} be the solution of a stochastic differential equation (SDE) in ^d which fixes 0, and let denote the Lyapunov exponent for the linear SDE obtained by linearizing the original SDE at 0. It is known that, under appropriate conditions, the sign of controls the stability/instability of 0 and the transience/recurrence of {x _t :t0} on ^d \{0}. In particular if the coefficients in the SDE depend on some parameterz which is varied in such a way that the corresponding Lyapunov exponent ^z changes sign from negative to positive the (almost-surely) stable fixed point at 0 is replaced by an (almost-surely) unstable fixed point at 0 together with an attracting invariant probability measure ^z on ^d \{0}. In this paper we investigate the limiting behavior of ^z as ^z converges to 0 from above. The main result is that the rescaled measures (1/ ^z ) ^z converge (in an appropriate weak sense) to a non-trivial -finite measure on ^d \{0}.Research supported in part by Office of Naval Research contract N00014-91-J-1526     

Galois points for a plane curve in characteristic two

Let C$C$ be an irreducible plane curve. A point P$P$ in the projective plane is said to be Galois with respect to C$C$ if the function field extension induced by the projection from P$P$ is Galois. We denote by δ^′(C)${\delta }^{\prime }\left(C\right)$ the number of Galois points contained in P²?C${\mathbb{P}}^2?C$. In this article we will present two results with respect to determination of δ^′(C)${\delta }^{\prime }\left(C\right)$ in characteristic two. First we determine δ^′(C)${\delta }^{\prime }\left(C\right)$ for smooth plane curves of degree a power of two. In particular, we give a new characterization of the Klein quartic in terms of δ^′(C)${\delta }^{\prime }\left(C\right)$. Second we determine δ^′(C)${\delta }^{\prime }\left(C\right)$ for a generalization of the Klein quartic, which is related to an example of Artin–Schreier curves whose automorphism group exceeds the Hurwitz bound. This curve has many Galois points.     

On duality between étale groupoids and Hopf algebroids

For any étale Lie groupoid G over a smooth manifold M, the groupoid convolution algebra of smooth functions with compact support on G has a natural coalgebra structure over the commutative algebra which makes it into a Hopf algebroid. Conversely, for any Hopf algebroid A over we construct the associated spectral étale Lie groupoid over M such that is naturally isomorphic to G. Both these constructions are functorial, and is fully faithful left adjoint to . We give explicit conditions under which a Hopf algebroid is isomorphic to the Hopf algebroid of an étale Lie groupoid G.     

Footnote to a paper by hartshorne

We prove that every curve in the biliaison class of an extremal curve in P³ can be connected to an extremal curve.     

Finite group actions on bordered surfaces of small genus

This paper considers finite group actions on compact bordered surfaces — quotients of unbordered orientable surfaces under the action of a reflectional symmetry. Classification of such actions (up to topological equivalence) is carried out by means of the theory of non-euclidean crystallographic groups, and determination of normal subgroups of finite index in these groups, up to conjugation within their automorphism group. A result of this investigation is the determination, up to topological equivalence, of all actions of groups of finite order 6 or more on compact (orientable or non-orientable) bordered surfaces of algebraic genus p for 2≤p≤6. We also study actions of groups of order less than 6, or of prime order, on bordered surfaces of arbitrary algebraic genus p≥2.     

Stability of homogeneous principal bundles with a classical groupas the structure group

Let G be a connected semisimple linear algebraic group defined over an algebraically closed field k and P ⊂ G a parabolic subgroup without any simple factor. Let H be a connected reductive linear algebraic group defined over the field k such that all the simple quotients of H are of classical type. Take any homomorphism π : P → H such that the image of p is not contained in any proper parabolic subgroup of H. Consider the corresponding principal H-bundle E_P(H) = (G × H)/P over G/P. We prove that E_P (H) is strongly stable with respect to any polarization on G/P.