Defining equations for cyclic prime covers of the Riemann sphere

We determine a method to find explicit defining equations for each compact Riemann surface which admits a cyclic group of automorphisms C _p of prime order p such that the quotient space has genus 0.     

Moduli of commutative and non-commutative finite covers

We classify affine, not necessarily commutative,n-coversB of commutativeK-algebrasA using data triples (A, M, α) consisting of the algebraA, a freeA-moduleM of rankn − 1, and an associative, unitary trace-zero structure constant tensorα. We construct a versal deformation space for the deformations of aK-algebraB ₀ as a section of the completion at the tensorα ₀ ofB ₀ of the structure-constant schemeC _n . In order to obtain concrete information about the algebraic structure ofC _n obtained as above.     

Almost geodesic mappings of Riemann spaces onto symmetric Riemann spaces

It is shown that if a Riemann space V_n admits a reduced almost geodesic mapping Π₂ onto a symmetric Riemann space ¯V_n, then ¯V_n has constant curvature, and V_n is itself a symmetric space.     

Multiple prime covers of the riemann sphere

A compact Riemann surface X of genus g≥2 which admits a cyclic group of automorphisms C _q of prime order q such that X/C _q has genus 0 is called a cyclic q-gonal surface. If a q-gonal surface X is also p-gonal for some prime p≠q, then X is called a multiple prime surface. In this paper, we classify all multiple prime surfaces. A consequence of this classification is a proof of the fact that a cyclic q-gonal surface can be cyclic p-gonal for at most one other prime p.     

Moduli spaces of quadratic differentials

The cotangent bundle ofJ (g, n) is a union of complex analytic subvarieties, V(π), the level sets of the function “singularity pattern” of quadratic differentials. Each V(π) is endowed with a natural affine complex structure and volume element. The latter contracts to a real analytic volume element, Μ_π, on the unit hypersurface, V₁(π), for the Teichmüller metric. Μ_π is invariant under the pure mapping class group, γ(g, n), and a certain class of functions is proved to be L^p(Μ_π), 0 <p < 1, over the moduli space V₁(π)/γ (g, n). In particular, Μ_π(V₁(π)/γ(g, n)) < ∞, a statement which generalizes a theorem by H. Masur. Research supported by NSF-MCS-8219148 and NSF-DMS-8521620.     

Symmetry types of cyclic covers of the sphere

We examine all compact Riemann surfaces of genus greater than one which admit a cyclic group of automorphisms that yields a covering of the Riemann sphere with exactly three branch points. We determine the number of non-conjugate symmetries of each of these surfaces. For each symmetry, we determine the number of ovals it fixes and whether the orbit space under the symmetry is orientable or not. This yields the species of each symmetry and the symmetry type of each surface. Explicit defining equations of each surface and symmetry are given.     

Branched covers of the sphere and the prime-degree conjecture

To a branched cover ${\widetilde{\Sigma} \to \Sigma}$ between closed, connected, and orientable surfaces, one associates a branch datum, which consists of Σ and ${\widetilde{\Sigma}}$ , the total degree d, and the partitions of d given by the collections of local degrees over the branching points. This datum must satisfy the Riemann–Hurwitz formula. A candidate surface cover is an abstract branch datum, a priori not coming from a branched cover, but satisfying the Riemann– Hurwitz formula. The old Hurwitz problem asks which candidate surface covers are realizable by branched covers. It is now known that all candidate covers are realizable when Σ has positive genus, but not all are when Σ is the 2-sphere. However, a long-standing conjecture asserts that candidate covers with prime degree are realizable. To a candidate surface cover, one can associate one ${\widetilde {X} \dashrightarrow X}$ between 2-orbifolds, and in Pascali and Petronio (Trans Am Math Soc 361:5885–5920, 2009), we have completely analyzed the candidate surface covers such that either X is bad, spherical, or Euclidean, or both X and ${\widetilde{X}}$ are rigid hyperbolic orbifolds, thus also providing strong supporting evidence for the prime-degree conjecture. In this paper, using a variety of different techniques, we continue this analysis, carrying it out completely for the case where X is hyperbolic and rigid and ${\widetilde{X}}$ has a 2-dimensional Teichmüller space. We find many more realizable and non-realizable candidate covers, providing more support for the prime-degree conjecture.     

Neighborhoods of the Riemann sphere in complex surfaces

Moscow State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 27, No. 3, pp. 29–41, July–September, 1993.     

Moduli spaces for principal bundles in arbitrary characteristic

In this article, we solve the problem of constructing moduli spaces of semistable principal bundles (and singular versions of them) over smooth projective varieties over algebraically closed ground fields of positive characteristic.     

On prime galois coverings of the Riemann sphere

Research supported by a grant of the CICYT, M.E.C., Spain.     

Universal covers for Hausdorff limits of noncompact spaces

We prove that if is the Gromov-Hausdorff limit of a sequence of complete manifolds, , with a uniform lower bound on Ricci curvature, then has a universal cover.

     

The Moduli Space of Riemann Surfaces of Large Genus

Let ${\mathcal{M}_{g,\epsilon}}$ be the ${\epsilon}$ -thick part of the moduli space ${\mathcal{M}_g}$ of closed genus g surfaces. In this article, we show that the number of balls of radius r needed to cover ${\mathcal{M}_{g,\epsilon}}$ is bounded below by ${(c_1g)^{2g}}$ and bounded above by ${(c_2g)^{2g}}$ , where the constants c ₁, c ₂depend only on ${\epsilon}$ and r, and in particular not on g. Using this counting result we prove that there are Riemann surfaces of arbitrarily large injectivity radius that are not close (in the Teichmüller metric) to a finite cover of a fixed closed Riemann surface. This result illustrates the sharpness of the Ehrenpreis conjecture.     

Compact covers and function spaces

For a Tychonoff space X  , we denote by _Cp(X)$C_p(X)$ and _Cc(X)$C_c(X)$ the space of continuous real-valued functions on X equipped with the topology of pointwise convergence and the compact-open topology respectively. Providing a characterization of the Lindelöf Σ-property of X   in terms of _Cp(X)$C_p(X)$, we extend Okunev?s results by showing that if there exists a surjection from _Cp(X)$C_p(X)$ onto _Cp(Y)$C_p(Y)$ (resp. from _Lp(X)$L_p(X)$ onto _Lp(Y)$L_p(Y)$) that takes bounded sequences to bounded sequences, then υY is a Lindelöf Σ-space (respectively K-analytic) if υX has this property. In the second part, applying Christensen?s theorem, we extend Pelant?s result by proving that if X is a separable completely metrizable space and Y   is first countable, and there is a quotient linear map from _Cc(X)$C_c(X)$ onto _Cc(Y)$C_c(Y)$, then Y   is a separable completely metrizable space. We study also a non-separable case, and consider a different approach to the result of J. Baars, J. de Groot, J. Pelant and V. Valov, which is based on the combination of two facts: Complete metrizability is preserved by _?p${?}_p$-equivalence in the class of metric spaces (J. Baars, J. de Groot, J. Pelant). If X   is completely metrizable and _?p${?}_p$-equivalent to a first-countable Y, then Y is metrizable (V. Valov). Some additional results are presented.