Positive Genus Research Articles

Local heights on Galois covers of the projective line

Let X be a smooth projective curve of positive genus defined over a number field K. Assume given a Galois covering map x : X → PK and a place v of K. We introduce a local canonical height on X(Kv) associated to x as an integral with logarithmic integrand, generalizing Tate’s local Neron function on an elliptic curve. The resulting global height can be viewed as a ‘Mahler measure’ associated to x. We prove that the local canonical height can be obtained by averaging, and taking a limit, over divisors of higher order Weierstrass points on X. This generalizes previous results by Everest-ni Fhlathuin and Szpiro-Tucker. Our construction of the local canonical height is an application of potential theory on Berkovich curves in the presence of a canonical measure.

Insecurity for compact surfaces of positive genus: commentary

Insecurity for compact surfaces of positive genus: commentary

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.

Elements and cyclic subgroups of finite order of the Cremona group

We give the classification of elements – respectively cyclic subgroups – of finite order of the Cremona group, up to conjugation. Natural parametrisations of conjugacy classes, related to fixed curves of positive genus, are provided.

The real section conjecture and Smith's fixed-point theorem for pro-spaces

We prove a topological version of the section conjecture for the profinite completion of the fundamental group of finite CW-complexes equipped with the action of a group of prime order $p$ whose $p$-torsion cohomology can be killed by finite covers. As an application we derive the section conjecture for the real points of a large class of varieties defined over the field of real numbers and the natural analogue of the section conjecture for fixed points of finite group actions on projective curves of positive genus defined over the field of complex numbers.

Supercritical Conformal Metrics on Surfaces with Conical Singularities

We study the problem of prescribing the Gaussian curvature on surfaces with conical singularities in supercritical regimes. Using a Morse-theoretical approach, we prove a general existence theorem on surfaces with positive genus, with a generic multiplicity result.

HIGHER-GENUS MEAN CURVATURE-ONE CATENOIDS IN HYPERBOLIC AND DE SITTER 3-SPACES

We show the existence of constant mean curvature-one surfaces in both hyperbolic 3-space and de Sitter 3-space with two complete embedded ends and any positive genus up to genus twenty. We also find another such family of surfaces in de Sitter 3-space, but with a different non-embedded end behavior.

Scaling Limits for Random Quadrangulations of Positive Genus

Abstract. We discuss scaling limits of large bipartite quadrangulations of positive genus. For a given $g$, we consider, for every positive integer $n$, a random quadrangulation $q_n$ uniformly distributed over the set of all rooted bipartite quadrangulations of genus $g$ with $n$ faces. We view it as a metric space by endowing its set of vertices with the graph distance. We show that, as $n$ tends to infinity, this metric space, with distances rescaled by the factor $n$ to the power of $-1/4$, converges in distribution, at least along some subsequence, toward a limiting random metric space. This convergence holds in the sense of the Gromov-Hausdorff topology on compact metric spaces. We show that, regardless of the choice of the subsequence, the Hausdorff dimension of the limiting space is almost surely equal to $4$. Our main tool is a bijection introduced by Chapuy, Marcus, and Schaeffer between the quadrangulations we consider and objects they call well-labeled $g$-trees. An important part of our study consists in determining the scaling limits of the latter.

Detecting rigid convexity of bivariate polynomials

Given a polynomial x ∈ R n ↦ p ( x ) in n = 2 variables, a symbolic-numerical algorithm is first described for detecting whether the connected component of the plane sublevel set P = { x : p ( x ) ⩾ 0 } containing the origin is rigidly convex, or equivalently, whether it has a linear matrix inequality (LMI) representation, or equivalently, if polynomial p ( x ) is hyperbolic with respect to the origin. The problem boils down to checking whether a univariate polynomial matrix is positive semidefinite, an optimization problem that can be solved with eigenvalue decomposition. When the variety C = { x : p ( x ) = 0 } is an algebraic curve of genus zero, a second algorithm based on Bézoutians is proposed to detect whether P has an LMI representation and to build such a representation from a rational parametrization of C . Finally, some extensions to positive genus curves and to the case n > 2 are mentioned.

Insecurity for compact surfaces of positive genus

A pair of points in a riemannian manifold $M$ is secure if the geodesics between the points can be blocked by a finite number of point obstacles; otherwise the pair of points is insecure. A manifold is secure if all pairs of points in $M$ are secure. A manifold is insecure if there exists an insecure point pair, and totally insecure if all point pairs are insecure. Compact, flat manifolds are secure. A standing conjecture says that these are the only secure, compact riemannian manifolds. We prove this for surfaces of genus greater than zero. We also prove that a closed surface of genus greater than one with any riemannian metric and a closed surface of genus one with generic metric are totally insecure.

Cauchy problem for one class of degenerate parabolic equations of Kolmogorov type with positive genus

We investigate properties of a fundamental solution and establish the correct solvability of the Cauchy problem for one class of degenerate Kolmogorov-type equations with $$ \left\{ {\overrightarrow p, \overrightarrow h } \right\} $$ -parabolic part with respect to the main group of variables and with positive vector genus in the case where solutions are infinitely differentiable functions and their initial values may be generalized functions of Gevrey ultradistribution type.

Strange duality and the Hitchin/WZW connection

For a compact Riemann surface $X$ of positive genus, the space of sections of a certain theta bundle on moduli of bundles of rank $r$ and level $k$ admits a natural map to (the dual of) a similar space of sections of rank $k$ and level $r$ (the strange duality isomorphism). Both sides of the isomorphism carry projective connections as $X$ varies in a family. We prove that this map is (projectively) flat.

On the Hasse principle for Shimura curves

Let C be an algebraic curve defined over a number field K, of positive genus and without K-rational points. We conjecture that there exists some extension field L over which C has points everywhere locally but not globally. We show that our conjecture holds for all but finitely many Shimura curves of the form X0D (N)/ℚ or X1D (N)/ℚ, where D > 1 and N are coprime squarefree positive integers. The proof uses a variation on a theorem of Frey, a gonality bound of Abramovich, and an analysis of local points of small degree.

Schnyder Woods for Higher Genus Triangulated Surfaces, with Applications to Encoding

Schnyder woods are a well-known combinatorial structure for plane triangulations, which yields a decomposition into three spanning trees. We extend here definitions and algorithms for Schnyder woods to closed orientable surfaces of arbitrary genus. In particular, we describe a method to traverse a triangulation of genus g and compute a so-called g-Schnyder wood on the way. As an application, we give a procedure to encode a triangulation of genus g and n vertices in 4n+O(glog (n)) bits. This matches the worst-case encoding rate of Edgebreaker in positive genus. All the algorithms presented here have execution time O((n+g)g) and hence are linear when the genus is fixed.

Kodaira dimension and symplectic sums

Modulo trivial exceptions, we show that symplectic sums of symplectic 4-manifolds along surfaces of positive genus are never rational or ruled, and we enumerate each case in which they have Kodaira dimension zero (i.e., are blowups of symplectic 4-manifolds with torsion canonical class). In particular, a symplectic four-manifold of Kodaira dimension zero arises by such a surgery only if it is diffeomorphic to a blowup either of the K3 surface, the Enriques surface, or a member of a particular family of T2-bundles over T2 each having b1 = 2.

The structure of unicellular maps, and a connection between maps of positive genus and planar labelled trees

A unicellular map is a map which has only one face. We give a bijection between a dominant subset of rooted unicellular maps of given genus and a set of rooted plane trees with distinguished vertices. The bijection applies as well to the case of labelled unicellular maps, which are related to all rooted maps by Marcus and Schaeffer’s bijection. This gives an immediate derivation of the asymptotic number of unicellular maps of given genus, and a simple bijective proof of a formula of Lehman and Walsh on the number of triangulations with one vertex. From the labelled case, we deduce an expression of the asymptotic number of maps of genus g with n edges involving the ISE random measure, and an explicit characterization of the limiting profile and radius of random bipartite quadrangulations of genus g in terms of the ISE.

The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems

In this paper, the Conley conjecture, which was recently proved by Franks and Handel [J. Franks, M. Handel, Periodic points of Hamiltonian surface diffeomorphism, Geom. Topol. 7 (2003) 713–756] (for surfaces of positive genus), Hingston [N. Hingston, Subharmonic solutions of Hamiltonian equations on tori, Ann. Math., in press] (for tori) and Ginzburg [V.L. Ginzburg, The Conley conjecture, arXiv: math.SG/0610956v1] (for closed symplectically aspherical manifolds), is proved for C 1 -Hamiltonian systems on the cotangent bundle of a C 3 -smooth compact manifold M without boundary, of a time 1-periodic C 2 -smooth Hamiltonian H : R × T * M → R which is strongly convex and has quadratic growth on the fibers. Namely, we show that such a Hamiltonian system has an infinite sequence of contractible integral periodic solutions such that any one of them cannot be obtained from others by iterations. If H also satisfies H ( − t , q , − p ) = H ( t , q , p ) for any ( t , q , p ) ∈ R × T * M , it is shown that the time-1-map of the Hamiltonian system (if exists) has infinitely many periodic points siting in the zero section of T * M . If M is C 5 -smooth and dim M > 1 , H is of C 4 class and independent of time t, then for any τ > 0 the corresponding system has an infinite sequence of contractible periodic solutions of periods of integral multiple of τ such that any one of them cannot be obtained from others by iterations or rotations. These results are obtained by proving similar results for the Lagrangian system of the Fenchel transform of H, L : R × T M → R , which is proved to be strongly convex and to have quadratic growth in the velocities yet.

Holomorphic mappings and basic extremal lengths of once-holed tori

We are concerned with the set of once-holed tori from which there are handle-preserving holomorphic mappings into a given Riemann surface of positive genus. We investigate how the extremal lengths of homology classes of simple closed curves change under those holomorphic mappings of once-holed tori.

A Negative Mass Theorem for Surfaces of Positive Genus

Let M be a closed surface. For a metric g on M, denote the Laplace-Beltrami operator by Δ = Δg. We define trace \({\Delta^{-1} = \int_M m(p)dA}\) , where dA is the area element for g and m(p) is the Robin constant at the point \({p \in M}\) , that is the value of the Green function G(p, q) at q = p after the logarithmic singularity has been subtracted off. Since trace Δ−1 can also be obtained by regularization of the spectral zeta function, it is a spectral invariant. Heuristically it represents the sum of squares of the wavelengths of the surface. We define the Δ-mass of (M, g) to equal \({({\rm trace} \Delta_g^{-1} - {\rm trace} \Delta_{S^{2},A}^{-1})/A}\) , where \({\Delta_{S^2,A}}\) is the Laplacian on the round sphere of area A. This is an analog for closed surfaces of the ADM mass from general relativity. We show that if M has positive genus, the minimum of the Δ-mass on each conformal class is negative and attained by a smooth metric. For this minimizing metric, there is a sharp logarithmic Hardy-Littlewood-Sobolev inequality and a Moser-Trudinger-Onofri type inequality.

The Fano surface of the Klein cubic threefold

We prove that the Klein cubic threefold $F$ is the only smooth cubic threefold which has an automorphism of order $11$. We compute the period lattice of the intermediate Jacobian of $F$ and study its Fano surface $S$. We compute also the set of fibrations of $S$ onto a curve of positive genus and the intersection between the fibres of these fibrations. These fibres generate an index $2$ sub-group of the Néron-Severi group and we obtain a set of generators of this group. The Néron-Severi group of $S$ has rank $25=h^{1,1}$ and discriminant $11^{10}$.