Branch Locus Research Articles

Lifting elementary Abelian covers of curves

Given a Galois cover of curves f over a field of characteristic p, the lifting problem asks whether there exists a Galois cover over a complete mixed characteristic discrete valuation ring whose reduction is f. In this paper, we consider the case where the Galois groups are elementary abelian p-groups. We prove a combinatorial criterion for lifting an elementary abelian p-cover, dependent on the branch loci of lifts of its p-cyclic subcovers. We also study how branch points of a lift coalesce on the special fiber. Finally, for p=2, we analyze lifts for several families of (Z/2)3-covers of various conductor types, both with equidistant branch locus geometry and non-equidistant branch locus geometry.

Cyclic‐Schottky strata of Schottky space

AbstractSchottky space , where is an integer, is a connected complex orbifold of dimension ; it provides a parametrization of the ‐conjugacy classes of Schottky groups of rank . The branch locus , consisting of those conjugacy classes of Schottky groups being a finite index proper normal subgroup of some Kleinian group, is known to be connected. If , then there is a Kleinian group containing as a normal subgroup of index some prime integer . The structural description, in terms of Klein–Maskit Combination Theorems, of such a group is completely determined by a triple , where are integers such that . For each such tuple , there is a corresponding cyclic‐Schottky stratum . It is known that is connected. In this paper, for , we study the connectivity of these .

INTERSECTING THE TORSION OF ELLIPTIC CURVES

Abstract Bogomolov and Tschinkel [‘Algebraic varieties over small fields’, Diophantine Geometry, U. Zannier (ed.), CRM Series, 4 (Scuola Normale Superiore di Pisa, Pisa, 2007), 73–91] proved that, given two complex elliptic curves $E_1$ and $E_2$ along with even degree- $2$ maps $\pi _j\colon E_j\to \mathbb {P}^1$ having different branch loci, the intersection of the image of the torsion points of $E_1$ and $E_2$ under their respective $\pi _j$ is finite. They conjectured (also in works with Fu) that the cardinality of this intersection is uniformly bounded independently of the elliptic curves. The recent proof of the uniform Manin–Mumford conjecture implies a full solution of the Bogomolov–Fu–Tschinkel conjecture. In this paper, we prove a generalisation of the Bogomolov–Fu–Tschinkel conjecture whereby, instead of even degree- $2$ maps, one can use any rational functions of bounded degree on the elliptic curves as long as they have different branch loci. Our approach combines Nevanlinna theory with the uniform Manin–Mumford conjecture. With similar techniques, we also prove a result on lower bounds for ranks of elliptic curves over number fields.

Dynamics of Chebyshev endomorphisms on some affine algebraic varieties

The Chebyshev polynomials Td in one variable are typical chaotic maps on C. Chebyshev endomorphisms PAnd: Cn→Cn are also chaotic. We consider the action of the dihedral group Dn+1 on Cn. The endomorphism PAnd maps any Dn+1-orbit of z∈Cn to a Dn+1-orbit of PAnd(z). The endomorphism PAnd induces a mapping on Cn∕Dn+1. Using invariant theory, we embed Cn∕Dn+1 as an affine subvariety X in Cm. Then we have morphisms gd on X. We study the cases n=2 and 3. In these cases, the morphisms gd are defined over Z. We find a class of affine subvarieties V of X which are invariant under gd. These varieties are concerned with branch loci or critical loci. The class contains C2, a cuspidal cubic, a parabola, a quadric hypersurface in C4, an affine algebraic surface in C4 which is birationally equivalent to an affine quadric cone in C3, and others. For each affine variety V in the class, there exists a polynomial parameterization PV satisfying gd|V(PV(y1,…,yk))=PV(Td(y1),…,Td(yk)), where Td(z) is a Chebyshev polynomial in one variable. Then we determine the set of bounded orbits of gd|V in each invariant set V and give relations between them.

Realizing Artin–Schreier covers with minimal a-numbers in positive characteristic

Suppose $X$ is a smooth projective connected curve defined over an algebraically closed field of characteristic $p>0$ and $B \subset X$ is a finite, possibly empty, set of points. Booher and Cais determined a lower bound for the $a$-number of a $\mathbf{Z}/p \mathbf{Z}$-cover of $X$ with branch locus $B$. For odd primes $p$, in most cases it is not known if this lower bound is realized. In this note, when $X$ is ordinary, we use formal patching to reduce that question to a computational question about $a$-numbers of $\mathbf{Z}/p\mathbf{Z}$-covers of the affine line. As an application, when $p=3$ or $p=5$, for any ordinary curve $X$ and any choice of $B$, we prove that the lower bound is realized for Artin-Schreier covers of $X$ with branch locus $B$.

Monodromy of the family of Cubic Surfaces branching over Smooth Cubic Curves

Consider the family of smooth cubic surfaces which can be realized as threefold-branched covers of $\mathbb{P}^{2}$, with branch locus equal to a smooth cubic curve. This family is parametrized by the space $\mathcal{U}_{3}$ of smooth cubic curves in $\mathbb{P}^{2}$ and each surface is equipped with a $\mathbb{Z}/3\mathbb{Z}$ deck group action. We compute the image of the monodromy map $\rho$ induced by the action of $\pi_{1}\left(\mathcal{U}_{3}\right)$ on the $27$ lines contained on the cubic surfaces of this family. Due to a classical result, this image is contained in the Weyl group $W\left(E_{6}\right)$. Our main result is that $\rho$ is surjective onto the centralizer of the image a of a generator of the deck group. Our proof is mainly computational, and relies on the relation between the $9$ inflection points in a cubic curve and the $27$ lines contained in the cubic surface branching over it.

On subvarieties of singular quotients of bounded domains

Let X $X$ be a quotient of a bounded domain in C n $\mathbb {C}^n$ . Under suitable assumptions, we prove that every subvariety of X $X$ not included in the branch locus of the quotient map is of log-general type in some orbifold sense. This generalizes a recent result by Boucksom and Diverio, which treated the case of compact, étale quotients. Finally, in the case where X $X$ is compact, we give a sufficient condition under which there exists a proper analytic subset of X $X$ containing all entire curves and all subvarieties not of general type (meant this time in in the usual sense as opposed to the orbifold sense).

TRIPLE COVERS OF K3 SURFACES

AbstractWe study triple covers of K3 surfaces, following Miranda (1985, American Journal of Mathematics 107, 1123–1158). We relate the geometry of the covering surfaces with the properties of both the branch locus and the Tschirnhausen vector bundle. In particular, we classify Galois triple covers computing numerical invariants of the covering surface and of its minimal model. We provide examples of non-Galois triple covers, both in the case in which the Tschirnhausen bundle splits into the sum of two line bundles and in the case in which it is an indecomposable rank 2 vector bundle. We provide a criterion to construct rank 2 vector bundles on a K3 surface S which determine a non-Galois triple cover of S. The examples presented are in any admissible Kodaira dimension, and in particular, we provide the constructions of irregular covers of K3 surfaces and of surfaces with geometrical genus equal to 2 whose transcendental Hodge structure splits in the sum of two Hodge structures of K3 type.

The generic isogeny decomposition of the Prym Variety of a cyclic branched covering

Let f:S'longrightarrow S be a cyclic branched covering of smooth projective surfaces over {mathbb {C}} whose branch locus Delta subset S is a smooth ample divisor. Pick a very ample complete linear system |{mathcal {H}}| on S, such that the polarized surface (S, |{mathcal {H}}|) is not a scroll nor has rational hyperplane sections. For the general member [C]in |{mathcal {H}}| consider the mu _{n}-equivariant isogeny decomposition of the Prym variety {{,mathrm{Prym},}}(C'/C) of the induced covering f:C'{:}{=}f^{-1}(C)longrightarrow C: Prym(C′/C)∼∏d|n,d≠1Pd(C′/C).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} {{\\,\\mathrm{Prym}\\,}}(C'/C)\\sim \\prod _{d|n,\\ d\\ne 1}{\\mathcal {P}}_{d}(C'/C). \\end{aligned}$$\\end{document}We show that for the very general member [C]in |{mathcal {H}}| the isogeny component {mathcal {P}}_{d}(C'/C) is mu _{d}-simple with {{,mathrm{End},}}_{mu _{d}}({mathcal {P}}_{d}(C'/C))cong {mathbb {Z}}[zeta _{d}]. In addition, for the non-ample case we reformulate the result by considering the identity component of the kernel of the map {mathcal {P}}_{d}(C'/C)subset {{,mathrm{Jac},}}(C')longrightarrow {{,mathrm{Alb},}}(S').

Branched Cauchy–Riemann structures on once-punctured torus bundles

Unlike in hyperbolic geometry, the monodromy ideal triangulation of a hyperbolic once-punctured torus bundle $M_f$ has no natural geometric realisation in Cauchy-Riemann (CR) space. By introducing a new type of $3$--cell, we construct a different cell decomposition $\mathcal{D}_f$ of $M_f$ that is always realisable in CR space. As a consequence, we show that every hyperbolic once-punctured torus bundle admits a branched CR structure, whose branch locus is the set of edges of $\mathcal{D}_f$. Furthermore, we explicitly compute the ramification order around each component of the branch locus and analyse the corresponding holonomy representations.

Double cover K3 surfaces of Hirzebruch surfaces

Abstract General K3 surfaces obtained as double covers of the n-th Hirzebruch surfaces with n = 0, 1, 4 are not double covers of other smooth surfaces. We give a criterion for such a K3 surface to be a double covering of another smooth rational surface based on the branch locus of double covers and fibre spaces of Hirzebruch surfaces.

Spectral maps associated to semialgebraic branched coverings

In this article we prove that a semialgebraic map \(\pi :M\rightarrow N\) is a branched covering if and only if its associated spectral map is a branched covering. In addition, such spectral map has a neat behavior with respect to the branching locus, the ramification set and the ramification index. A crucial fact to prove the preceding result is the characterization of the prime ideals whose fibers under the previous spectral map are singletons.

Liftable homeomorphisms of cyclic and rank two finite abelian branched covers over the real projective plane

In this note, we investigate the property for regular branched finite abelian covers of the real projective plane, where each homeomorphism of the base (preserving the branch locus) lifts to a homeomorphism of the covering surface.

Liftable homeomorphisms of rank two finite abelian branched covers

We investigate branched regular finite abelian A-covers of the 2-sphere, where every homeomorphism of the base (preserving the branch locus) lifts to a homeomorphism of the covering surface. In this study, we prove that if A is a finite abelian p-group of rank k and $$\Sigma \rightarrow S^2$$ is a regular A-covering branched over n points such that every homeomorphism $$f:S^2 \rightarrow S^2$$ lifts to $$\Sigma $$ , then $$n=k+1$$ . We will also give a partial classification of such covers for rank two finite p-groups. In particular, we prove that for a regular branched A-covering $$\pi :\Sigma \rightarrow S^2$$ , where $$A={{\mathbb {Z}}}_{p^r}\times {{\mathbb {Z}}}_{p^t}, \ 1\le r\le t$$ , all homeomorphisms $$f:S^2 \rightarrow S^2$$ lift to those of $$\Sigma $$ if and only if $$t=r$$ or $$t=r+1$$ and $$p=3$$ .

Examples of compact Einstein four-manifolds with negative curvature

We give new examples of compact, negatively curved Einstein manifolds of dimension 4 4 . These are seemingly the first such examples which are not locally homogeneous. Our metrics are carried by a sequence of four-manifolds ( X k ) (X_k) previously considered by Gromov and Thurston (Pinching constants for hyperbolic manifolds, Invent. Math. 89 (1987), no. 1, 1–12). The construction begins with a certain sequence ( M k ) (M_k) of hyperbolic four-manifolds, each containing a totally geodesic surface Σ k \Sigma _k which is nullhomologous and whose normal injectivity radius tends to infinity with k k . For a fixed choice of natural number l l , we consider the l l -fold cover X k → M k X_k \to M_k branched along Σ k \Sigma _k . We prove that for any choice of l l and all large enough k k (depending on l l ), X k X_k carries an Einstein metric of negative sectional curvature. The first step in the proof is to find an approximate Einstein metric on X k X_k , which is done by interpolating between a model Einstein metric near the branch locus and the pull-back of the hyperbolic metric from M k M_k . The second step in the proof is to perturb this to a genuine solution to Einstein’s equations, by a parameter dependent version of the inverse function theorem. The analysis relies on a delicate bootstrap procedure based on L 2 L^2 coercivity estimates.

A SAGE Package for n-Gonal Equisymmetric Stratification of

In this work, we present an algorithm running over SAGE, which allows users to deal with group actions on Riemann surfaces up to topological equivalence. Our algorithm allows us to study the equisymmetric stratification of the branch locus of the moduli space of compact Riemann surfaces of genus corresponding to group actions with orbit genus 0. That is, it works for actions on surfaces of any genus in the case the genus of the quotient surface is zero, except for obvious hardware constraints. Our approach is toward studying inclusions and intersections of (closed) strata of We apply our algorithm to describe part of the geometry of the branch locus in terms of equisymmetric stratification. We also use it to compute all group actions up to topological equivalence for genus 5–10, this completes the lists. Finally, we add an optimized version of an algorithm, which allows us to identify Jacobian varieties of CM-type. As a byproduct, we obtain a Jacobian variety of dimension 11 which is isogenous to where Ei and are elliptic curves with complex multiplication.

Realizing Artin-Schreier covers of curves with minimal Newton polygons in positive characteristic

Suppose X is a smooth projective connected curve defined over an algebraically closed field k of characteristic p>0 and B⊂X(k) is a finite, possibly empty, set of points. The Newton polygon of a degree p Galois cover of X with branch locus B depends on the ramification invariants of the cover. When X is ordinary, for every possible set of branch points and ramification invariants, we prove that there exists such a cover whose Newton polygon is minimal or close to minimal.

Cyclic covers and Ihara’s question

Let $\ell$ be a rational prime and $k$ a number field. Given a superelliptic curve $C/k$ of $\ell$-power degree, we describe the field generated by the $\ell$-power torsion of the Jacobian variety in terms of the branch set and reduction type of $C$ (and hence, in terms of data determined by a suitable affine model of $C$). If the Jacobian is good away from $\ell$ and the branch set is defined over a pro-$\ell$ extension of $k(\mu_{\ell^\infty})$ unramified away from $\ell$, then the $\ell$-power torsion of the Jacobian is rational over the maximal such extension. By decomposing the covering into a chain of successive cyclic $\ell$-coverings, the mod $\ell$ Galois representation attached to the Jacobian is decomposed into a block triangular form. The blocks on the diagonal of this form are further decomposed in terms of the Tate twists of certain subgroups $W_s$ of the quotients of the Jacobians of successive coverings. The result is a natural extension of earlier work by Anderson and Ihara, who demonstrated that a stricter condition on the branch locus guarantees the $\ell$-power torsion of the Jacobian is rational over the fixed field of the kernel of the canonical pro-$\ell$ outer Galois representation attached to an open subset of $\mathbf{P}^1$.

BRANCHED CAUCHY–RIEMANN STRUCTURES ON ONCE-PUNCTURED TORUS BUNDLES

Unlike in hyperbolic geometry, the monodromy ideal triangulation of a hyperbolic once-punctured torus bundle $M_f$ has no natural geometric realisation in Cauchy-Riemann (CR) space. By introducing a new type of $3$--cell, we construct a different cell decomposition $\mathcal{D}_f$ of $M_f$ that is always realisable in CR space. As a consequence, we show that every hyperbolic once-punctured torus bundle admits a branched CR structure, whose branch locus is the set of edges of $\mathcal{D}_f$. Furthermore, we explicitly compute the ramification order around each component of the branch locus and analyse the corresponding holonomy representations.

Isomonodromy deformations at an irregular singularity with coalescing eigenvalues

We consider an $n\times n$ linear system of ODEs with an irregular singularity of Poincar\'e rank 1 at $z=\infty$, holomorphically depending on parameter $t$ within a polydisc in $\mathbb{C}^n$ centred at $t=0$. The eigenvalues of the leading matrix at $z=\infty$ coalesce along a locus $\Delta$ contained in the polydisc, passing through $t=0$. Namely, $z=\infty$ is a resonant irregular singularity for $t\in \Delta$. We analyse the case when the leading matrix remains diagonalisable at $\Delta$. We discuss the existence of fundamental matrix solutions, their asymptotics, Stokes phenomenon and monodromy data as $t$ varies in the polydisc, and their limits for $t$ tending to points of $\Delta$. When the deformation is isomonodromic away from $\Delta$, it is well known that a fundamental matrix solution has singularities at $\Delta$. When the system also has a Fuchsian singularity at $z=0$, we show under minimal vanishing conditions on the residue matrix at $z=0$ that isomonodromic deformations can be extended to the whole polydisc, including $\Delta$, in such a way that the fundamental matrix solutions and the constant monodromy data are well defined in the whole polydisc. These data can be computed just by considering the system at fixed $t=0$. Conversely, if the $t$-dependent system is isomonodromic in a small domain contained in the polydisc not intersecting $\Delta$, if the entries of the Stokes matrices with indices corresponding to coalescing eigenvalues vanish, then we show that $\Delta$ is not a branching locus for the fundamental matrix solutions. The importance of these results for the analytic theory of Frobenius Manifolds is explained. An application to Painlev\'e equations is discussed.