Thomas Peternell, Andrew Sommese
In continuation of our work in Comm. in Algebra, vol. 28 (2000), we study ramified coverings of projective manifolds, in particular over Fano manifolds and investigate positivity properties of the associated vector bundle. Moreover we study the topology of low degree coverings and the structure of the ramification divisor.
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Given a covering f: X \to Y of projective manifolds, we consider the vector bundle E on Y given as the dual of f_*(Ø_X) / Ø_Y. This vector bundles often has positivity properties, e.g. E is ample when Y is projective space by a theorem of Lazarsfeld. In general however E will not be ample due to the geometry of Y. We prove various results when E is spanned, nef or generically nef, under some assumptions on the base Y.
Let $f\colon S'\longrightarrow S$ be a cyclic branched covering of smooth projective surfaces over $\mathbb{C}$ whose branch locus $Δ\subset S$ 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 $μ_{n}$-equivariant isogeny decomposition of the Prym variety $Prym(C'/C)$ of the induced covering $f\colon C'= f^{-1}(C)\longrightarrow C$:\[Prym(C'/C)\sim\prod_{d|n,\ d\neq1}\mathcal{P}_{d}(C'/C).\…
In this article we prove that a semialgebraic map 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.
We study nonsingular branched coverings of a homogeneous space X. There is a vector bundle associated with such a covering which was conjectured by O. Debarre to be ample when the Picard number of X is one. We prove this conjecture, which implies Barth-Lefschetz type theorems, for lagrangian grassmannians, and for quadrics up to dimension six. We propose a conjectural extension to homogeneous spaces of Picard number larger than one and prove a weaker version.
Quadratic Euler Characteristic of Geometrically Cyclic Branched Coverings
本文利用Levine的二次Riemann-Hurwitz公式，计算几何循环分支覆盖的二次欧拉特征，并在特定情况下将其与基底方案的特征联系起来。
Prym loci of branched double coverings and generalized Andreotti-Mayer loci
本文将安德烈奥蒂-迈耶轨迹推广至非主极化阿贝尔簇模空间，证明分支二重覆盖的普里姆轨迹是广义安德烈奥蒂-迈耶轨迹的不可约分支。
In this paper, we construct a natural probability measure on the space of real branched coverings from a real projective algebraic curve $(X,c_X)$ to the projective line $(\mathbb{C}\mathbb{P}^1,\textrm{conj})$. We prove that the space of degree $d$ real branched coverings having "many" real branched points (for example more than $\sqrt{d}^{1+α}$, for any $α>0$) has exponentially small measure. In particular, maximal real branched coverings, that is real branched coverings such that all the branched points are real, are exponentially rare.
In this paper, we prove the irreducibility of the monodromy action on the anti-invariant part of the vanishing cohomology on a double cover of a very general element in an ample hypersurface of a complex smooth projective variety branched at an ample divisor. As an application, we study dominant rational maps from a double cover of a very general surface $S$ of degree$\geq 7$ in ${\mathbb P}^3$ branched at a very general quadric surface to smooth projective surfaces $Z$. Our method combines the classification theory of algebraic surfaces, deformation theory, and Hodge theory.
We give a group-theoretic description of the parity of a pull-back of a theta characteristic under a branched covering. It involves lifting monodromy of the covering to the semidirect product of the symmetric and Clifford groups, known as the Sergeev group. As an application, we enumerate torus coverings with respect to their ramification and parity and, in particular, show that the corresponding all-degree generating functions are quasimodular forms.
We compute the asymptotics of the number of connected branched coverings of a torus as their degree goes to infinity and the ramification type stays fixed. These numbers are equal to the volumes of the moduli spaces of pairs (curve, holomorphic differential) with fixed multiplicities of zeros of the differential and have several applications in ergodic theory.
For any $n>1$, we construct examples branched Galois coverings from $M$ to the nth projective space ${\mathbb P}^n$ where $M$ is one of $({\mathbb P}^1)^n$, ${\mathbb C}^n$ or $(B_1)^n$, and $B_1$ is the 1-ball. In terms of orbifolds, this amounts to giving examples of orbifolds over ${\mathbb P}^n$ uniformized by $M$.We also discuss the related "orbifold braid groups".
On branched coverings of the projective line over the integers
该研究探讨整数射影直线水平除子补集的平展基本群，证明其无非平凡有限可解商群的充要条件，还给出特定条件下不存在同构于$\boldsymbol{\text{PSL}}_{2}(q)$商群的结论。
A Decomposition Theorem for Topological Branched Coverings
本研究提出拓扑分支覆盖的分解定理，通过构造无分支覆盖映射的分解方法，并推广到非流形目标空间的情形。
We show that the vector bundle associated to a smooth projective connected finite covering of a simple complex abelian variety is ample (under a simple necessary condition). This result is obtained by showing that this bundle is M-regular in the sense of Pareschi-Popa, and that any M-regular sheaf is ample.
For a closed oriented surface $ Σ$ let $X_{Σ,n}$ be the space of isomorphism classes of orientation preserving $n$-fold branched coverings $ Σ\rightarrow S^2 $ of the two-dimensional sphere. At a previous paper, the authors constructed a compactification $\bar{X}_{Σ,n}$ of the space that coincides with the Diaz-Edidin-Natanzon-Turaev compactification of the Hurwitz space $H(Σ,n)\subset X_{Σ,n}$ consisting of isomorphism classes of branched coverings with all critical values being simple. Using Grothendieck's dessins d'enfants we construct a cell structure of the compactification. The obtained …
An algebraic variety X is embedded to the order k via a line bundle L if the global sections of L generate all (simultaneous) jets of order k on X or if they separate all zero-dimensional subschemes of length at most k+1. Even though we refer to both situations as "higher order embeddings", the first notion (in which case L is said to be k-jet ample) is stronger than the second one (when L is k-very ample). The purpose of this paper is to study higher order embeddings of cyclic coverings π:Y\to X via line bundles given by pulling back "sufficiently positive" line bundles on X. Given a line bun…
Given a closed Riemann surface $Σ$ equipped with a volume form $ω$, we construct a natural probability measure on the space $\mathcal{M}_d(Σ)$ of degree $d$ branched coverings from $Σ$ to the Riemann sphere $\mathbb{C}\mathbb{P}^1.$ We prove a large deviations principle for the number of critical points in a given open set $U\subset Σ$: given any sequence $ε_d$ of positive numbers, the probability that the number of critical points of a branched covering deviates from $2d\cdot\textrm{Vol}(U)$ more than $ε_d$ is smaller than $\exp(-C_Uε^3_d d)$, for some positive constant $C_U$. In particular, …
Braid group actions on branched coverings and full exceptional sequences
本文通过将完全例外序列与圆盘上的分支覆盖相联系，利用Birman–Hilden理论和Hurwitz系统研究辫群作用，并构造了Bondal–Polishchuk传递性猜想的反例。
The Prym variety for a branched double covering of a nonsingular projective curve is defined as a polarized abelian variety. We prove that any double covering of an elliptic curve which has more than $4$ branch points is recovered from its Prym variety.
Motivated by the problem of finding algebraic constructions of finite coverings in commutative algebra, the Steinitz realization problem in number theory, and the study of Hurwitz spaces in algebraic geometry, we investigate the vector bundles underlying the structure sheaf of a finite flat branched covering. We prove that, up to a twist, every vector bundle on a smooth projective curve arises from the direct image of the structure sheaf of a smooth, connected branched cover.
Esnault-Viehweg developed the theory of cyclic branched coverings $\tilde X\to X$ of smooth surfaces providing a very explicit formula for the decomposition of $H^1(\tilde X,\mathbb{C})$ in terms of a resolution of the ramification locus. Later, the first author applies this to the particular case of coverings of $\mathbb{P}^2$ reducing the problem to a combination of global and local conditions on projective curves. In this paper we extend the above results in three directions: first, the theory is extended to surfaces with quotient singularities, second the ramification locus can be partiall…
A $K3$ surface with an ample divisor of self-intersection 2 is a double cover of the plane branched over a sextic curve. We conjecture that a similar statement holds for the generic couple $(X,H)$ with $X$ a deformation of $(K3)^{[n]}$ and $H$ an ample divisor of square 2 for Beauville's quadratic form. If $n=2$ then according to the conjecture $X$ is a double cover of a (singular) sextic 4-fold in $\PP^5$. It follows from the conjecture that a deformation of $(K3)^{[n]}$ carrying a divisor (not necessarily ample) of degree 2 has an anti-symplectic birational involution. We test the conjecture…
In this paper we deal with branched coverings over the complement to finitely many exceptional points on the Riemann sphere having the property that the local monodromy around each of the branching points is of finite order. To such a covering we assign its \textit{signature}, i.e. the set of its exceptional and branching points together with the orders of local monodromy operators around the branching points. What can be said about the monodromy group of a branched covering if its signature is known? It seems at first that the answer is nothing or next to nothing. Indeed, generically it is so…
Every compact symplectic 4-manifold can be realized as a branched cover of the complex projective plane branched along a symplectic curve with cusp and node singularities; the covering map is induced by a triple of sections of a "very ample" line bundle. In this paper, we give an explicit formula describing the behavior of the braid monodromy invariants of the branch curve upon degree doubling of the linear system (from a very ample bundle $L^k$ to $L^{2k}$). As a consequence, we derive a similar degree doubling formula for the mapping class group monodromy of Donaldson's symplectic Lefschetz …
Let $S$ be a rational surface with $\dim|-K_S|\ge 1$ and let $π: X\rightarrow S$ be a ramified cyclic covering from a nonruled smooth surface $X$. We show that for any integer $k\ge 3$ and ample divisor $A$ on $S$, the adjoint divisor $K_X+kπ^*A$ is very ample and normally generated. Similar result holds for minimal (possibly singular) coverings.
Using a hyperKähler rotation on complex structures of a Calabi-Yau 2-fold and rolling of an isotropic 2-submanifold in a symplectic 6-manifold, we construct, by gluing, a natural family of immersed Lagrangian deformations of a branched covering of a special Lagrangian 3-sphere in a Calabi-Yau 3-fold and study how they deviate from being deformable to a family of special Lagrangian deformations by examining in detail Joyce's criteria on this family. The result suggests a potential image-support rigidity of A-branes that wrap around a special Lagrangian 3-sphere in a Calabi-Yau 3-fold, which res…
In this paper we construct 206 examples of Calabi-Yau manifolds with different Euler numbers. All constructed examples are smooth models of double coverings of $P^3$ branched along an octic surface. We allow 11 types of (not necessary isolated) singularities in the branch locus. Thus we broaden the class of examples studied in math.AG/9902057. For every considered example we compute the Euler number and give a precise description of a resolution of singularities.
Let $Y$ be a smooth projective surface defined over an algebraically closed field $k$ with ${\rm Char}\ k\nmid n$, and let $π:X\rightarrow Y$ be a $n$-cyclic covering branched along a smooth divisor $B$. We show that under some conditions $\mathscr{T}_X$ is semi-stable with respect to $π^*\mathcal {H}$ if the tangent bundle $\mathscr{T}_Y$ is semi-stable with respect to some ample line bundle $\mathcal {H}$ on $Y$.
Isometric embeddings of Teichmüller spaces are covering constructions
该研究证明维数≥2时泰希米勒空间的所有等距嵌入均源自分支覆叠，推广了Royden定理，并给出全测地子流形性质及黎曼曲面模空间局部等距嵌入分类。
Let $X$ be a compact Riemann surface, $Σ$ a finite set of points and $M = X\setminus Σ$. We study the $L^2$ cohomology of a polarized complex variation of Hodge structure on a Galois covering of the Riemann surface of finite type $M$. In this article we treat the case when the covering comes from a branched covering of $X$, and where $M$ is endowed with a metric asymptotic to a Poincaré metric. We prove that after tensorisation with the algebra of affiliated operators, the $L^2$ cohomology admits a pure Hodge structure.
Graber, Harris and Starr proved, when n >= 2d, the irreducibility of the Hurwitz space H^0_{d,n}(Y) which parametrizes degree d coverings of a smooth, projective curve Y of positive genus, simply branched in n points, with full monodromy group S_d (math.AG/0205056). We sharpen this result and prove that H^0_{d,n}(Y) is irreducible if n >= max{2,2d-4} and in the case of elliptic Y if n >= max{2,2d-6}. We extend the result to coverings simply branched in all but one point of the discriminant. Fixing the ramification multiplicities over the special point we prove that the corresponding Hurwitz sp…
We introduce a one-parameter deformation of the 2-Toda tau-function of (weighted) Hurwitz numbers, obtained by deforming Schur functions into Jack symmetric functions. We show that its coefficients are polynomials in the deformation parameter $b$ with nonnegative integer coefficients. These coefficients count generalized branched coverings of the sphere by an arbitrary surface, orientable or not, with an appropriate $b$-weighting that "measures" in some sense their non-orientability. Notable special cases include non-orientable dessins d'enfants for which we prove the most general result so fa…
We study the Kähler geometry of the classical Hurwitz space $\mathcal{H}^{n,b}$ of simple branched coverings of the Riemann sphere $\mathbb{P}^1$ by compact hyperbolic Riemann surfaces. A generalized Weil-Petersson metric on the Hurwitz space was recently introduced. Deformations of simple branched coverings fit into the more general framework of Horikawa's deformation theory of holomorphic maps, which we equip with distinguished representatives in the presence of hermitian metrics. In the article we will investigate the curvature of the generalized Weil-Petersson Kähler metric on the Hurwitz …
In the context of orientable circuits and subcomplexes of these as representing certain singular spaces, we consider characteristic class formulas generalizing those classical results as seen for the Riemann-Hurwitz formula for regulating the topology of branched covering maps and that for monoidal transformations which include the standard blowing-up process. Here the results are presented as cap product pairings, which will be elements of a suitable homology theory, rather than characteristic numbers as would be the case when taking Kronecker products once Poincaré duality is defined. We fur…
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