Linear series and existence of branched covers

Simply branched covers of an elliptic curve and the moduli space of curves

Consider genus g curves that admit degree d covers to an elliptic curve simply branched at 2g-2 points. Vary a branch point and the locus of such covers forms a one-parameter family W. We investigate the geometry of W by using admissible covers to study its slope, genus and components. The results can also be applied to study slopes of effective divisors on the moduli space of genus g curves.

P¹的分支覆盖与类群中的可除性

Branched covers of $\mathbb{P}^1$ and divisibility in class group

研究通过m-正则曲线的雅可比群n- torsion生成数域K的类群n- torsion，探讨分支覆盖与数论中类群可除性关系。

Toric vector bundles, branched covers of fans, and the resolution property

We associate to each toric vector bundle on a toric variety X(Delta) a "branched cover" of the fan Delta together with a piecewise-linear function on the branched cover. This construction generalizes the usual correspondence between toric Cartier divisors and piecewise-linear functions. We apply this combinatorial geometric technique to investigate the existence of resolutions of coherent sheaves by vector bundles, using singular nonquasiprojective toric threefolds as a testing ground. Our main new result is the construction of complete toric threefolds that have no nontrivial toric vector bun…

D-brane probes, branched double covers, and noncommutative resolutions

This paper describes D-brane probes of theories arising in abelian gauged linear sigma models (GLSMs) describing branched double covers and noncommutative resolutions thereof, via nonperturbative effects rather than as the critical locus of a superpotential. As these theories can be described as IR limits of Landau-Ginzburg models, technically this paper is an exercise in utilizing (sheafy) matrix factorizations. For Landau-Ginzburg models which are believed to flow in the IR to smooth branched double covers, our D-brane probes recover the structure of the branched double cover (and flat nontr…

Galois branched covers with fixed ramification locus

We examine conditions under which there exists a non-constant family of Galois branched covers of curves over an algebraically closed field $k$ of fixed degree and fixed ramification locus, under a notion of equivalence derived from considering linear series on a fixed smooth proper source curve $X$. We show such a family exists precisely when the following conditions are satisfied: $\operatorname{char}(k)=p>0$, $X$ is isomorphic to $\mathbb{P}^1_k$, there is a unique ramification point, and the Galois group is $(\mathbb{Z}/p\mathbb{Z})^m$ for some integer $m>0$.

分支覆盖的一个同伦后果

A homotopical consequence of branched covers

研究证明伪流形的余有限完成等于其分支覆盖的Artin-Mazur层叠同伦类型构造，揭示了伪流形中足够多的K(π,1)稠密子空间的存在性。

Hodge theory of cyclic covers branched over a union of hyperplanes

Suppose that Y is a cyclic cover of projective space branched over a hyperplane arrangement D, and that U is the complement of the ramification locus in Y. The first theorem implies that the Beilinson-Hodge conjecture holds for U if certain multiplicities of D are coprime to the degree of the cover. For instance this applies when D is reduced with normal crossings. The second theorem shows that when D has normal crossings and the degree of the cover is a prime number, the generalized Hodge conjecture holds for any toroidal resolution of Y. The last section contains some partial extensions to m…

Modularity of two double covers of ${\mathbb P}^5$ branched along $12$ hyperplanes

For two varieties of dimension $5$ constructed as double covers of ${\mathbb P}^5$ branched along the union of $12$ hyperplanes, we prove that the number of points over ${\mathbb F}_p$ can be expressed in terms of Artin symbols and the $p$th Fourier coefficients of modular forms. Many analogous results are known in dimension $\le 3$, but very few in higher dimension. In addition, we use an idea of Burek to construct quotients of our varieties for which the point counts mod $p$ are expressible in terms of Artin symbols and the coefficients of a single modular form of weight $6$.

关于 $\mathbf{P}^1$ 在三个点处分支的阿贝尔覆盖的牛顿多边形与埃克萨尔-奥尔特类型

Ekedahl-Oort Types and Newton Polygons of Abelian Covers of $\mathbf{P}^1$ Branched at Three Points

本文研究了$\mathbf{P}^1$在三个点处分支的阿贝尔覆盖的牛顿多边形与埃克萨尔-奥尔特类型，揭示了超奇异和超特殊曲线的高频出现，并为Oort猜想提供了新的证据。

Completions, branched covers, Artin groups and singularity theory

We study the curvature of metric spaces and branched covers of Riemannian manifolds, with applications in topology and algebraic geometry. Here curvature bounds are expressed in terms of the CAT(k) inequality. We prove a general CAT(k) extension theorem, giving sufficient conditions on and near the boundary of a locally CAT(k) metric space for the completion to be CAT(k). We use this to prove that a branched cover of a complete Riemannian manifold is locally CAT(k) if and only if all tangent spaces are CAT(0) and the base has sectional curvature bounded above by k. We also show that the branch…

三次分支覆盖的复射影直线的数值计算

Numerical calculation of three-point branched covers of the projective line

本文提出一种数值方法来计算复射影直线的三次分支覆盖，通过开发处理Fuchsian三角群及其有限指数子群的算法，计算这些群上的模形式的幂级数展开。