Daniele Faenzi, Janis Stipins
Given a triple cover p: X --> Y of varieties, we produce a new variety Z and a birational morphism f: Z --> X which is an isomorphism away from the fat-point ramification locus of p. The variety Z has a natural interpretation in terms of the data describing the triple cover, and the morphism f has an elegant geometric description.
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We consider surfaces of geometric genus $3$ with the property that their transcendental cohomology splits into $3$ pieces, each piece coming from a $K3$ surface. For certain families of surfaces with this property, we can show there is a similar splitting on the level of Chow groups (and Chow motives).
Special triple covers of algebraic surfaces
本文研究代数曲面的特殊三重覆盖，通过刻画 Tschirnhausen 丛为分裂秩三向量丛的商，给出存在性的充要条件，并完成特殊三重平面的分类，发现两个 K3 曲面族。
Studying coverings over algebraic varieties is an effective method in algebraic geometry. By combining the technique of triple cover from Miranda and Tan, we proved that if the degree of the branch divisor of a normal triple cover over $¶^2$ is no more than $18$ and not equal to $16$, then the trace-free bundle of the triple cover is uniform. Moreover, we totally classified all these trace-free bundles.
We study triple covers of K3 surfaces, following Miranda's theory of triple covers. 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 vec…
Let $X$ be a smooth algebraic curve. Suppose that there exists a triple covering $f : X \to Y$ where $Y$ is a smooth algebraic curve. In this paper, we investigate the existence of morphisms from $X$ to the projective line $\mathbf{P}^1$ which do not factor through the covering $f$. For this purpose, we generalize the classical results of Maroni concerning base-point-free pencils on trigonal curves to the case of triple covers of arbitrary smooth irrational curves.
We construct a surface of general type with canonical map of degree 12 which factors as a triple cover and a bidouble cover of $\mathbb P^2$. We also show the existence of a smooth surface with $q=0,$ $χ=13$ and $K^2=9χ$ such that its canonical map is either of degree 3 onto a surface of general type or of degree 9 onto a rational surface.
We show that the Prym variety associated to a triple covering f: Y --> X of curves is principally polarized of dimension > 1, if and only if f is non-cyclic, etale and X is of genus 2. We investigate some properties of these Prym varieties and their moduli.
In this paper, we show that if $X$ is a smooth variety of general type of dimension $m \geq 3$ for which the canonical map induces a triple cover onto $Y$, where $Y$ is a projective bundle over $\mathbf P^1$ or onto a projective space or onto a quadric hypersurface, embedded by a complete linear series, then the general deformation of the canonical morphism of $X$ is again canonical and induces a triple cover. The extremal case when $Y$ is embedded as a variety of minimal degree is of interest, due to its appearance in numerous situations. This is especially interesting as well, since it has n…
Let $S$ and $T$ be reduced divisors on $\mathbb{P}^2$ which have no common components, and $Δ=S+2\,T.$ We assume $\degΔ=6.$ Let $π:X\to\mathbb{P}^2$ be a normal triple cover with branch divisor $Δ,$ i.e. $π$ is ramified along $S$ (resp. $T$) with the index 2 (resp. 3). In this note, we show that $X$ is either a $\mathbb{P}^1$-bundle over an elliptic curve or a normal cubic surface in $\mathbb{P}^3.$ Consequently, we give a necessary and sufficient condition for $Δ$ to be the branch divisor of a normal triple cover over $\mathbb{P}^2.$
In a previous paper, the authors proved that the Prym variety of any non-cyclic etale triple cover of a smooth curve of genus 2 is a Jacobian variety of dimension 2. This gives a map from the moduli space of such covers to the moduli space of Jacobian varieties of dimension 2. We extend this map to a proper map of a certain moduli space of admissible $S_3$-covers of genus 7 to the moduli space of principally polarized abelian surfaces. The main result is that this map is finite surjective of degree 10.
Triple intersections for algebraic curves on the real torus
本文受Corvaja和Zannier问题启发，研究实环面上三条实代数曲线投影的三重交点，对三条直线给出完整分类，对高次曲线用Siegel和Levin定理得有限性结果，并构造无穷多三重交点的显式例子。
We study projective varieties whose universal cover is biholomorphic to a semialgebraic open subset of a projective variety.
Given a triple covering $X$ of genus $g$ of a general (in the sense of Brill-Noether) curve $C$ of genus $h$, we show the existence of base-point-free pencils of degree $d$ which are not composed with the triple covering for any $d\ge g-[{3h+1\over 2}]-1$ by utilizing some enumerative methods and computations. We also discuss about the sharpness of our main result and the so-called Castelnuovo-Severi bound byexhibiting some examples.
We prove that the universal cover of a normal, projective variety X is quasi-projective if and only if a finite, étale cover of X is a fiber bundle over an Abelian variety with simply connected fiber.
Non-cycle triple planes with branch curve of degree at most 10
本文研究了分支曲线次数不超过10的正常非循环三重覆盖的分类问题。
The aim of this paper is to show that using some natural curve arrangements in algebraic surfaces and Hirzebruch-Kummer covers one cannot construct new examples of ball-quotients, i.e., minimal smooth complex projective surfaces of general type satisfying equality in the Bogomolov-Miyaoka-Yau inequality.
Let $f:X\to Y$ be a finite ramified Galois covering of algebraic varieties defined over the complex numbers. In this paper, we prove some structure theorems for such coverings in the case that the non-abelian Galois group of the cover is metabelian (in particular metacyclic). Our results extend the previous ones, obtained by several authors, in the case of abelian and dihedral Galois coverings. In analogy with the abelian (and generalizing the dihedral) case, we find "building data" of metabelian covers from which the cover can be reconstructed. In addition to analyzing several examples, we co…
We discover suprising connections between three seemingly different problems: finding right triangles with rational sides in a non-Euclidean geometry, finding three integers such that the difference of the squares of any two is a square, and the problem of finding rational points on an algebraic surface in algebraic geometry. We will also reinterpret Euler's work on the second problem with a modern point of view.
In this article we study pluriregular varieties X of general type with base-point-free canonical bundle whose canonical morphism has degree 3 and maps X onto a variety of minimal degree Y. We carry out our study from two different perspectives. First we study in Section 2 the canonical ring of X describing completely the degrees of its minimal generators. We apply this to the study of the projective normality of the images of the pluricanonical morphisms of X. Our study of the canonical ring of $X$ also shows that, if the dimension of X is greater than or equal to 3, there does not exist a con…
Let $X$ be a normal noetherian scheme and $Z \subseteq X$ a closed subset of codimension $\geq 2$. We consider here the local obstructions to the map $\hatπ_{1}(X\backslash Z) \to \hatπ_{1}(X)$ being an isomorphism. Assuming $X$ has a regular alteration, we prove the equivalence of the obstructions being finite and the existence of a Galois quasi-étale cover of $X$, where the corresponding map on fundamental groups is an isomorphism.
Flop between algebraically integrable foliations on potentially klt varieties
该研究证明潜在klt簇上lc代数可积叶状三元组的极小模型可经翻转变换连接，还探讨了三维非代数可积叶状结构极小模型的联系。
We prove the birational superrigidity and the nonrationality of a cyclic triple cover of $\mathbb{P}^{2n}$ branched over a nodal hypersurface of degree $3n$ for $n\ge 2$. In particular, the obtained result solves the problem of the birational superrigidity of smooth cyclic triple spaces. We also consider certain relevant problems.
Let G be a connected algebraic group over an algebraically closed field of characteristic p (possibly 0), and X a variety on which G acts transitively with connected stabilizers. We show that any étale Galois cover of X of degree prime to p is also homogeneous, and that the maximal prime-to-p quotient of the étale fundamental group of X is commutative. We moreover obtain an explicit bound for the number of topological generators of the said quotient. When G is commutative, we also obtain a description of the prime-to-p torsion in the Brauer group of G.
Non-reduced components of the Hilbert scheme of curves using triple covers
本文利用曲线理想自由分解与 Ciliberto 形变技术，证明锥上三重覆盖曲线的形变仍位于锥上，并构造出高亏格光滑曲线 Hilbert 概形的一个非约化分支。
Hurwitz moduli varieties parameterizing pointed covers of an algebraic curve with a fixed monodromy group
本文构造并证明了参数化固定单值群点态代数曲线覆盖的 Hurwitz 空间及其泛族。
The goal of this paper is to give a method to compute the space of infinitesimal deformations of a double cover of a smooth algebraic variety. The space of all infinitesimal deformations has a representation as a direct sum of two subspaces. One is isomorphic to the space of simultaneous deformations of the branch locus and the base of the double covering. The second summand is the subspace of deformations of the double covering which induce trivial deformations of the branch divisor. The main result of the paper is a description of the effect of imposing singularities in the branch locus. As …
We show that infinitesimal Torelli for $n$-forms holds for abelian covers of algebraic varieties of dimension $n\ge 2$, under some explicit ampleness assumptions on the building data of the cover. Moreover, we prove a variational Torelli result for some families of abelian covers.
A well-known and difficult problem in computational number theory and algebraic geometry is to write down equations for branched covers of algebraic curves with specified monodromy type. In this article, we present a technique for computing such covers in the case of totally ramified covers of elliptic curves branched at one point.
Hurwitz moduli varieties parameterizing Galois covers of an algebraic curve
该研究针对代数曲线的伽罗瓦覆盖模问题，利用代数拓扑与复代数几何工具，证明了不同条件下Hurwitz空间的精细/粗模簇性质并构造万有族。
Let $Y$ be an algebraic variety, $\mathcal{F}$ a locally free sheaf of $\mathcal{O}_Y$-modules, and $\mathcal{R}(\mathcal{F})$ the $\mathcal{O}_Y$-algebra $\operatorname{Sym}^\bullet \mathcal{F}$. In this paper we study local properties of sheaves of $\mathcal{O}_{\mathcal{R}(\mathcal{F})}$-ideals $\mathcal{I}$ such that $\mathcal{R}(\mathcal{F}))/\mathcal{I}$ is an algebraic cover of $Y$. Following the work of Miranda for triple covers, for $\mathcal{Q}$ a direct summand of $\mathcal{R}(\mathcal{F})$, we say that a morphism $Φ\colon \mathcal{Q}\rightarrow\mathcal{R}(\mathcal{F})/\langle\mathc…
Cyclic covers of an algebraic curve from an adelic viewpoint
本文从adelic视角提出代数方法，用于研究曲线的循环覆盖，确定循环环扩展与曲线覆盖之间的关系，并解决等价性问题。
It is known that the monodromy group of each cover of a general curve of genus g>3 equals either the symmetric or the alternating group. The classical Catalan numbers count the minimal degree covers (with symmetric monodromy) of a general curve of even genus. We solve the analogous problem for the alternating group and we determine the number of alternating covers of minimal degree 2g+1 of a general curve of genus g.
One dimensional metrical geometry may be developed in either an affine or projective setting over a general field using only algebraic ideas and quadratic forms. Some basic results of universal geometry are already present in this situation, such as the Triple quad formula, the Triple spread formula and the Spread polynomials, which are universal analogs of the Chebyshev polynomials of the first kind. Chromogeometry appears here, and the related metrical and algebraic properties of the projective line are brought to the fore.
In this article we investigate the algebra and geometry of dihedral covers of smooth algebraic varieties. To this aim we first describe the Weil divisors and the Picard group of divisorial sheaves on normal double covers. Then we provide a structure theorem for dihedral covers, that is, given a smooth variety Y, we describe the algebraic "building data" on Y which are equivalent to the existence of such covers X --> Y. We introduce then two special very explicit classes of dihedral covers: the simple and the almost simple dihedral covers, and we determine their basic invariants. For the simple…
In topology, the notions of the fundamental group and the universal cover are closely intertwined. By importing usual notions from topology into the algebraic and arithmetic setting, we construct a fundamental group family from a universal cover, both of which are schemes. A geometric fiber of the fundamental group family (as a topological group) is canonically the 'etale fundamental group. The constructions apply to all connected quasicompact quasiseparated schemes. With different methods and hypotheses, this fundamental group family was already constructed by Deligne.
We suggest a general method of computation of the homology of certain smooth covers $\hat{\mathcal{M}}_{g,1}(\mathbb{C})$ of moduli spaces $\mathcal{M}_{g,1}\br{\mathbb{C}}$ of pointed curves of genus $g$. Namely, we consider moduli spaces of algebraic curves with level $m$ structures. The method is based on the lifting of the Strebel-Penner stratification $\mathcal{M}_{g,1}\br{\mathbb{C}}$. We apply this method for $g\leq 2$ and obtain Betti numbers; these results are consistent with Penner and Harer-Zagier results on Euler characteristics.
Relating Hodge Atoms, Spectral Triples, and BPS Flows
研究非交换Hodge结构对四次立方流形上quiver规范场理论的影响，提出K3 Hodge原子代表一种受保护的量子相，其谱在非微扰隧道过程中保持不变。
Let $Z \to X$ be a finite branched Galois cover of normal projective geometrically integral varieties of dimension $d \geq 2$ over a perfect field $k$. For such a cover, we prove a Chebotarev-type density result describing the decomposition behaviour of geometrically integral Cartier divisors. As an application, we classify Galois covers among all finite branched covers of a given normal geometrically integral variety $X$ over $k$ by the decomposition behaviour of points of a fixed codimension $r$ with $0 < r < \dim X$.
A Criterion for the Algebraic Density Property of Affine $SL_2$-Manifolds
本文针对仿射$SL_2$流形，提出了判断其代数密度性质的代数准则，并将该准则应用于物理学中三类$SL_2$簇的代数密度性质证明。
We prove that the set of non-degenerate second order maximally superintegrable systems in the complex Euclidean plane carries a natural structure of a projective variety, equipped with a linear isometry group action. This is done by deriving the corresponding system of homogeneous algebraic equations. We then solve these equations explicitly and give a detailed analysis of the algebraic geometric structure of the corresponding projective variety. This naturally associates a unique planar line triple arrangement to every superintegrable system, providing a geometric realisation of this variety …
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