Bernhard Köck
We give an elementary, self-contained and quick proof of Belyi's theorem. As a by-product of our proof we obtain an explicit bound for the degree of the defining number field of a Belyi surface.
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We give a Belyi-type characterisation of smooth complete intersections of general type over $\mathbb{C}$ which can be defined over $\bar{\mathbb{Q}}$. Our proof uses the higher-dimensional analogue of the Shafarevich boundedness conjecture for families of canonically polarized varieties, finiteness results for maps to varieties of general type, and rigidity theorems for Lefschetz pencils of complete intersections.
We show that every smooth projective curve over a finite field k admits a finite tame morphism to the projective line over k. Furthermore, we construct a curve with no such map when k is an infinite perfect field of characteristic two. Our work leads to a refinement of the tame Belyi theorem in positive characteristic, building on results of Saïdi, Sugiyama-Yasuda, and Anbar-Tutdere.
Belyi's theorem for smooth complete intersections of general type in generalised Grassmannians and weighted projective spaces
本文将Belyi定理从普通射影空间中的一般型光滑完全交推广到广义Grassmann流形和加权射影空间，并提出向更一般Mori dream空间推广的思路。
The purpose of the present paper is to give an effective version of the noncritical $p$-tame Belyi theorem. That is to say, we compute explicitly an upper bound of the minimal degree of tamely ramified Belyi maps in positive characteristic which are unramified at a prescribed finite set of points.
On Belyi's Theorem in positive characteristic
本文证明在q≥5时，F_q(T)是唯一允许一维刚性解析空间存在类似Belyi定理刻画的正特征函数域，并指出Drinfel'd模曲线为此类空间的实例。
Given an object over the algebraic closure Qbar of Q, there is often no reason for invariants of the corresponding holomorphic object to be preserved by the absolute Galois group Gal(Qbar/Q), and in general this is not true, although it is sometimes surprising to observe in practice. The case of covers of the projective line branched only over the points 0, 1, and infinity, through Belyi's theorem, leads to Grothendieck's dessins d'enfants program for understanding the absolute Galois group through its faithful action on such covers. This note is motivated by Catanese's question about a higher…
Belyi's theorem: coverings, dessins, and fields of definition
本文完整证明Belyi定理，给出Belyi多项式的结构定理与极值映射的质量公式，并展示非伽罗瓦三次域上的显式轨道。
We prove an analog of Belyi's theorem for the algebraic surfaces. Namely, any non-singular algebraic surface can be defined over a number field if and only it covers the complex projective plane with ramification at three knotted two-dimensional spheres.
For a field of characteristic zero, M. Levine has proved that his category of triangulated motives is equivalent to the one constructed by V. Voevodsky. In this paper we show that the strategy of Levine's proof can be applied on every perfect field to the categories of triangulated motives with rational coefficients.
We revisit the nef curve cone theorem and use it to reprove the pseudo-effectivity of the second Chern classes for terminal weak Q-Fano varieties.
Bertini theorems revisited
本文将经典Bertini定理推广至任意域与离散赋值环，证明了若干新定理。
We revisit some of the basic results of generic vanishing theory, as pioneered by Green and Lazarsfeld, in the context of constructible sheaves. Using the language of perverse sheaves, we give new proofs of some of the basic results of this theory. Our approach is topological/arithmetic, and avoids Hodge theory.
Bridgeland stability conditions allow for a new generalization of Thaddeus pairs to surfaces and a new interpretation of Reider's theorem as a consequence of "Schur's lemma" for stable objects (Hom(E,F) = 0 if E,F are stable objects and the slope of E exceeds the slope of F). One improvement of Reider's theorem results (Proposition 3.8/Corollary 3.9), and wall-crossings for the new Thaddeus pairs are discussed. This paper was submitted to the CMI conference proceedings celebrating the 65th birthday of Peter Newstead.
A theorem of O. Haupt, rediscovered by M. Kapovich and celebrated by his proof invoking Ratner theory, describes the set of de Rham cohomology classes on a topological orientable surface, which can be realized by an abelian differential in some respective complex structure, in purely topological terms. We make an attempt to describe similarly pairs and triples of cohomology classes, which can be realized by abelian differentials in some complex structure. This leads us to some interesting problems in algebraic geometry of curves, and gives an unexpected local description of the Teichmüller spa…
Von Staudt's theorem revisited
本文在具有足够多单位（尤其 2 为单位）的非交换环上，建立了保持调和四元组的映射的 von Staudt 定理版本。
The spectral gluing theorem revisited
本研究强化了几何朗兰兹猜想谱侧的粘合定理，明确刻画了$IndCoh_N(LS_G)$到由标准抛物型参数化的傅里叶系数粘合范畴之嵌入的本质像，完善了该定理的结论。
After recalling basic definitions and constructions for a finite group $G$ action on a $k$-linear category we give a concise proof of the following theorem of Elagin: if $\mathcal{C} = \langle \mathcal{A}, \mathcal{B} \rangle$ is a semiorthogonal decomposition of a triangulated category which is preserved by the action of $G$, and $\mathcal{C}^G$ is triangulated, then there is a semiorthogonal decomposition $\mathcal{C}^G = \langle \mathcal{A}^G, \mathcal{B}^G \rangle$. We also prove that any $G$-action on $\mathcal{C}$ is weakly equivalent to a strict $G$-action which is the analog of the Coh…
Quantum Lefschetz theorem revisited
本文通过引入容许级数并利用扩展变量表述超几何修正，证明了任意光滑 Deligne-Mumford 栈中光滑完全交的亏格零量子莱夫谢茨定理，从而证实了 Coates-Corti-Iritani-Tseng 在超越凸性情形下的预测。
Saito's theorem revisited and application to free pencils of hypersurfaces
本文通过多重特征概形重新解释斋藤自由性准则，构造出$\mathbb P^n$中大量新的自由约化超曲面族，解决了已知自由超曲面例子少的问题。
We present a proof of Chow's theorem using two results of Errett Bishop retated to volumes and limits of analytic varieties. We think this approach suggested a long time ago in the beautiful book by Gabriel Stolzenberg, is very attractive and easier for students and newcomers to understand, also the theory presented here is linked to areas of mathematics that are not usually associated with Chow's theorem. Furthermore, Bishop's results imply both Chow's and Remmert-Stein's theorems directly, meaning that this approach is more economic and just as profound as Remmert-Stein's proof. At the end o…
We give a simple generalisation of a theorem of Morita, which leads to a great number of relations among tautological classes on moduli spaces of curves.
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