In this work we present new results to produce an algorithm that returns, for any fixed pair of natural integers $K^2$ and $χ$, all regular surfaces $S$ of general type with self-intersection $K_S^2=K^2$ and Euler characteristic $χ(\mathcal O_S)=χ$, that are product-quotient surfaces. The key result we obtain is an algebraic characterization of all families of regular product-quotients surfaces, up to isomorphism, arising from a pair of $G$-coverings of $\mathbb P^1$. As a consequence of our work, we provide a classification of all regular product-quotient surfaces of general type with $23\leq K^2\leq 32$ and $χ(\mathcal O_S)=4$. Furthermore, we study their canonical map and present several new examples of surfaces of general type with a high degree of the canonical map.
In this paper, we shall discuss Hilbert property of ${\rm Hilb}^{G}(\mathbb{C}^{3})$, Fujiki-Oka resolutions and iterated Fujiki-Oka resolutions for three dimensional canonical cyclic quotient singularities by using the classification shown by Ishida and Iwashita\cite{II}. Finally, we shall prove that there exists a Hilbert desingularization for any three dimensional canonical cyclic quotient singularity.
Quotient singularities by permutation actions are canonical
Graduate School of Science, the University of Osaka（大阪大学理学研究科）；Kavli Institute for the Physics and Mathematics of the Universe, The University of Tokyo（东京大学宇宙物理数学研究所）
The wild McKay correspondence, a variant of the McKay correspondence in positive characteristics, shows that stringy motives of quotient varieties equal some motivic integrals on the moduli space of of the Galois covers of a formal disk. In this paper, we determine when the integrals converge for the the case of cyclic groups of prime power order. As an application, we give a criterion for the quotient variety being canonical or log canonical.
We provide a description of the fundamental group of the quotient of a product of topological spaces $X_i$, each admitting a universal cover, by a finite group $G$, provided that there is only a finite number of path-connected components in $X_i^g$ for every $g\in G$. This generalizes previous work of Bauer-Catanese-Grunewald-Pignatelli and Dedieu-Perroni.
We prove Qingyuan Jiang's conjecture on semiorthogonal decompositions of derived categories of Quot schemes of locally free quotients. The author's result on categorified Hall products for Grassmannian flips is applied to prove the conjecture.
In this note we present examples of complex algebraic surfaces of general type with canonical maps of degree $10$, $11$ and $14$. They are constructed as quotients of a product of two Fermat septics using certain free actions of the group $\mathbb Z_7^2$.
In this paper the authors study quotients of the product of elliptic curves by a rigid diagonal action of a finite group $G$. It is shown that only for $G = \operatorname{He(3)}, \mathbb Z_3^2$, and only for dimension $\geq 4$ such an action can be free. A complete classification of the singular quotients in dimension 3 and the smooth quotients in dimension $4$ is given. For the other finite groups a strong structure theorem for rigid quotients is proven.
In this note we present examples of complex algebraic surfaces with canonical maps of degree $12$, $13$, $15$, $16$ and $18$. They are constructed as quotients of a product of two curves of genus $10$ and $19$ using certain non-free actions of the group $S_3\times \mathbb Z_3^2$. To our knowledge there are no other examples in literature of surfaces with canonical map of degree $13$, $15$ and $18$.
设X是具有对合g的K3曲面，该对合g具有非空固定轨迹X^g并且非平凡地作用在非零全纯2-形式上。我们将从指数至多为2的log del Pezzo曲面或有理椭圆曲面的一些更好的双覆盖中，以规范的方式构造所有这样的对（X，g），并构造X^g包含10（最大可能）条曲线的三种极值情形中的每一种的唯一族。我们还分类合理的日志Enriques表面的指数为2。我们的方法更多的是几何的，而不是晶格理论的（见Nikulin的论文，后者的方法）。
