A survey on the bicanonical map of surfaces with $p_g=0$ and $K^2\ge 2$

On surfaces with $p_g=2$, $q=1$ and non-birational bicanonical map

In the present note we show that any surface of general type with $p_g=2$,$q=1$ and non-birational bicanonical map has a pencil of curves of genus 2. Combining this result with previous ones, one obtains thatan irregular surface $S$ of general type with $χ(S)\geq 2$ and non-birational bicanonical map has a pencil of curves of genus 2.

The bicanonical map of the Cartwright-Steger surface

We prove that the bicanonical map of the Cartwright-Steger surface is an embedding. We also discuss two minimal surfaces of general type, both covered by the Cartwright-Steger surface. One has $K^2=2$, $p_g=1$, $π_1=\{1\}$ and the other has $K^2=1$, $p_g=0$, $π_1=\mathbb{Z}/2\mathbb{Z}$.

The Bicanonical map of fake projective planes with an automorphism

We show, for several fake projective planes with nontrivial automorphism group, that the bicanonical map is an embedding.

The bicanonical map of surfaces with $p_g=0$ and $K^2\ge 7$, II

We study the minimal complex surfaces of general type with $p_g=0$ and $K^2=7$ or 8 whose bicanonical map is not birational. In the paper 'The bicanonical map of surfaces with $p_g=0$ and $K^2\ge 7$' we have shown that if $S$ is such a surface, then the bicanonical map has degree 2. Here we describe precisely such surfaces showing that there is a fibration $f\colon S\to \pp^1$ such that: i) the general fibre $F$ of $f$ is a genus 3 hyperelliptic curve; ii) the involution induced by the bicanonical map of $S$ restricts to the hyperelliptic involution of $F$. Furthermore, if $K^2_S=8$, then $f$ …

关于一般型三维流形大体积的双CANONICAL映射

On bicanonical maps of threefolds of general type with large volumes

研究一般型三维流形大体积的双CANONICAL映射的维度以及大canonical体积的pluricanonical映射的纤维结构

一般型代数三维簇的双典范映射研究

On the bicanonical map of algebraic threefolds of general type

本研究解答了一则代数几何领域的公开问题，证明几何亏格大于201的非奇异射影一般型三维簇的双典范系统不构成曲面束。

On surfaces with $p_g=q=2$ and non-birational bicanonical map

This paper is devoted to the classification of irregular surfaces of general type with $p_g=q=2$ and non birational bicanonical map. The main result is that, if $S$ is such a surface and if $S$ is minimal with no pencil of curves of genus 2, then $S$ is a double cover of a principally polarized abelian surface $(A,Θ)$, with $Θ$ irreducible. The double cover $S\to A$ is branched along a divisor $B\in |2Θ|$, having at most double points and so $K_S^2=4$.

The bicanonical map of surfaces with $p_g=0$ and $K^2 \geq 7$

A minimal surface of general type with $p_g(S)=0$ satisfies $1\le K^2\le 9$ and it is known that the image of the bicanonical map $\fie$ is a surface for $K_S^2\geq 2$, whilst for $K^2_S\geq 5$, the bicanonical map is always a morphism. In this paper it is shown that $\fie$ is birational if $K_S^2=9$ and that the degree of $\fie$ is at most 2 if $K_S^2=7$ or $K_S^2=8$. By presenting two examples of surfaces $S$ with $K_S^2=7$ and 8 and bicanonical map of degree 2, it is also shown that this result is sharp. The example with $K_S^2=8$ is, to our knowledge, a new example of a surface of general …

The classification of surfaces with $p_g=0$, $K^2=6$ and non birational bicanonical map

Let $S$ be a minimal surface of general type with $p_g=0$ and $K^2=6$, such that its bicanonical map $\fie\colon S\to\pp^6$ is not birational. The map $\fie$ is a morphism of degree $\le 4$ onto a surface. The case of $°\fie=4$ is completely characterized in [Topology, {\bf 40} (5) (2001), 977--991] and the present paper completes the classification of these surfaces. It is proven that the degree of $\fie$ cannot be equal to 3, and the geometry of surfaces with $°\fie=2$ is analysed in detail. The last section contains three examples of such surfaces, two of which appear to be new.

On the bicanonical map of irregular varieties

From the point of view of uniform bounds for the birationality of pluricanonical maps, irregular varieties of general type and maximal Albanese dimension behave similarly to curves. In fact Chen-Hacon showed that, at least when their holomorphic Euler characteristic is positive, the tricanonical map of such varieties is always birational. In this paper we study the bicanonical map. We consider the natural subclass of varieties of maximal Albanese dimension formed by primitive varieties of Albanese general type. We prove that the only such varieties with non-birational bicanonical map are the n…

Surfaces with $p_g = q= 1$, $K^2 = 7$ and non-birational bicanonical mpas

Let $S$ be a minimal surface of general type with $p_g = q = 1, K_S^2 = 7$. We prove that the degree of the bicanonical map is 1 or 2. Furthermore, if the degree is 2, we describe $S$ by a double cover.

Bicanonical and adjoint linear systems on surfaces of general type

This note contains a new proof of a theorem of Gang Xiao saying that the bicanonical map of a surface S of general type is generically finite if and only if the second plurigenus of S is strictly larger than 2. Such properties are also studied for adjoint linear systems |K_S+L|, where L is any divisor with at least 2 linearly independent sections.

Surfaces of general type with $p_g=q=1, K^2=8$ and bicanonical map of degree 2

We classify the minimal algebraic surfaces of general type with $p_g=q=1, K^2=8$ and bicanonical map of degree 2. It will turn out that they are isogenous to a product of curves, so that if $S$ is such a surface then there exist two smooth curves $C, F$ and a finite group $G$ acting freely on $C \times F$ such that $S = (C \times F)/G$. We describe the $C, F$ and $G$ that occur. In particular the curve $C$ is a hyperelliptic-bielliptic curve of genus 3, and the bicanonical map $ϕ$ of $S$ is composed with the involution $σ$ induced on $S$ by $τ\times id: C \times F \longrightarrow C \times F$, …

A twisted bicanonical system with base points

By a theorem of Reider, a twisted bicanonical system, that means a linear system of divisors numerically equivalent to a bicanonical divisor, on a minimal surface of general type, is base point free if $K^2_S \geq 5$. Twisted bicanonical systems with base points are known in literature only for $K^2=1,2$. We prove in this paper that all surfaces in a family of surfaces with $K^2=3$ constructed in a previous paper with G. Bini and J. Neves have a twisted bicanonical system (different from the bicanonical system) with two base points. We show that the map induced by the above twisted bicanonical…

On the bicanonical morphism of quadruple Galois canonical covers

In this article we study the bicanonical map $ϕ_2$ of quadruple Galois canonical covers X of surfaces of minimal degree. We show that $ϕ_2$ has diverse behavior and exhibit most of the complexities that are possible for a bicanonical map of surfaces of general type, depending on the type of X. There are cases in which $ϕ_2$ is an embedding, and if so happens, $ϕ_2$ embeds $X$ as a projectively normal variety, and cases in which $ϕ_2$ is not an embedding. If the latter, $ϕ_2$ is finite of degree 1, 2 or 4. We also study the canonical ring of X, proving that it is generated in degree less than o…

Exceptional collections and the bicanonical map of Keum's fake projective planes

We apply the recent results of Galkin et al. [GKMS15] to study some geometrical features of Keum's fake projective planes. Among other things, we show that the bicanonical map of Keum's fake projective planes is always an embedding. Moreover, we construct a nonstandard exceptional collection on the unique fake projective plane $X$ with $H_{1}(X; \mathbb{Z})=(\mathbb{Z}/2\mathbb{Z})^{4}$.

A new family of surfaces with $p_g=0$ and $K^2=3$

Let S be a minimal complex surface of general type with p_g=0 such that the bicanonical map of S is not birational and let Z be the bicanonical image. In [M.Mendes Lopes, R.Pardini, "Enriques surfaces with eight nodes", Math. Zeit. 241 (4) (2002), 673-683] it is shown that either: i) Z is a rational surface, or ii) K^2_S=3, the bicanonical map is a degree two morphism and Z is birational to an Enriques surface. Up to now no example of case ii) was known. Here an explicit construction of all such surfaces is given. Furthermore it is shown that the corresponding subset of the moduli space of sur…

On surfaces with $p_g=q=1$ and non-ruled bicanonical involution

This paper classifies surfaces of general type $S$ with $p_g=q=1$ having an involution $i$ such that $S/i$ has non-negative Kodaira dimension and that the bicanonical map of $S$ factors through the double cover induced by $i.$ It is shown that $S/i$ is regular and either: a) the Albanese fibration of $S$ is of genus 2 or b) $S$ has no genus 2 fibration and $S/i$ is birational to a $K3$ surface. For case a) a list of possibilities and examples are given. An example for case b) with $K^2=6$ is also constructed.

Involutions on surfaces with $p_g=q=0$ and $K^2=3$

We study surfaces of general type $S$ with $p_g=0$ and $K^2=3$ having an involution $i$ such that the bicanonical map of $S$ is not composed with $i$. It is shown that, if $S/i$ is not rational, then $S/i$ is birational to an Enriques surface or it has Kodaira dimension $1$ and the possibilities for the ramification divisor of the covering map $S\rightarrow S/i$ are described. We also show that these two cases do occur, providing an example. In this example $S$ has a hyperelliptic fibration of genus $3$ and the bicanonical map of $S$ is of degree $2$ onto a rational surface.

Involutions on surfaces with $p_g=q=1$

In this paper some numerical restrictions for surfaces with an involution are obtained. These formulas are used to study surfaces of general type $S$ with $p_g=q=1$ having an involution $i$ such that $S/i$ is a non-ruled surface and such that the bicanonical map of $S$ is not composed with $i.$ A complete list of possibilities is given and several new examples are constructed, as bidouble covers of surfaces. In particular the first example of a minimal surface of general type with $p_g=q=1$ and $K^2=7$ having birational bicanonical map is obtained.

Enriques surfaces with eight nodes

A nodal Enriques surface can have at most 8 nodes. We give an explicit description of Enriques surfaces with 8 nodes, showing that they are quotients of products of elliptic curves by a group isomorphic to $\Z_2^2$ or to $\Z_2^3$ acting freely in codimension 1. We use this result to show that if $S$ is a minimal surface of general type with $p_g=0$ such that the image of the bicanonical map is birational to an Enriques surface then $K^2_S=3$ and the bicanonical map is a morphism of degree 2.

A connected component of the moduli space of surfaces of general type with $p_g=0$

Let S be a minimal surface of general type with $p_g(S)=0$ and such that the bicanonical map $ϕ:S\to \pp^{K^2_S}$ is a morphism: then the degree of $ϕ$ is at most 4 and if it is equal to 4 then $K^2_S\le 6$. Here we prove that if $K^2_S=6$ and $°ϕ=4$ then S is a so-called {\em Burniat surface}. In addition we show that minimal surfaces with $p_g=0$, $K^2=6$ and bicanonical map of degree 4 form a 4-dimensional irreducible connected component of the moduli space of surfaces of general type.

Surfaces of general type with p_g=q=0 having a pencil of hyperelliptic curves of genus 3

We prove that the bicanonical map of a surfaces of general S type with p_g=q=0 is non birational if there exists a pencil on S whose general member is an hyperelliptic curve of genus 3.

Complex surfaces of general type: some recent progress

Chapters : Old and new inequalities; Surfaces with $χ=1$ and the bicanonical map; Surfaces with $p_g=4$; Surfaces isogeneous to a product, Beauville surfaces and the absolute Galois group;Lefschetz pencils and braid monodromies;DEF, DIFF and other equivalence relations.

Some bidouble planes with $p_g=q=0$ and $4\leq K^2\leq 7$

We give a list of possibilities for surfaces of general type with $p_g=0$ having an involution $i$ such that the bicanonical map of $S$ is not composed with $i$ and $S/i$ is not rational. Some examples with $K^2=4,...,7$ are constructed as double coverings of an Enriques surface. These surfaces have a description as bidouble coverings of the plane.

A characterization of Burniat surfaces with $K^{2}=4$ and of non nodal type

Let $S$ be a minimal surface of general type with $p_{g}(S)=0$ and $K^{2}_{S}=4$. Assume the bicanonical map $φ$ of $S$ is a morphism of degree $4$ such that the image of $φ$ is smooth. Then we prove that the surface $S$ is a Burniat surface with $K^{2}=4$ and of non nodal type.

On surfaces of general type with $p_g=q=1, K^2=3$

The moduli space $\mathscr{M}$ of surfaces of general type with $p_g=q=1, K^2=g=3$ (where $g$ is the genus of the Albanese fibration) was constructed by Catanese and Ciliberto in \cite{CaCi93}. In this paper we characterize the subvariety $\mathscr{M}_2 \subset \mathscr{M}$ corresponding to surfaces containing a genus 2 pencil, and moreover we show that there exists a non-empty, dense subset $\mathscr{M}^0 \subset \mathscr{M}$ which parametrizes isomorphism classes of surfaces with birational bicanonical map.

曲率2的曲面上的pencil，包络线以及与平面三次曲线相切的圆锥线

Genus $2$ pencils on surfaces with $p_g=K^2=1$, envelopes, and conics tangent to plane cubic curves

本文研究了具有p_g=K^2=1的曲面中曲率2 pencil的存在性，证明其在模空间中形成不可约子流形，并探讨了与平面三次曲线相切的圆锥线几何关系。

Algebraic Surfaces of General Type with $p_g=q=1$ and Genus 2 Albanese Fibrations

In this paper, we study algebraic surfaces of general type with $p_g=q=1$ and genus 2 Albanese fibrations. We first study the examples of surfaces with $p_g=q=1, K^2=5$ and genus 2 Albanese fibrations constructed by Catanese using singular bidouble covers of $\mathbb{P}^2$. We prove that these surfaces give an irreducible and connected component of $\mathcal{M}_{1,1}^{5,2}$, the Gieseker moduli space of surfaces of general type with $p_g=q=1, K^2=5$ and genus 2 Albanese fibrations. Then by constructing surfaces with $p_g=q=1,K^2=3$ and a genus 2 Albanese fibration such that the number of the s…

关于多项式映射的拓扑分类问题：综述

Around the topological classification problem of polynomial maps: A survey

本综述回顾多项式映射拓扑分类问题的历史与进展，涵盖雅可比猜想及托姆猜想，并聚焦当前分类研究的核心问题与相关仿射几何议题。

On some differential-geometric aspects of the Torelli map

In this note we survey recent results on the extrinsic geometry of the Jacobian locus inside $\mathsf{A}_g$. We describe the second fundamental form of the Torelli map as a multiplication map, recall the relation between totally geodesic subvarieties and Hodge loci and survey various results related to totally geodesic subvarieties and the Jacobian locus.