We survey some old and new results about finite subgroups of the Cremona group $\mathrm{Cr}_2 (k)$ of birational automorphisms of the projective plane over a field $k$.

This survey paper concerns elliptic surfaces with section. We give a detailed overview of the theory including many examples. Emphasis is placed on rational elliptic surfaces and elliptic K3 surfaces. To this end, we particularly review the theory of Mordell–Weil lattices and address arithmetic questions.

This article is the expanded version of a talk given at the conference: Algebraic geometry in East Asia 2008. In this notes, I intend to give a brief survey of results on the behavior of semi-stable bundles under the Frobenius pullback and direct images. Some results are new.

Let $S$ be a complex minimal nonsingular projective irregular surface of general type with $K_S^2 \leq 4_{\mathcal{X}} (\mathcal{O_S})$ and ${}_\mathcal{X} (\mathcal{O_S}) \gt 12$. Then the group of automorphisms of $S$ acts faithfully on the cohomology ring $H^* (S, \mathbb{Q})$ with the exceptional case that $S$ is as in [Ca3, Theorem 2.5].

Let $X$ be a Fano variety with at worst isolated quotient singularities. Our result asserts that if $C \cdot (-K_X) \gt max \{\frac{n}{2} + 1, \frac{2n}{3}\}$ for every curve $C \subset X$, then $\rho_X = 1$.

We extend Kollár's technique to look for an explicit function $h(n)$ with $\varphi_m$ birational onto its image for all integers $m \geq h(n)$ and for all $n$-dimensional nonsingular projective varieties of general type.

For $X$ a complete, reduced, geometrically connected scheme over a perfect field of characteristic $p \gt 0$, we analyze the decomposition of Nori's fundamental group scheme into its local and étale parts and raise the question of the relation between the geometry and the splitting of the group scheme. We also describe in categorial terms the functor which corresponds to the inclusion of the maximal reduced subgroup scheme.

We prove that a smooth projective complex surface $X$, not necessarily minimal, contains $h^{1,1} (X) - 1$ disjoint $(-2)$-curves if and only if $X$ is isomorphic to a relatively minimal ruled rational surface $\mathbf{F}_2$ or $\mathbf{P}^2$ or a fake projective plane.

We also describe smooth projective complex surfaces $X$ with $h^{1,1} (X) - 2$ disjoint $(-2)$-curves.

We show that there is a complex structure on the symplectic 4-manifold $W_{4,k}$ obtained from the elliptic surface E(4) by rationally blowing down $k$ sections for $2 \leq k \leq 9$. And we interpret it via $\mathbb{Q}$-Gorenstein smoothing. This answers affirmatively to a question raised by R. Gompf.

In the joint paper [8] with Y.-P. Lee and C.-L. Wang, we have shown that the big quantum ring is invariant under $\mathbb{P}^r$ flops of splitting type, after an analytic continuation over the extended Kähler moduli space. It is a generalization of our previous work for the case of simple $\mathbb{P}^r$ flops [7]. In this note, I would like to outline the results and concentrate mainly on the detailed study of a simple type, called $\mathbb{P}^1$ flops of type $(k + 2, k)$.

The structure of non-singular projective surfaces admitting nonisomorphic surjective separable endomorphisms is studied in the positive characteristic case. The case of characteristic zero is treated in [2], [16] (cf. [3]). Many similar classification results are obtained also in this case; on the other hand, some examples peculiar to the positive characteristic are given explicitly.

We realize the logarithm of the third smallest known Salem number as the topological entropy of a K3 surface automorphism with a Siegel disk and a pointwise fixed curve at the same time. We also show that the logarithm of the Lehmer number, the smallest known Salem number, is not realizable as the topological entropy of any Enriques surface automorphism. These results are entirely inspired by McMullen's works and Mathematica programs.

Let $S$ be a smooth simply-connected complex projective surface, and let $A$ be a finite abelian group. We define invariants $T_A$, $F_A$ and $\sigma_A$ for curves $B$ on $S$ by means of étale Galois coverings of the complement of $B$ with the Galois group $A$, and show that they are useful in finding examples of Zariski pairs of curves on $S$. We also investigate the relation between these invariants and the fundamental group of the complement of $B$.