Let $V$ be a complex algebraic variety, homogeneous under the action of a complex algebraic group. We show that the log Kodaira dimension of $V$ is non-negative if and only if $V$ is a semi-abelian variety.

We determine the Newton trees of the rational polynomials of simple type, thus filling a gap in the proof of the classification of these polynomials given by Neumann and Norbury.

We study certain pencils $\overline{f}:\mathbb{P}\dashrightarrow\mathbb{P}^{1}$ of del Pezzo surfaces generated by a smooth del Pezzo surface $S$ of degree less than or equal to 3 anti-canonically embedded into a weighted projective space $\mathbb{P}$ and an appropriate multiple of a hyperplane $H$. Our main observation is that every minimal model program relative to the morphism $\tilde{f}:\tilde{\mathbb{P}}\rightarrow\mathbb{P}^{1}$ lifting $\overline{f}$ on a suitable resolution $\sigma:\tilde{\mathbb{P}}\rightarrow\mathbb{P}$ of its indeterminacies preserves the open subset $\sigma^{-1}(\mathbb{P}\setminus H)\simeq\mathbb{A}^{3}$. As an application, we obtain projective completions of $\mathbb{A}^{3}$ into del Pezzo fibrations over $\mathbb{P}^{1}$ of every degree less than or equal to 4. We also obtain completions of $\mathbb{A}^{3}$ into Mori conic bundles, whose restrictions to $\mathbb{A}^{3}$ are twisted $\mathbb{A}_{*}^{1}$-fibrations over $\mathbb{A}^{2}$.

We discuss the (twisted) weak positivity theorem. We also treat some applications.

The correspondence between $G_a$-actions on affine varieties and locally nilpotent derivations of the coordinate algebras is generalized in the projective case to the correspondence between stratified $G_a$-actions on smooth projective varieties $V$ and regular vector fields on $V$ which are effectively locally nilpotent with stratification. These notions with stratifications are inspired by explicit computations of $G_a$-actions on the projective space $\mathbb{P}^n$ as well as the Hirzebruch surface $\mathbb{F}_n$ and the associated regular vector fields. Using partly these observations, we investigate the existence of $\mathbb{A}^1$-cylinders in Fano threefolds with rank one.

We study cusp singularities from the viewpoint of locally polyhedral sets and reflection groups. Following the definition of quasi-polyhedral sets by Grünbaum, we consider a special kind of rational quasi-polyhedral sets with group action and refer to the relation with cusp singularities. We give also some examples of such quasi-polyhedral sets by using discrete groups generated by reflections.

We extract the Abhyankar-Moh-Suzuki theorem from the Lin-Zaidenberg theorem.

We study irrational open algebraic surfaces of non-negative logarithmic Kodaira dimension in any characteristic. We give a structure theorem for the irrational open surfaces of logarithmic Kodaira dimension zero. Then, by using this result and the results in [7], we prove that, for an irrational ruled open surface, its logarithmic Kodaira dimension is non-negative if and only if its logarithmic bigenus is positive.

This is a survey on the automorphism groups in various classes of affine algebraic surfaces and the algebraic group actions on such surfaces. Being infinite-dimensional, these automorphism groups share some important features of algebraic groups. At the same time, they can be studied from the viewpoint of the combinatorial group theory, so we put a special accent on group-theoretical aspects (ind-groups, amalgams, etc.). We provide different approaches to classification, prove certain new results, and attract attention to several open problems.

As a variant of Shokurov's criterion of toric surface, we give a criterion of two new classes of normal projective surfaces, called pseudo-toric surfaces of defect one and half-toric surfaces. A typical example of pseudo-toric surface of defect one is a projective toric surface blown up at a non-singular point of the boundary divisor. A half-toric surface is the quotient of a projective toric surface by an almost free involution preserving the boundary divisor. The structure of pseudo-toric surface of defect one and that of half-toric surface are also studied in detail.

In this paper we give first examples of $\mathbb{Q}$-Fano threefolds whose birational Mori fiber structures consist of exactly three $\mathbb{Q}$-Fano threefolds. These examples are constructed as weighted hypersurfaces in a specific weighted projective space. We also observe that the number of birational Mori fiber structures does not behave upper semi-continuously in a family of $\mathbb{Q}$-Fano threefolds.

Exploring Bass' Triangulability Problem on unipotent algebraic subgroups of the affine Cremona groups, we prove a triangulability criterion, the existence of nontriangulable connected solvable affine algebraic subgroups of the Cremona groups, and stable triangulability of such subgroups; in particular, in the stable range we answer Bass' Triangulability Problem in the affirmative. To this end we prove a theorem on invariant subfields of 1-extensions. We also obtain a general construction of all rationally triangulable subgroups of the Cremona groups and, as an application, classify rationally triangulable connected one-dimensional unipotent affine algebraic subgroups of the Cremona groups up to conjugacy.

We produce new families of smooth Fano fourfolds with Picard rank 1, which contain cylinders, i.e., Zariski open subsets of form $Z\times{\mathbb A}^1$, where $Z$ is a quasiprojective variety. The affine cones over such a fourfold admit effective $\mathbb{G}_{\operatorname{a}}$-actions. Similar constructions of cylindrical Fano threefolds and fourfolds were done previously in [KPZ11, KPZ14, PZ16].

Many types of automorphism groups in algebra have nice structures arising from actions on combinatoric spaces. We recount some examples including Nagao's Theorem, the Jung-Van der Kulk Theorem, and a new structure theorem for the tame subgroup $\text{TA}_3(K)$ of the group $\text{GA}_3(K)$ of polynomial automorphisms of $\mathbb{A}_K^3$, for $K$ a field of characteristic zero. We also ask whether a larger collection of automorphism groups possess a similar kind of structure.