In this article, we prove that a $\mathbb{Q}$-homology plane $X$ with two algebraically independent $G_a$-actions is isomorphic to either the affine plane or a quotient of an affine hypersurface $xy=z^m-1$ in the affine $3$-space via a free $\mathbb{Z}/m\mathbb{Z}$-action, where $m$ is the order of a finite group $H_1(X;\mathbb{Z})$.

We construct del Pezzo surfaces of degree $4$ violating the Hasse principle explained by the Brauer-Manin obstruction. Using these del Pezzo surfaces, we show that there are algebraic families of $K3$ surfaces violating the Hasse principle explained by the Brauer-Manin obstruction. Various examples are given.

Let S be a ruled surface in P3 with no multiple generators. Let d and q be nonnegative integers. In this paper we determine which pairs (d,q) correspond to the degree and irregularity of a ruled surface, by considering these surfaces as curves in a smooth quadric hypersurface in P5.

We describe the set of points over which a dominant polynomial map $f=(f_1,...,f_n):{ℂ}^n\to {ℂ}^n$ is not a local analytic covering. We show that this set is either empty or it is a uniruled hypersurface of degree bounded by $({\prod }_{i=1}^{n}deg{f}_i-\mu \left(f\right))/\left(mi{n}_{i=1,...,n}deg{f}_i\right)$.

Let $X$ be any rational surface. We construct a tilting bundle $T$ on $X$. Moreover, we can choose $T$ in such way that its endomorphism algebra is quasi-hereditary. In particular, the bounded derived category of coherent sheaves on $X$ is equivalent to the bounded derived category of finitely generated modules over a finite dimensional quasi-hereditary algebra $A$. The construction starts with a full exceptional sequence of line bundles on $X$ and uses universal extensions. If $X$ is any smooth projective variety...

The theory of slice-regular functions of a quaternion variable is applied to the study of orthogonal complex structures on domains $\Omega$ of ${ℝ}^4$. When $\Omega$ is a symmetric slice domain, the twistor transform of such a function is a holomorphic curve in the Klein quadric. The case in which $\Omega$ is the complement of a parabola is studied in detail and described by a rational quartic surface in the twistor space $ℂP^3$.