Related papers: Real forms of some Gizatullin surfaces and Koras-R…
Koras-Russell threefolds are certain smooth contractible complex hypersurfaces in affine complex four-space which are not algebraically isomorphic to affine three-space. One of the important examples is the cubic Russell threefold, defined…
For any positive integer $r$, we construct a smooth complex projective rational surface which has at least $r$ real forms not isomorphic over $\mathbb{R}$.
We present the topological classification of real parts of real regular elliptic surfaces with a real section.
This note is concerned with the zeros of holomorphic Hecke cusp forms of large weight on the modular surface. The zeros of such forms are symmetric about three geodesic segments and we call those zeros that lie on these segments, real. Our…
The aim of this paper is to find all algebraic threefolds admitting quasi-regular Poisson structure. There are three types of such varieties: abelian varieties, smooth flat conic bundles over abelian surfaces and quotients of the product of…
The Koras-Russell threefold is the hypersurface X of the complex affine four-space defined by the equation x^2y+z^2+t^3+x=0. It is well-known that X is smooth contractible and rational but that it is not algebraically isomorphic to affine…
We present a computational study of smooth curves of degree six in the real projective plane. In the Rokhlin-Nikulin classification, there are 56 topological types, refined into 64 rigid isotopy classes. We developed software that…
We classify the topological types of surfaces in the 3-dimensional unit sphere that contain both a great and a small circle through each point. In particular, these surfaces are homeomorphic to one of five normal forms and are either the…
We give a complete deformation classification of real Zariski sextics, that is of generic apparent contours of nonsingular real cubic surfaces. As a by-product, we observe a certain "reversion" duality in the set of deformation classes of…
We survey some results on real rational surfaces focused on their topology and their birational geometry.
We construct a smooth complex projective rational surface with infinitely many mutually non-isomorphic real forms. This gives the first definite answer to a long standing open question if a smooth complex projective rational surface has…
We exhibit a smooth complex rational affine surface with uncountably many nonisomorphic real forms.
The aim of this paper is twofold. First of all, we confirm a few basic criteria of the finiteness of real forms of a given smooth complex projective variety, in terms of the Galois cohomology set of the discrete part of the automorphism…
It is classically known that a real cubic surface in the real projective 3-space cannot have more than one solitary point (locally given by x^2+y^2+z^2=0) whereas it can have up to four nodes (x^2+y^2-z^2=0). We show that on any surface of…
Gizatullin surfaces of type $[[0,0,-r_2,-r_3]]$ can be described by the equations $yu = x P(x)$, $xv = u Q(u)$ and $yv = P(x) Q(u)$ in $\mathbb{C}^4_{x,y,u,v}$ where $P$ and $Q$ are non-constant polynomials. We establish the algebraic…
Totally real surfaces in the nearly K\"ahler $\mathbb{C}P^3$ are investigated and are completely classified under various additional assumptions, resulting in multiple new examples. Among others, the classification includes totally real…
In this paper, we investigate the geometries associated with 3-forms of various orbital types on a symplectic 6-manifold. We show that there are extremely rich geometric structures attached to certain unstable 3-forms arising naturally from…
We construct several rigid (i.e., unique in their deformation class) surfaces which have particular behavior with respect to real structures: in one example the surface has no any real structure, in the other one it has a unique real…
We study a construction, which produces surfaces $Y \subset P_3$ with cusps. For example we obtain surfaces of degree six with 18, 24 or 27 three-divisible cusps. For sextic surfaces in a particular family of up to 30 cusps the codes of…
In this article we fully classify regular tubular surfaces in Euclidean, Lorentzian and hyperbolic 3-spaces whose Gaussian and mean curvatures $K$ and $H$ verify a polynomial relation. More precisely, we determine the set $S(Q)$ of all…