Related papers: Real Rational Surfaces Are Quasi-Simple
In this paper, we give a complete description of the deformation classes of real structures on minimal ruled surfaces. In particular, we show that these classes are determined by the topology of the real structure, which means that real…
We survey some results on real rational surfaces focused on their topology and their birational geometry.
We give necessary and sufficient topological conditions for a simple closed curve on a real rational surface to be approximable by smooth rational curves. We also study approximation by smooth rational curves with given complex…
Smooth real cubic surfaces are birationally trivial (over $\R$) if and only if their real locus is connected or, equivalently, if and only if they have two skew real lines or two skew complex conjugate lines. In such a case a…
By the famous ADE classification rational double points are simple. Rational triple points are also simple. We conjecture that the simple normal surface singularities are exactly those rational singularities, whose resolution graph can be…
For any positive integer $r$, we construct a smooth complex projective rational surface which has at least $r$ real forms not isomorphic over $\mathbb{R}$.
Segre proved that a smooth cubic surface over Q is unirational iff it has a rational point. We prove that the result also holds for cubic hypersurfaces over any field, including finite fields.
We classify $G$-solid rational surfaces over the field of complex numbers.
It is proved that a smooth rational surface in projective four-space, which is ruled by cubics or quartics has degree at most 12. It is also proved that a smooth rational surface in projective four-space which is the image of Fn by a linear…
We prove analogues of several well-known results concerning rational morphisms between quadrics for the class of so-called quasilinear $p$-hypersurfaces. These hypersurfaces are nowhere smooth over the base field, so many of the geometric…
This article proves hypersurfaces of degree d in projective n-space are "rationally simply-connected" if $d^2 \leq n$. In a forthcoming paper, de Jong and I prove a slightly weaker result when $d^2 \leq n+1$.
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 give a complete equisingular deformation classification of simple spatial quartic surfaces which are in fact $K3$-surfaces.
This is an expository paper which presents the holomorphic classification of rational complex surfaces from a simple and intuitive point of view, which is not found in the literature. Our approach is to compare this classification with the…
It is constructed a normal form for a class of real-smooth surfaces M\subset\mathbb{C}^{2} defined near a degenerate CR singularity.
Among geometrically rational surfaces, del Pezzo surfaces of degree two over a field k containing at least one point are arguably the simplest that are not known to be unirational over k. Looking for k-rational curves on these surfaces, we…
The ruled surface is a typical modeling surface in computer aided geometric design. It is usually given in the standard parametric form. However, it can also be in the forms than the standard one. For these forms, it is necessary to…
We exhibit a smooth complex rational affine surface with uncountably many nonisomorphic real forms.
If a smooth, geometrically rational surface over a finite field is not rational over that field, then over some finite extension of that field the Brauer group of the surface is nonzero. In particular such a surface is not stably rational.…
The ruled surfaces, i.e., surfaces generated by one parametric set of lines, are widely used in the~field of applied geometry. An~isophote on a surface is a curve consisting of surface points whose normals form a constant angle with some…