Related papers: Rational cuspidal curves with four cusps on Hirzeb…
We list up all the candidates for the real isotopy types of real anti-bicanonical curves with one real nondegenerate double point on the 4-th real Hirzebruch surface RF_4 by enumerating the connected components of the moduli space of real…
The classical Kummer construction attaches to an abelian surface a K3 surface. As Shioda and Katsura showed, this construction breaks down for supersingular abelian surfaces in characteristic two. Replacing supersingular abelian surfaces by…
We complete the topological classification of real algebraic non-singular curves of bidegree $(5, 5)$ on the quadric ellipsoid. We show in particular that previously known restrictions form a complete system for this bidegree. Therefore,…
We generalize results by Wakabayashi and Orevkov about rational cuspidal curves on the projective plane to that on $\mathbb{Q}$-homology projective planes. It turns out that the result is exactly the same as the projective plane case under…
This paper deals with rational curves and birational contractions on irreducible holomorphically symplectic manifold. We survey some recent results about minimal rational curves, their deformations, extremal rays associated with these…
We realize every closed flat 3-manifold as a cusp section of a complete, finite-volume hyperbolic 4-manifold whose symmetry group acts transitively on the set of cusps. Moreover, for every such 3-manifold, a dense subset of its flat metrics…
Building on work of Segre and Koll'ar on cubic hypersurfaces, we construct over imperfect fields of characteristic p\geq 3 particular hypersurfaces of degree p, which show that geometrically rational schemes that are regular and whose…
We prove that any two irreducible cuspidal Hurwitz curves $C_0$ and $C_1$ (or more generally, curves with A-type singularities) in the Hirzebruch surface $F_N$ with coinciding homology classes and sets of singularities are regular…
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…
Building on recent work of Bhargava--Elkies--Schnidman and Kriz--Li, we produce infinitely many smooth cubic surfaces defined over the field of rational numbers that contain rational points.
In the open problem of classification of rational cuspidal plane curves it is essential to find good necessary conditions on the type of singularities of a curve C in order C to exit. Motivated by the study of the Seiberg-Witten invariant…
This paper proposes a simple technique of curve and surface construction with B-splines. Given a control polygon or a control mesh together with node ordinates corresponding to all control points, a rational curve or surface is obtained by…
We classify special self-birational transformations of the smooth quadric threefold and fourfold, $Q^3$ and $Q^4$. It turns out that there is only one such example in each dimension. In the case of $Q^3$, it is given by the linear system of…
A `trinomial hyper surface' is defined in \S 1 below. In this article, I provide a supportive reasoning towards the fact that there can be examples of trinomial hyper surfaces (at least over fields of characteristic 2) for which the…
Consider the scheme parametrizing non-constant morphisms from a fixed projective curve to a projective surface. There is a rational map between this scheme and the Chow variety of $1$-cycles on the surface. We prove that, if the curve is…
We present a method for computing projective isomorphisms between rational surfaces that are given in terms of their parametrizations. The main idea is to reduce the computation of such projective isomorphisms to five base cases by…
In this paper, we deal with plane curves with cusps. It is well known that there are various types of cusps. Among them, we investigate criteria for $(n, n+1)$ cusps with respect to several differential conditions and relations between…
We show a counterexample to a conjecture of de Bobadilla, Luengo, Melle-Hern\'{a}ndez and N\'{e}methi on rational cuspidal projective plane curves. The counterexample is a tricuspidal curve of degree 8. On the other hand, we show that if…
In this survey we discuss the problem of the existence of rational curves on complex surfaces, both in the K\"ahler and non-K\"ahler setup. We systematically go through the Enriques--Kodaira classification of complex surfaces to highlight…
We study the moduli spaces of rational curves on cubic hypersurfaces in characteristic $\neq2,3$. As a result, we prove that for every integer $d\geq1$ the Kontsevich moduli space of stable maps on a smooth cubic hypersurface $X$ of degree…