Related papers: Triple lines on a cubic threefold
We prove the sharp upper bound of at most $52$ lines on a complex K3-surface of degree four with a non-empty singular locus. We also classify the configurations of more than $48$ lines on smooth complex quartics.
We classify smooth Fano threefolds that admit degenerations to toric Fano threefolds with ordinary double points.
In a series of two articles Kebekus studied deformation theory of minimal rational curves on contact Fano manifolds. Such curves are called contact lines. Kebekus proved that a contact line through a general point is necessarily smooth and…
We describe the possible 3-divisible $A_2^n$ configurations of smooth rational curves on K3 surfaces in characteristic 3 and fully classify the resulting triple covers.
We study those Lagrangian surfaces in complex Euclidean space which are foliated by circles or by straight lines. The former, which we call cyclic, come in three types, each one being described by means of, respectively, a planar curve, a…
We prove that the space of pairs $(X,l)$ formed by a real non-singular cubic hypersurface $X\subset P^4$ with a real line $l\subset X$ has 18 connected components and give for them several quite explicit interpretations. The first one…
We study the tropical lines contained in smooth tropical surfaces in R^3. On smooth tropical quadric surfaces we find two one-dimensional families of tropical lines, like in classical algebraic geometry. Unlike the classical case, however,…
Cubic surfaces in characteristic two are investigated from the point of view of prime characteristic commutative algebra. In particular, we prove that, the non-Frobenius split cubic surfaces form a linear subspace of codimension four in the…
We address the following question: When an affine cone over a smooth Fano threefold admits an effective action of the additive group? In this paper we deal with Fano threefolds of index 1 and Picard number 1. Our approach is based on a…
We study the geometry and codes of quartic surfaces with many cusps. We apply Gr\"obner bases to find examples of various configurations of cusps on quartics.
We classify smooth Fano threefolds with infinite automorphism groups.
In this paper we prove that any smooth prime Fano threefold, different from the Mukai-Umemura threefold, contains a 1-dimensional family of intersecting lines. Combined with a result of the second author (see J. Algebr. Geom. 8:2 (1999),…
The topological structure of the lines of principal curvature, the umbilic and partially umbilic singularities of all tridimensional ellipsoids of ${\mathbb R}^4$ is described.
This note aims to improve known numerical bounds proved earlier by Chen \cite{PAMS} and Chen-Hacon \cite{Chen-Hacon} and to present some new examples of smooth minimal 3-folds canonically fibred by surfaces (resp. curves) of geometric genus…
We prove that the enumerative geometry of lines on smooth cubic surfaces is governed by the arithmetic of the base field. In 1949, Segre proved that the number of lines on a smooth cubic surface over any field is 0, 1, 2, 3, 5, 7, 9, 15, or…
In the paper, we investigate properties of the nine-dimensional variety of the inflection points of the plane cubic curves. The description of local monodromy groups of the set of inflection points near singular cubic curves is given. Also,…
We study an irreducible component H(X) of the Hilbert scheme Hilb^{2t+2}(X) of a smooth cubic hypersurface X containing two disjoint lines. For cubic threefolds, H(X) is always smooth, as shown in arXiv:2010.11622. We provide a second proof…
A complex contact threefold is a threefold with a two-dimensional non-integrable holomorphic distribution. A contact curve on a contact threefold is an integrable curve of the distribution. This work was inspired by two papers of Bryant, in…
We study double line structures in projective spaces and quadric hypersurfaces, and investigate the geometry of irreducible components of Hilbert scheme of curves and moduli of stable sheaves of pure dimension 1 on a smooth quadric…
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…