Related papers: On curves and surfaces with projectively equivalen…
This paper gives an explicit formula for the Ehrhart quasi-polynomial of certain 2-dimensional polyhedra in terms of invariants of surface quotient singularities. Also, a formula for the dimension of the space of quasi-homogeneous…
We address the problem of the maximal finite number of real points of a real algebraic curve (of a given degree and, sometimes, genus) in the projective plane. We improve the known upper and lower bounds and construct close to optimal…
The main goal of this work is to prove that every entire curve in a smooth hypersurface of degree greater than or equal to 97 in the complex projective space of dimension 4 must satisfy an algebraic differential equation of order 3. A…
The Legendre curve in the unit tangent bundle over Euclidean plane is a plane curve with a moving frame. We have the (Legendre) curvature of the Legendre curve, and the existence and uniqueness theorems for the curvature are valid. In this…
Based on the fact that projective monomial curves in the plane are complete intersections, we give an effective inductive method for creating infinitely many monomial curves in the projective $n$-space that are set theoretic complete…
Motivated by the analogies between the projective and the almost quaternionic geometries, we study the generalized planar curves and mappings. We follow, recover, and extend the classical approach as developed by Mikes and Sinyukov. Then we…
Surfaces and curves play an important role in geometric design. In recent years, problem of finding a surface passing through a given curve have attracted much interest. In the present paper, we propose a new method to construct a surface…
In this work we characterize branch data of branched coverings of even degree over the projective plane which are realizable by indecomposable branched coverings.
The main theorem of this paper is a result of estimated transversality with respect to stratifications of jet spaces in the approximately holomorphic category over an almost-complex manifold. The notion of asymptotic ampleness of complex…
We give necessary conditions on the invariants (d,g) of a smooth, integral curve self-linked by a complete intersection of type (a,b) in projective three space. Similar conditions are given for s.t.c.i. curves with a multiplicity three…
We describe the behaviour of a free reduced plane projective curve with respect to the deletion, respectively addition, of a smooth conic. These results apply in particular to conic-line arrangements. We present some obstructions to the…
We obtain a complete list of smooth projective threefolds over $\mathbb C$ for which the dimension of the space of vanishing cycles (in $H^2$ of the smooth hyperplane section) equals $2$. We also obtain a complete list of rank 2 very ample…
Lasell and Ramachandran show that the existence of rational curves of positive self-intersection on a smooth projective surface $X$ implies that all the finite dimensional linear representations of the fundamental group $\pi_1(X)$ are…
We find the first examples of real hypersurfaces with two nonconstant principal curvatures in complex projective and hyperbolic planes, and we classify them. It turns out that each such hypersurface is foliated by equidistant Lagrangian…
A projectability result is proved for surfaces of prescribed mean curvature (shortly called $H$-surfaces) spanned in a partially free boundary configuration. Hereby, the $H$-surface is allowed to meet the support surface along its free…
We determine non-Hopf hypersurfaces with constant mean curvature in the complex projective plane which attain equality in a basic inequality between the maximum Ricci curvature and the squared mean curvature.
Using a recent description of the geometric stability manifold, we show the geometric stability manifold associated to any smooth projective complex surface is contractible. We then use this result to demonstrate infinitely many new…
We find upper bounds, sharp in most cases, on the number of real hyperplane sections of real smooth polarized $K3$-surfaces that split into lines. Most bounds coincide with their complex counterparts.
We prove that the variety of flexes of algebraic curves of degree $3$ in the projective plane is an ideal theoretic complete intersection in the product of a two-dimensional and a nine-dimensional projective spaces.
Ballico proved that a smooth projective variety $X$ of degree $d$ over a finite field of $q$ elements admits a smooth hyperplane section if $q\geq d(d-1)^{\dim X}$. In this paper, we refine this criterion for higher codimensional linear…