Related papers: The genus of projective curves on complete interse…

We determine the crossing number of polynomial size curve systems on standard surfaces, in terms of the genus, up to high precision.

We give an upper bound for the degree of rational curves in a family that covers a given birational ruled surface in projective space. The upper bound is stated in terms of the degree, sectional genus and arithmetic genus of the surface. We…

We describe each multiple curve on the orientable surface of genus-$g$ with $n$ punctures and one boundary component by using this multiple curve's geometric intersection number with the embedded curves in this surface.

In this paper we describe projective curves and surfaces such that almost all their hyperplane sections are projectively equivalent. Our description is complete for curves and close to being complete for smooth surfaces. In the appendix we…

We obtain a coarse relationship between geometric intersection numbers of curves and the sum of their subsurface projection distances with explicit quasi-constants. By using this relationship, we give applications in the studies of the…

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…

We prove a symmetric version of B\'ezout's theorem. More precisely, we show that the symmetric orbit type of a transverse intersection of complex symmetric hypersurfaces in projective space is determined by the degrees. In the projective…

We consider systems of simple closed curves on surfaces and their total number of intersection points, their so-called crossing number. For a fixed number of curves, we aim to minimise the crossing number. We determine the minimal crossing…

In this work we study neighborhoods of curves in surfaces with positive self-intersection that can be embeeded as a germ of neighborhood of a curve on the projective plane.

In this paper, we compute the number of self-intersections of a plane projection of a generic complete intersection curve defined by polynomials with the given support. Moreover, we discuss the tropical counterpart of this problem.

We give an exponential upper and a quadratic lower bound on the number of pairwise non-isotopic simple closed curves can be placed on a closed surface of genus g such that any two of the curves intersects at most once. Although the gap is…

We define and study analogs of curve graphs for infinite type surfaces. Our definitions use the geometry of a fixed surface and vertices of our graphs are infinite multicurves which are bounded in both a geometric and a topological sense.…

We obtain a sharp bound on the number of self-intersections of a closed planar curve with trigonometric parameterization. Moreover, we show that a generic curve of this form is normal in the sense of Whitney.

We give a sharp bound on the number of automorphisms of a stable curve of a given genus and describe all curves attaining this bound.

We give effective upper bounds for the number of purely inseparable points on non isotrivial curves over function fields of positive characteristic and of transcendence degree one. These bounds depend on the genus of the curve, the genus of…

Given an ordered sequence of $N$-choose-2 integers, we give necessary and sufficient conditions to have an ordered collection of $N$ simple closed curves on a torus such that the algebraic pairwise intersections of those curves are the…

We construct pairs of elliptic curves over number fields with large intersection of projective torsion points.

In the article, we exhibit a series of new examples of rigid plane curves, that is, curves, whose collection of singularities determines them almost uniquely up to a projective transformation of the plane.

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.

We present an algorithm for computing curves and families of curves of prescribed degree and geometric genus on real rational surfaces.