Surfaces that are covered by two families of circles

We list up to M\"obius equivalence all possible degrees and embedding dimensions of real surfaces that are covered by at least two pencils of circles, together with the number of such pencils. In addition, we classify incidences between the contained circles, complex lines and isolated singularities. Such geometric characteristics are encoded in the Neron-Severi lattices of such surfaces and is of potential interest to geometric modelers and architects. As an application we confirm Blum's conjecture in higher dimensional space and we address the Blaschke-Bol problem by classifying surfaces that are covered by hexagonal webs of circles. In particular, we find new examples of such webs that cannot be embedded in 3-dimensional space

Algorithms for singularities and real structures of weak Del Pezzo surfaces

In this paper we consider the classification of singularities (Du Val) and real structures (Wall) of weak Del Pezzo surfaces from an algorithmic point of view. It is well known that the singularities of weak Del Pezzo surfaces correspond to root subsystems. We present an algorithm which computes the classification of these root subsystems. We represent equivalence classes of root subsystems by unique labels. These labels allow us to construct examples of weak Del Pezzo surfaces with the corresponding singularity configuration. Equivalence classes of real structures of weak Del Pezzo surfaces are also represented by root subsystems. We present an algorithm which computes the classification of real structures. This leads to an alternative proof of the known classification for Del Pezzo surfaces and extends this classification to singular weak Del Pezzo surfaces. As an application we classify families of real conics on cyclides.Comment: Journal of Algebra and its Applications, World Scientific, 201

Families of bitangent planes of space curves and minimal non-fibration families

We define a cone curve to be a reduced sextic space curve which lies on a quadric cone and does not go through the vertex. We classify families of bitangent planes of cone curves. The methods we apply can be used for any space curve with ADE singularities, though in this paper we concentrate on cone curves. An embedded complex projective surface which is adjoint to a degree one weak Del Pezzo surface contains families of minimal degree rational curves, which cannot be defined by the fibers of a map. Such families are called minimal non-fibration families. Families of bitangent planes of cone curves correspond to minimal non-fibration families. The main motivation of this paper is to classify minimal non-fibration families. We present algorithms wich compute all bitangent families of a given cone curve and their geometric genera. We consider cone curves to be equivalent if they have the same singularity configuration. For each equivalence class of cone curves we determine the possible number of bitangent families and the number of rational bitangent families. Finally we compute an example of a minimal non-fibration family on an embedded weak degree one Del Pezzo surface

Euclidean sums and Hamiltonian products of circles in the 3-sphere

We classify the singular loci of surfaces in the 3-sphere that are the pointwise Euclidean sum or Hamiltonian product of circles. Such surfaces are the union of circles in at least two ways. As an application we classify surfaces that are covered by both great circles and little circles up to homeomorphism

Minimal families of curves on surfaces

A minimal family of curves on an embedded surface is defined as a 1-dimensional family of rational curves of minimal degree, which cover the surface. We classify such minimal families using constructive methods. This allows us to compute the minimal families of a given surface. The classification of minimal families of curves can be reduced to the classification of minimal families which cover weak Del Pezzo surfaces. We classify the minimal families of weak Del Pezzo surfaces and present a table with the number of minimal families of each weak Del Pezzo surface up to Weyl equivalence. As an application of this classification we generalize some results of Schicho. We classify algebraic surfaces which carry a family of conics. We determine the minimal lexicographic degree for the parametrization of a surface which carries at least 2 minimal families

M\"obius automorphisms of surfaces with many circles

We classify real two-dimensional orbits of conformal subgroups such that the orbits contain two circular arcs through a point. Such surfaces must be toric and admit a M\"obius automorphism group of dimension at least two. Our theorem generalizes the classical classification of Dupin cyclides

A degree bound for families of rational curves on surfaces

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 introduce an algorithm for constructing examples where the upper bound is tight. As an application of our methods we improve an inequality on lattice polygons

Computing basepoints of linear series in the plane

We present an algorithm for detecting basepoints of linear series of curves in the plane. Moreover, we give an algorithm for constructing a linear series of curves in the plane for given basepoints. The underlying method of these algorithms is the classical procedure of blowing up points in the plane. We motivate the algorithmic version of this procedure with several applications

Webs of rational curves on real surfaces and a classification of real weak del Pezzo surfaces

We classify webs of minimal degree rational curves on surfaces and give a criterion for webs being hexagonal. In addition, we classify Neron-Severi lattices of real weak del Pezzo surfaces. These two classifications are related to root subsystems of E8

Kinematic generation of Darboux cyclides

We state a relation between two families of lines that cover a quadric surface in the Study quadric and two families of circles that cover a Darboux cyclide