19 research outputs found
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