Related papers: Geometry on ruled surfaces with finite multiplicit…

The generating functions of the Severi degrees for sufficiently ample line bundles on algebraic surfaces are multiplicative in the topological invariants of the surface and the line bundle. Recently new proofs of this fact were given for…

We study linearizability of actions of finite groups on cubic threefolds with nonnodal isolated singularities.

We give a sharp lower bound on the area of the domain enclosed by an embedded curve lying on a two-dimensional sphere, provided that geodesic curvature of this curve is bounded from below. Furthermore, we prove some dual inequalities for…

These lectures give a short introduction to the study of curves on algebraic varieties. After an elementary proof of the dimension formula for the space of curves, we summarize the basic properties of uniruled and of rationally connected…

In \cite{X-Z DCS1}, we introduced discrete conformal structures on surfaces with boundary via an axiomatic framework, and provided a classification of such discrete conformal structures. The present work focuses on the rigidity and…

We characterise the form of all simple, finite graphs for which the girth of the graph is equal to the circumference of the graph. We apply this to prove a bound on the number of edges in such a graph.

We give a bound, linear in the complexity of the surface, on the asymptotic dimension of the curve complex as well as the capacity dimension of the ending lamination space.

We generalise surface cluster algebras to the case of infinite surfaces where the surface contains finitely many accumulation points of boundary marked points. To connect different triangulations of an infinite surface, we consider infinite…

We study ruled surfaces in R3 which are obtained from dual spher- ical indicatrix curves of dual Frenet vector fields. We find the Gaussian and mean curvatures of the ruled surfaces and give some results of being Wein- garten surface.

We study filling sets of simple closed curves on punctured surfaces. In particular we study lower bounds on the cardinality of sets of curves that fill and that pairwise intersect at most k times on surfaces with given genus and number of…

In terms of the number of triangles, it is known that there are more than exponentially many triangulations of surfaces, but only exponentially many triangulations of surfaces with bounded genus. In this paper we provide a first geometric…

In this paper we study nodal deformations of singular surfaces $S\subset \mathbb P^3$. In particular we consider the case in which $S$ has an isolated singularity of multiplicity $m$ and the case in which $S$ has only ordinary singularities…

We bound the genus of a projective curve lying on a complete intersection surface in terms of its degree and the degrees of the defining equations of the surface on which it lies.

The set of volumes of stable surfaces does have accumulation points. In this paper, we study this phenomenon for surfaces with one cyclic quotient singularity, towards answering the question under which conditions we can still have…

We consider singularities of frontal surfaces of corank one and finite frontal codimension. We look at the classification under left-right-equivalence and introduce the notion of frontalisation for singularities of fold type. We define the…

We make a systematic study of the focal surface of a congruence of lines in the projective space. Using differential techniques together with techniques from intersection theory, we reobtain in particular all the invariants of the focal…

We prove that rigid representations of the fundamental group of a surface into the group of oreintation-preserving homeomorphisms of the circle are geometric, thereby establishing a converse statement of a theorem by the first author.

In this paper are studied the nets of principal curvature lines on surfaces embedded in Euclidean $3-$space near their end points, at which the surfaces tend to infinity. This is a natural complement and extension to smooth surfaces of the…

We study the topological configurations of the lines of principal curvature, the asymptotic and characteristic curves on a cuspidal edge, in the domain of a parametrization of this surface as well as on the surface itself. Such…

This paper studies properties of tilings of the plane by parallelograms. In particular it is established that in parallelogram tilings using a finite number of shapes all tiles occur in only finitely many orientations.