Related papers: Notes on Projective, Contact, and Null Curves
These notes outline some basic notions of Tropical Geometry and survey some of its applications for problems in classical (real and complex) geometry. To appear in the Proceedings of the Madrid ICM.
We use contact geometry to describe the monoid of projectively equivariant meromorphic differential operators on a complex curve, quantization of which generalizes known constructions of classical equivariants to non-commutative function…
This is a revision of some expository lecture notes written originally for a 5-hour minicourse on the intersection theory of punctured holomorphic curves and its applications in 3-dimensional contact topology. The main lectures are aimed…
These notes loosely follow an introductory course on graph complexes, held at Humboldt-Universit\"at zu Berlin in summer 23. Instead of simply typing up my lecture notes I decided to give here an overview over (parts of) the topic (lecture…
The algebras obtained as fixed points of the action of the cyclic group $Z_N$ on the coordinate algebra of the quantum disc are studied. These can be understood as coordinate algebras of quantum or non-commutative cones. The following…
Many classical results in algebraic geometry arise from investigating some extremal behaviors that appear among projective varieties not lying on any hypersurface of fixed degree. We study two numerical invariants attached to such…
Classification theory and the study of projective varieties which are covered by rational curves of minimal degrees naturally leads to the study of families of singular rational curves. Since families of arbitrarily singular curves are hard…
These notes are an expanded version of the author's lectures at the graduate workshop "Noncommutative Algebraic Geometry" at the Mathematical Sciences Research Institute in June 2012. The main topics discussed are Artin-Schelter regular…
Classical contact Lie algebras are the fundamental algebraic structures on the manifolds of contact elements of configuration spaces in classical mechanics. In this paper, we determine the structure of the currently largest known category…
This note is devoted to a trick which yields almost trivial proofs that certain complexes associated to topological surfaces are connected or simply connected. Applications include new proofs that the complexes of curves, separating curves,…
The minimal surface equation $Q$ in the second order contact bundle of $R^3$, modulo translations, is provided with a complex structure and a canonical vector-valued holomorphic differential form $Omega$ on $Q\0$. The minimal surfaces $M$…
In this paper are studied the simplest qualitative properties of asymptotic lines of a plane field in Euclidean space. These lines are the integral curves of the null directions of the normal curvature of the plane field, on the closure of…
The authors study smooth lines on projective planes over the algebra C of complex numbers, the algebra C^1 of double numbers, and the algebra C^0 of dual numbers. In the space RP^5, to these smooth lines there correspond families of…
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…
In this paper we give Weierstrass-type representation formulas for the null curves and for the minimal Lorentz surfaces in the Minkowski 3-space $\mathbb R^3_1$ using real-valued functions. Applying the Weierstrass-type representations for…
This paper is the text of a lecture given at the Catania conference on Commutative Algebra and Algebraic Geometry, in honor of the 60th birthday of Silvio Greco. It analyses the correspondences between equivalence classes of objects related…
In this (mostly) survey article, we give a synopsis of a number of results relating to Brill--Noether theory on curves and metric graphs, together with some speculations about the behavior of one-dimensional linear series on a class of…
We study the space of real rational curves of low degree in the quadric of signature $(3,2)$ and provides a classificaton of real rational knots and nodal curves. Apart from the classification, we also study the relationship between the…
We make explicit the equations of any projective $PGL_2(\C)$-variety defined by quadrics. We study their zero-locus and their relationship with the geometry of the Veronese curve.
We study holomorphic curves $f:\C\longrightarrow \C^3$ avoiding four complex hyperplanes and a real subspace of real dimension four or five in $\C^3$. We show that the projection of $f$ into the complex projective space $\C P^2$ is not…