Related papers: Real algebraic curves, the moment map and amoebas
We produce examples of complex algebraic surfaces with isolated singularities such that these singularities are not metrically conic, i.e. the germs of the surfaces near singular points are not bi-Lipschitz equivalent, with respect to the…
We investigate the geometry of holomorphic curves and complex surfaces from the perspective of singularity theory. We show that, with a suitable choice of a complex bilinear symmetric form, the families of functions and mappings that…
We give a constructive proof using tropical modifications of the existence of a family of real algebraic plane curves with asymptotically maximal numbers of even ovals.
We prove a correspondence theorem for singular tropical surfaces in real three space, which recovers singular algebraic surfaces in an appropriate toric three-fold that tropicalize to a given singular tropical surface. Furthermore, we…
We find a sharp bound for the order of the automorphism group of a stable curve of genus $g$ with $3g-3$ nodes, and a sharp bound for the order of the automorphism group of such a curve with all smooth components. Combined with the results…
Connected components of real algebraic sets are semi-algebraic, i.e. they are described by a boolean formula whose atoms are polynomial constraints with real coefficients. Computing such descriptions finds topical applications in optical…
Amoebas are projections of complex algebraic varieties in the algebraic torus under a Log-absolute value map, which have connections to various mathematical subjects. While amoebas of hypersurfaces have been intensively studied in recent…
The canonical projections of the unit spheres are generalized to special generic maps and round fold maps, for example. They are generalizations from the viewpoint of singularity theory of differentiable maps and these maps restrict the…
We use Morse theoretical arguments to study algebraic curves in C^2. We take an algebraic curve C in C^2 and intersect it with a family of spheres with fixed origin and varying radii. We explain in detail how does the resulting link change…
The aim of this paper is to provide a direct link between maximizing curves that occur in the construction of smooth algebraic surfaces having the maximal possible Picard numbers and reduced free plane curves with simple singularities. We…
We will use toric degenerations of the projective plane ${{\mathbb{P}}^ 2}$ to give a new proof of the triple points interpolation problems in the projective plane. We also give a complete list of toric surfaces that are useful as…
We show that mapping class groups associated to all types of real algebraic curves are virtual duality groups. We also deduce some results about the orbifold homotopy groups of the moduli spaces of real algebraic curves. We achieve these…
We prove a new patchworking theorem for singular algebraic curves, which states the following. Given a complex toric threefold $Y$ which fibers over ${\mathbb C}$ with a reduced reducible zero fiber $Y_0$ and other fibers $Y_t$ smooth, and…
We study the curvature of a smooth algebraic surface $X\subset \mathbb R^3$ of degree $d$ from the point of view of algebraic geometry. More precisely, we consider umbilical points and points of critical curvature. We prove that the number…
The traditional study of plane and space algebraic curves by looking at their tangent vectors, curvatures and torsions provides geometric, but unfortunately not sufficient information about individual curves in order to be able to…
We give necessary and sufficient topological conditions for a simple closed curve on a real rational surface to be approximable by smooth rational curves. We also study approximation by smooth rational curves with given complex…
Morse functions with exactly two singular points on homotopy spheres and canonical projections of spheres are generalized as special generic maps. A special generic map is, roughly, a smooth map represented as the composition of a smooth…
Modern methods of graph theory describe a graph up to isomorphism, which makes it difficult to create mathematical models for visualizing graph drawings on a plane. The topological drawing of the planar part of a graph allows representing…
We prove that Viro's patchworking produces real algebraic curves with the maximal number of real inflection points. In particular this implies that maximally inflected real algebraic $M$-curves realize many isotopy types. The strategy we…
We present algorithms to compute the topology of 2D and 3D hyperelliptic curves. The algorithms are based on the fact that 2D and 3D hyperelliptic curves can be seen as the image of a planar curve (the Weierstrass form of the curve), whose…