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In this article, we investigate some properties of cyclic coverings of complex surfaces of general type branched along smooth curves that are numerically equivalent to a multiple of the canonical class. The main results concern coverings of…
We present efficient algorithms for detecting central and mirror symmetry for the case of algebraic curves defined by means of polynomial parametrizations. The algorithms are based on the existence of a linear relationship between two…
A fake quadric is a smooth minimal surface of general type with the same invariants as the quadric in P^3, i.e. K^2=2c_2=8 and q=p_g=0. We study here quaternionic fake quadrics i.e. fake quadrics constructed arithmetically by using some…
An upper bound on the first S^1 invariant eigenvalue of the Laplacian for invariant metrics on the 2-sphere is used to find obstructions to the existence of isometric embeddings of such metrics in (R^3,can). As a corollary we prove: If the…
Most genuine multi-sided surface representations depend on a 2D domain that enables a mapping between local parameters and global coordinates. The shape of this domain ranges from regular polygons to curved configurations, but the simple…
In this work we provide a novel approach for computing the coefficients of the characteristic polynomial of a square matrix. We demonstrate that each coefficient can be efficiently represented by a set of circle graphs. Thus, one can employ…
We show how Seifert surfaces, so useful for the understanding of the Alexander polynomial \Delta_L(t), can be generalized in order to study the multivariable Alexander polynomial \Delta_L(t_1,...,t_\mu). In particular, we give an elementary…
In this paper we introduce the concept of polynomial diagrams and its area for special polynomials.We study the properties of polynomial area diagrams. The formula for the area of an arbitrary polynomial diagram.
We present a constructive approach for approximating the conformal map (uniformization) of a polyhedral surface to a canonical domain in the plane. The main tool is a characterization of convex spaces of quasiconformal simplicial maps and…
Canonical principal parameters are introduced for surfaces in $\mathbb R^3$ without umbilical points. It is proved that in these parameters the surface is determined (up to position in space) by a pair of invariants satisfying a partial…
We study the structure of collections of algebraic curves in three dimensions that have many curve-curve incidences. In particular, let $k$ be a field and let $\mathcal{L}$ be a collection of $n$ space curves in $k^3$, with…
Duality of curves is one of the important aspects of the ``classical'' algebraic geometry. In this paper, using this foundation, the duality of tropical polynomials is constructed to introduce the duality of Non-Archimedean curves. Using…
This paper studies geometric properties of the Iterated Matrix Multiplication polynomial and the hypersurface that it defines. We focus on geometric aspects that may be relevant for complexity theory such as the symmetry group of the…
It is well-known that theta characteristics on smooth plane curves over a field of characteristic different from two are in bijection with certain smooth complete intersections of three quadrics. We generalize this bijection to possibly…
We investigate geometric properties of surfaces given by certain formulae. In particular, we calculate the singular curvature and the limiting normal curvature of such surfaces along the set of singular points consisting of singular points…
We make a systematic investigation of quadrature properties for quadrics, namely integration of holomorphic functions over planar domains bounded by second degree curves. A full understanding requires extending traditional settings by…
This paper describes the foundations of a differential geometry of a quaternionic curves. The Frenet-Serret equations and the evolutes and evolvents of a particular quaternionic curve are accordingly determined. This new formulation takes…
Smooth real cubic surfaces are birationally trivial (over $\R$) if and only if their real locus is connected or, equivalently, if and only if they have two skew real lines or two skew complex conjugate lines. In such a case a…
We survey some aspects of the theory of elliptic surfaces and give some results aimed at determining the Picard number of such a surface. For the surfaces considered, this will be equivalent to determining the Mordell-Weil rank of an…
A surface of revolution is created by taking a curve in the $xy$-plane and rotating it about some axis. We develop a program which automatically generates crochet patterns for surfaces by revolution when they are obtained by rotating about…