Related papers: Amoebas of maximal area
The most useful and interesting line bundles over algebraic curves of a very high genus have the ratio \delta of the degree to the genus close to half-integer values, usually \delta \approx 0, \delta \approx 1/2, or \delta \approx 1; the…
A coamoeba is the image of a subvariety of a complex torus under the argument map to the real torus. We describe the structure of the boundary of the coamoeba of a variety, which we relate to its logarithmic limit set. Detailed examples of…
Given a real algebraic curve in the projective 3-space, its hyperbolicity locus is the set of lines with respect to which the curve is hyperbolic. We give an example of a smooth irreducible curve whose hyperbolicity locus is disconnected…
The number of topologically different plane real algebraic curves of a given degree $d$ has the form $\exp(C d^2 + o(d^2))$. We determine the best available upper bound for the constant $C$. This bound follows from Arnold inequalities on…
The paper initiates a systematic study of Moebius structures and Ptolemy spaces. We conjecture that every compact Ptolemy space with circles and many space inversions is Moebius equivalent to the boundary at infinity of a rank one symmetric…
Recall that the moduli space of smooth (that is, stable) cubic curves is isomorphic to the quotient of the upper half plane by the group of fractional linear transformations with integer coefficients. We establish a similar result for…
It is well known that plane curves with the same endpoints are homotopic. An analogous claim for plane curves with the same endpoints and bounded curvature still remains open. In this work we find necessary and sufficient conditions for two…
A basic arbitrariness in the determination of the topology of a manifold at the Planck length is discussed. An explicit example is given of a `smooth' change in topology from the 2-sphere to the 2-torus through a sequence of noncommuting…
The amoebas associated to algebraic varieties are certain concave regions in the Euclidean space whose shape reminds biological amoebas. This term was formally introduced to Mathematics in 1994 by Gelfand, Kapranov and Zelevinski. Some…
The study of hypersurfaces in a torus leads to the beautiful zoo of amoebas and their contours, whose possible configurations are seen from combinatorial data. There is a deep connection to the logarithmic Gauss map and its critical points.…
The author has been interested in regions surrounded by real algebraic curves of degree $1$ or $2$ in the plane. The author is mainly interested in their shapes and combinatorics. This is a fundamental and natural problem in mathematics…
Aougab and Gaster [Math. Proc. Cambridge Philos. Soc. 174 (2023), 569-584] proved that any set of simple closed curves on the torus, where any two are non-homotopic and intersect at most k times, has a maximum size of $k+O(\sqrt{k}\log k)$.…
We show that $\mathbb{C}^2$ contains pairs of properly embedded, smooth complex curves that are isotopic through homeomorphisms but not diffeomorphisms of $\mathbb{C}^2$. The construction is based on realizing corks as branched covers of…
Computing the topology of an algebraic plane curve $\mathcal{C}$ means to compute a combinatorial graph that is isotopic to $\mathcal{C}$ and thus represents its topology in $\mathbb{R}^2$. We prove that, for a polynomial of degree $n$ with…
We study arithmetical and geometrical properties of maximal curves, that is, curves defined over the finite field F_{q^2} whose number of F_{q^2}-rational points reaches the Hasse-Weil upper bound. Under a hypothesis on non-gaps at a…
This paper generalizes Bismut's equivariant Chern character to the setting of abelian gerbes. In particular, associated to an abelian gerbe with connection, an equivariantly closed differential form is constructed on the space of maps of a…
We classify, up to isomorphism, maximal curves covered by the Hermitian curve \mathcal H by a prime degree Galois covering. We also compute the genus of maximal curves obtained by the quotient of \mathcal H by several automorphisms groups.…
A (positive) locally convex curve in the 2-sphere is a curve with positive geodesic curvature (i.e., which always turns left). In the 3-sphere, it is a curve with positive torsion. In this work we discussed the topology of spaces of such…
This article discusses inequalities on lengths of curves on hyperbolic surfaces. In particular, a characterization is given of which topological types of curves and multicurves always have a representative that satisfies a length inequality…
We describe the relationship between dimer models on the real two-torus and coamoebas of curves in (\CC^\times)^2. We show, inter alia, that the dimer model obtained from the shell of the coamoeba is a deformation retract of the closed…