Narrow Results

We prove that the numerical invariant $3\mu -4\tau$ of a reduced irreducible plane curve singularity germ is non-negative, non-decreasing under blowups and strictly increasing unless the curve is non-singular. This provides a new perspective to understand the question posed by Dimca and Greuel. Moreover, our work can be put in the general framework of discovering monotonic invariants under blowups.

The Minkowski sum of two plane curves can be regarded as the area generated by sweeping one curve along the other. The boundary of the Minkowski sum consists of translated portions of the given curves and/or portions of a more complicated curve, the “envelope” of translates of the swept curve. We show that the Minkowski-sum boundary is describable as an algebraic curve (or subset thereof) when the given curves are algebraic, and illustrate the computation of its implicit equation. However, such equations are typically of high degree and do not offer a practical basis for tracing the boundary. For the case of polynomial parametric curves, we formulate a simple numerical procedure to address the latter problem, based on constructing the Gauss maps of the given curves and using them to identifying “corresponding” curve segments that are to be summed. This yields a set of discretely-sampled arcs that constitutes a superset of the Minkowski-sum boundary, and can be regarded as a planar graph. To extract the true boundary, we present a method for identifying and “trimming” away extraneous arcs by systematically traversing this graph.

In [A2] V.I. Arnold introduced three basic invariants St, J⁺ and J^- of plane curves and proposed some interesting conjectures concerning the extremal value of these invariants on a given set of curves. Partial answers have been obtained by O. Viro and A. N. Shumakovich. We give explicit formulas for these extremal values of sets of plane curves with fixed number of double points and of Whitney index and we determine on which curves these extremal values are attained (Theorems 3-6). Our arguments are based on understanding of the fine structures of generic curves and some surgery operations on curves.

Motivated by Arnold's theory of invariants of plane curves, we introduce the semi-group of equivalence classes of arrangements of nested curves. There exists a natural invariant of plane curves without inverse self-tangencies with values in this semi-group. We show that the associated Grothendieck group is ℤ × ℤ. These two factors correspond to previously known invariants of plane curves without inverse self-tangencies, namely Whitney's index and Arnold's J^- invariant. We show that arrangements of nested curves are not classified by their finite type invariants.

We study the degree of polynomial representations of knots. We obtain the lexicographic degree for two-bridge torus knots and generalized twist knots. The proof uses the braid theoretical method developed by Orevkov to study real plane curves, combined with previous results from [Chebyshev diagrams for two-bridge knots, Geom. Dedicata150 (2010) 405–425; E. Brugallé, P.-V. Koseleff, D. Pecker, Untangling trigonal diagrams, to appear in J. Knot Theory and its Ramifications]. We also give a sharp lower bound for the lexicographic degree of any knot, using real polynomial curves properties.

The problem concerning which Gauss diagrams can be realized by knots is an old one and has been solved in several ways. In this paper, we present a direct approach to this problem. We show that the needed conditions for realizability of a Gauss diagram can be interpreted as follows “the number of exits = the number of entrances” and the sufficient condition is based on the Jordan curve theorem. Further, using matrices, we redefine conditions for realizability of Gauss diagrams and then we give an algorithm to construct meanders.

Two recent publications describe realizable Gauss diagrams using conditions stating that the number of chords in certain sets of chords is even or odd. We demonstrate that these descriptions are incorrect by finding multiple counter-examples. However, the idea of having a parity-based description of realizable Gauss diagrams is attractive. We recall that realizability of Gauss diagrams as touch curves can be described via bipartite graphs. We show that realizable Gauss diagrams can be described via bipartite graphs.

In this paper we classify non-commutative quadrics and study their homological properties. In fact we find all non-commutative algebras of degree 2 up to isomorphism and we study these algebras via their homomorphic images onto the polynomial algebra k[x,y] as well as the Ext¹(k(p), k(q))-groups, where k(p) and k(q) are one-dimensional simple modules. Moreover some general results on simple finite-dimensional modules are obtained. Some of these results are applied to the special cases of non-commutative quadrics.

In this paper, we construct the first examples of what we call fake ES-irreducible components; Definition 2.8. In our way to do so, we classify the automorphism groups of smooth plane sextics that only have automorphisms of order $\le 3$; Theorems 2.1, 2.4 and 2.5, Corollaries 2.9 and 2.11.

We formulate an isoperimetric deformation of curves on the Minkowski plane, which is governed by the defocusing modified Korteweg-de Vries (mKdV) equation. Two classes of exact solutions to the defocusing mKdV equation are also presented in terms of the $\tau$ functions. By using one of these classes, we construct an explicit formula for the corresponding motion of curves on the Minkowski plane even though those solutions have singular points. Another class gives regular solutions to the defocusing mKdV equation. Some pictures illustrating the typical dynamics of the curves are presented.

Let $C$ be a smooth plane curve of degree $d\ge 4$ defined over a global field $k$ of characteristic $p=0$ or $p>(d-1)(d-2)/2$ (up to an extra condition on $\mathrm{\text{Jac}}(C)$). Unless the curve is bielliptic of degree four, we observe that it always admits finitely many quadratic points. We further show that there are only finitely many quadratic extensions $k(\sqrt{D})$ when $k$ is a number field, in which we may have more points of $C$ than these over $k$. In particular, we have this asymptotic phenomenon valid for Fermat’s and Klein’s equations. Second, we conjecture that there are two infinite sets ${ℰ}$ and ${𝒟}$ of isomorphism classes of smooth projective plane quartic curves over $k$ with a prescribed automorphism group, such that all members of ${ℰ}$ (respectively ${𝒟}$) are bielliptic and have finitely (respectively infinitely) many quadratic points over a number field $k$. We verify the conjecture over $k=\mathbb{Q}$ for $G=\mathbb{Z}/6\mathbb{Z}$ and $\mathrm{\text{GAP}}(16,13)$. The analog of the conjecture over global fields with $p>0$ is also considered.

Let the curvature of a plane curve parametrized by arclength s be given as a function of the Cartesian co-ordinates of the points the curve is passing through in the Euclidean plane. Then, the co-ordinates of its position vector are determined by a system of equations arising from the Frenet-Serret relations that can be regarded as a dynamical system of two degrees of freedom determining the motion (trajectories) of a particle of unit mass, s playing the role of time. Here, two classes of integrable systems of the foregoing type are identified. For that purpose, we explore the variational symmetries of a generic system of this kind with respect to Lie groups of point transformations of the involved variables. As a result, a set of sufficient conditions are found which ensure that such a system possesses two functionally independent integrals of motion and, consecutively, is integrable by quadratures. In each such case, we achieve either an explicit parameterization of the corresponding trajectory curves in terms of their curvatures or, at least, a separation of the dependent variables.

Let C be an irreducible singular plane curve of type (d, ν). Let G denote the gonality of C. Criteria for the equality: G = d – ν have been discussed ([2], [3], [4], [9]). The purpose of this paper is to improve the criteria given in our previous paper [9]. In particular, in case ν = 3, we obtain an effective criterion which seems to be optimal. However, for many singular plane curves, the equality is not the case. We therefore at the same time discuss the lower bounds of G. We generalize our previous methods employed in [9].