Related papers: The Analytic Classification of Plane Curves with T…
We completely classify all plane curves of degree at most 30 with a unique cuspidal (locally unibranch) singular point and rational normalization in terms of the Newton pairs parameterizing the cusp. We distinguish between prime and…
In a series of papers, Aluffi and Faber computed the degree of the $GL_3$ orbit closure of an arbitrary plane curve. We attempt to generalize this to the equivariant setting by studying how orbits degenerate under some natural…
For affine algebraic plane curves we reduce a calculation of its invariants to calculation of the intersection of kernels of some derivations.
We introduce new invariants of a class of toric surfaces (including the projective plane) that arise from appropriate enumeration of real curves of genus one and two. These invariants admit a refinement similar to the one introduced by…
In this survey we discuss the problem of the existence of rational curves on complex surfaces, both in the K\"ahler and non-K\"ahler setup. We systematically go through the Enriques--Kodaira classification of complex surfaces to highlight…
A novel and deterministic algorithm is presented to detect whether two given rational plane curves are related by means of a similarity, which is a central question in Pattern Recognition. As a by-product it finds all such similarities, and…
In this paper we present a framework which provides an analytical (i.e., infinitely differentiable) transformation between spatial coordinates and orbital elements for the solution of the gravitational two-body problem. The formalism omits…
In this article, we study the invariant differential forms which a correspondence of curves admits. We also try to classify the correspondences of $\mathbb{P}^1$ that admits such invariant differential forms.
At any point of a surface in the four-dimensional Euclidean space we consider the geometric configuration consisting of two figures: the tangent indicatrix, which is a conic in the tangent plane, and the normal curvature ellipse. We show…
We give a method of counting the number of curves with a given type of singularity in a suitably ample linear series on a smooth surface using punctual Hilbert schemes. The types of singulaties for which our methods suffice include the…
In this paper we obtain an explicit formula for the number of degree d curves in two dimensional complex projective space, passing through (d(d+3)/2 -k) generic points and having a codimension k singularity, where k is at most 7. In the…
We present some results concerning the Saito module and the torsion submodule of an analytic plane curve, and we provide a method for computing them. Using this algorithm, we compute analytic invariants for plane curves with multiplicity…
The article provides a local classification of singularities of meromorphic second order linear differential equation with respect to analytic/meromorphic linear point transformations. It also addresses the problem of determining the Lie…
We study integral plane curves meeting at a single unibranch point and show that such curves must satisfy two equivalent conditions. A numeric condition: the local invariants of the curves at the contact point must be arithmetically…
In the article, we exhibit a series of new examples of rigid plane curves, that is, curves, whose collection of singularities determines them almost uniquely up to a projective transformation of the plane.
We consider flat families of reduced curves on a smooth surface S such that each member C has the same number of singularities of fixed singularity types and the corresponding (locally closed) subscheme H of the Hilbert scheme of S. We are…
We give unique analytic "normal forms" for germs of a holomorphic vector field of the complex plane in the neighborhood of an isolated singularity of saddle-node type having a convergent formal separatrix. We specifically address the…
We define a computable topological invariant $\mu(\gamma)$ for generic closed planar regular curves $\gamma$, which gives an effective lower bound for the number of inflection points on a given generic closed planar curve. Using it, we…
This is the paper as published. The topology of a complex plane curve singularity with real branches is deduced from any real deformation having delta crossings. An example of the computation of the global geometric monodromy of a…
We apply classical invariant theory of binary forms to explicitly characterize isomorphism classes of hyperelliptic curves of small genus and, conversely, propose algorithms for reconstructing hyperelliptic models from given invariants. We…