The computation of the logarithmic cohomology for plane curves

We will give algorithms of computing bases of logarithmic cohomology groups for square-free polynomials in two variables.     

Fundamental groups of complements of plane curves and symplectic invariants

Introducing the notion of stabilized fundamental group for the complement of a branch curve in , we define effectively computable invariants of symplectic 4-manifolds that generalize those previously introduced by Moishezon and Teicher for complex projective surfaces. Moreover, we study the structure of these invariants and formulate conjectures supported by calculations on new examples.     

On multi-Frobenius non-classical plane curves

In this paper, we characterize the plane curves over \mathbb F_q{\mathbb {F}_q} which are Frobenius non-classical for different powers of q.     

On a property of plane curves

Let γ:[0,1]→²[0,1] be a continuous curve such that γ(0)=(0,0), γ(1)=(1,1), and γ(t)∈²(0,1) for all t∈(0,1). We prove that, for each n∈N, there exists a sequence of points Ai, 0?i?n+1, on γ such that A₀=(0,0), An+1=(1,1), and the sequences and , 0?i?n, are positive and the same up to order, where π₁, π₂ are projections on the axes.     

On the intersection number of two plane curves

The intersection number of two algebraic curves in a plane over any infinite field is shown to be uniquely defined by a fairly weak set of postulates. These postulates also imply some of the basic properties of that number. The treatment is completely elementary.     

On the realization problem of real algebraic plane curves as Hessian curves

In this paper we study some problems in Hessian topology. We prove that certain real plane curves satisfy the requirements of the Hessian curve of a differential function. The real plane curves we consider are those with k outer ovals and also those which only have one nest of depth k, with \({k \in \mathbb{N}}\) .     

On families of singular plane projective curves

Summary In this article it is studied, in modern form, the classical concept of «complete continuous system of projective plane curves of degree n, with singularities of a certain type». Properties of these schemes are studied (universal properties, techniques to compute tangent spaces, smoothness, special points, etc.). As an application, a modern presentation of the classical theories of systems of curves with nodes, and with nodes and cusps, is given.     

On the curve diffusion flow of closed plane curves

In this paper, we consider the steepest descent H ^?1-gradient flow of the length functional for immersed plane curves, known as the curve diffusion flow. It is known that under this flow there exist both initially immersed curves that develop at least one singularity in finite time and initially embedded curves that self-intersect in finite time. We prove that under the flow closed curves with initial data close to a round circle in the sense of normalised L ² oscillation of curvature exist for all time and converge exponentially fast to a round circle. This implies that for a sufficiently large ‘waiting time’, the evolving curves are strictly convex. We provide an optimal estimate for this waiting time, which gives a quantified feeling for the magnitude to which the maximum principle fails. We are also able to control the maximum of the multiplicity of the curve along the evolution. A corollary of this estimate is that initially embedded curves satisfying the hypotheses of the global existence theorem remain embedded. Finally, as an application we obtain a rigidity statement for closed planar curves with winding number one.     

On 3-syzygy and unexpected plane curves

Geometriae Dedicata - We prove a new inequality relating volume to length of closed geodesics on area minimizers for generic metrics on the complex projective plane. We exploit recent regularity...     

On plane models for Drinfeld modular curves

In this work, we find plane models for certain Drinfeld modular curves X₀(n) which have better properties than the plane models derived from the usual Drinfeld modular equations. As an application, we construct ring class fields over imaginary quadratic fields by using singular values of generators of the function field of X₀(n).     

On a class of rational cuspidal plane curves

We obtain new examples and the complete list of the rational cuspidal plane curvesC with at least three cusps, one of which has multiplicitydegC-2. It occurs that these curves are projectively rigid. We also discuss the general problem of projective rigidity of rational cuspidal plane curves.     

On the equi-normalizable deformations of singularities of complex plane curves

We study a specific class of deformations of curve singularities: the case when the singular point splits to several ones, such that the total δ invariant is preserved. These are also known as equi-normalizable or equi-generic deformations. We restrict primarily to the deformations of singularities with smooth branches. A natural invariant of the singular type is introduced: the dual graph. It imposes severe restrictions on the possible collisions/deformations. And allows to prove some bounds on the variation of classical invariants in equi-normalizable families. We consider in details deformations of ordinary multiple point, the deformations of a singularity into the collections of ordinary multiple points and deformations of the type x ^p  + y ^pk into the collections of A _k ’s. The research was constantly supported by the Skirball postdoctoral fellowship of the Center of Advanced Studies in Mathematics (Mathematics Department of Ben Gurion University, Israel). Part of the work was done in Mathematische Forschungsinsitute Oberwolfach, during the author’s stay as an OWL-fellow. Some results were published in the preprint [17].