Plane curves of genus p and degree 2p as projections of space curves of the same degree

We characterize plane curves Γ of genus p and degree 2p with respect to the possibility of obtaining them as projections of space curves C′ of the same degree. When Γ is hyperelliptic, we link this characterization with the configuration of the singularities of Γ and with the position of C′ on certain scrolls. Supported by the M.U.R.S.T. of the Italian Government     

On lifting singular curves

The aim of this note is to lift singular curves of a certain type to characteristic zero. The liftings are obtained as suitable pushouts and the corresponding relative jacobians are identified as rigidifed Picard functors.     

Theta-characteristics on singular curves

On a smooth curve a theta-characteristic is a line bundle L,the square of which is the canonical line bundle . The equivalentcondition om(L, ) L generalizes well to singular curves, asapplications show. More precisely, a theta-characteristic isa torsion-free sheaf of rank 1 with om(, ) . If the curvehas non-ADE singularities, then there are infinitely many theta-characteristics.Therefore, theta-characteristics are distinguished by theirlocal type. The main purpose of this article is to compute thenumber of even and odd theta-characteristics (that is withh⁰(C, ) 0 and h⁰(C, ) 1 modulo 2, respectively) in terms ofthe geometric genus of the curve and certain discrete invariantsof a fixed local type.     

Frobenius non classical curves

Partially supported by CNP_q. This research was concluded while the first author was Visiting Professor at the Math. Department of the University of Campinas.     

Homotopically invisible singular curves

Given a smooth manifold M and a totally nonholonomic distribution \(\Delta \subset TM \) of rank \(d\ge 3\), we study the effect of singular curves on the topology of the space of horizontal paths joining two points on M. Singular curves are critical points of the endpoint map \(F\,{:}\,\gamma \mapsto \gamma (1)\) defined on the space \(\Omega \) of horizontal paths starting at a fixed point x. We consider a sub-Riemannian energy \(J\,{:}\,\Omega (y)\rightarrow \mathbb R\), where \(\Omega (y)=F^{-1}(y)\) is the space of horizontal paths connecting x with y, and study those singular paths that do not influence the homotopy type of the Lebesgue sets \(\{\gamma \in \Omega (y)\,|\,J(\gamma )\le E\}\). We call them homotopically invisible. It turns out that for \(d\ge 3\) generic sub-Riemannian structures in the sense of Chitour et al. (J Differ Geom 73(1):45–73, 2006) have only homotopically invisible singular curves. Our results can be seen as a first step for developing the calculus of variations on the singular space of horizontal curves (in this direction we prove a sub-Riemannian minimax principle and discuss some applications).     

Enumeration of singular algebraic curves

We enumerate plane complex algebraic curves of a given degree with one singularity of any given topological type. Our approach is to compute the homology classes of the corresponding equisingular strata in the parameter spaces of plane curves. We suggest an inductive procedure, which is based on the intersection theory combined with liftings and degenerations. The procedure computes the homology class in question whenever a given singularity type is defined. Our method does not require knowledge of all the possible deformations of a given singularity, as it was in previously known procedures.     

Exact regularity of Bergman,Szegö and Sobolev space projections in non pseudoconvex domains

Mathematische Zeitschrift -     

Peaked singular wave solutions associated with singular curves

We present new types of singular wave solutions with peaks in this paper. When a heteroclinic orbit connecting two saddle points intersects with the singular curve on the topological phase plane for a generalized KdV equation, it may be divided into segments. Different combinations of these segments may lead to different singular wave solutions, while at the intersection points, peaks on the waves can be observed. It is shown for the first time that there coexist different types of singular waves corresponding to one heteroclinic orbit.     

Patchworking singular algebraic curves I

In this paper we present a general patchworking procedure for the construction of reduced singular curves having prescribed singularities and belonging to a given linear system on algebraic surfaces. It originates in the Viro “gluing” method for the construction of real non-singular algebraic hypersurfaces. The general procedure includes almost all known particular modifications, and goes far beyond. Some applications and examples illustrate the construction. Both authors were partially supported by the Herman Minkowsky-Minerva Center for Geometry at Tel Aviv University, and by grant no. G-616-15.6/99 from the German-Israeli Foundation for Research and Development. The first author was also supported by the Bessel Research Award from the Alexander von Humboldt Foundation. The second author was also partially supported by the EC-network ‘Algebraic Lie Representations” contract no. ERB-FMRX-CT97-0100.     

Koszul cohomology and singular curves

We investigate Koszul cohomology on irreducible nodal curves following the lines of [2]. In particular, we prove both Green and Green-Lazarsfeld conjectures for any k-gonal nodal curve which is general in the sense of [4].     

Patchworking singular algebraic curves II

In this paper we present various particular versions of the general patch-working procedurefor the construction of reduced algebraic curves with prescribed singularities, on algebraic surfaces. Among the main examples are a deformation of reducible algebraic curves on reducible algebraic surfaces in the presence of non-transverse intersections of a curve with the singular locus of a surface, and a deformation of curves with multiple components. As an application we deduce, a significant asymptotical improvement for the sufficient existence criterion of algebraic curves with arbitrary prescribed singlarities in given linear systems on smooth projective algebraic surfaces. Both authors were partially supported by the Herman Minkowsky-Minerva Center for Geometry at Tel Aviv University, and by grant no. G-616-15.6/99 from the German-Israeli Foundation for Research and Development. The first author was also supported by the Bessel Research Award from the Alexander von Humboldt Foundation.     

Linear systems on singular curves

We start this work by studying free linear systems on singular curves and related base point free linear systems on the non-singular model. We apply these results to the study of pencils of small degree on non-singular curves. We also prove a “base point free pencil trick” which holds for any (possibly) singular curve. Received: 15 June 1998     

Pythagorean-hodograph space curves

We investigate the properties of polynomial space curvesr(t)={x(t), y(t), z(t)} whose hodographs (derivatives) satisfy the Pythagorean conditionx′²(t)+y′²(t)+z′²(t)≡σ²(t) for some real polynomial σ(t). The algebraic structure of thecomplete set of regular Pythagorean-hodograph curves in ℝ³ is inherently more complicated than that of the corresponding set in ℝ². We derive a characterization for allcubic Pythagoreanhodograph space curves, in terms of constraints on the Bézier control polygon, and show that such curves correspond geometrically to a family of non-circular helices. Pythagorean-hodograph space curves of higher degree exhibit greater shape flexibility (the quintics, for example, satisfy the general first-order Hermite interpolation problem in ℝ³), but they have no “simple” all-encompassing characterization. We focus on asubset of these higher-order curves that admits a straightforward constructive representation. As distinct from polynomial space curves in general, Pythagorean-hodograph space curves have the following attractive attributes: (i) the arc length of any segment can be determined exactly without numerical quadrature; and (ii) thecanal surfaces based on such curves as spines have precise rational parameterizations.     

Sweeping algebraic curves for singular solutions

Many problems give rise to polynomial systems. These systems often have several parameters and we are interested to study how the solutions vary when we change the values for the parameters. Using predictor-corrector methods we track the solution paths. A point along a solution path is critical when the Jacobian matrix is rank deficient. The simplest case of quadratic turning points is well understood, but these methods no longer work for general types of singularities. In order not to miss any singular solutions along a path we propose to monitor the determinant of the Jacobian matrix. We examine the operation range of deflation and relate the effectiveness of deflation to the winding number. Computational experiments on systems coming from different application fields are presented.