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In this paper we prove that braid monodromy of an affine plane curve determines the topology of a related projective plane curve.

For a smooth quartic plane curve $C$ we show an existence of a family of Galois closure curves $\phi : S \longrightarrow C$, where $S$ is a nonsingular projective surface and $\phi ^{-1}(P)$ is isomorphic to the Galois closure curve $C_{P}$ for a general point $P \in C$. Moreover we determine the types of singular fibers. As a corollary we can say that $C_{P}$ is not isomorphic to $C_{Q}$ if $P$ is close to $Q$.

Since J.Wahl ([27]), it is known that degree $d$ plane curves having some fixed numbers of nodes and cusps as its only singularities can be represented by a scheme, let say $H$, which can be singular. In Wahl's example, $H$ is singular along a subscheme $F$ but the induced reduced scheme $H_{\rm{red}}$ is smooth along $F$. In this work, we construct explicitly a family of plane curves with nodes and cusps which are represented by singular points of $H_{\rm{red}}$. ¶ To this end, we begin to show that the Hilbert scheme of smooth and connected space curves of degree 12 and genus 15 is irreducible and generically smooth. It follows that it is singular along a hypersurface (3.10). This example is minimal in the sense that the Hilbert scheme of smooth and connected space curves is regular in codimension 1 for $d

Here we prove that a reduced curve $X \subset {\bf {P}}^N$ either canonically embedded or with a high degree complete embedding is arithmetically Cohen - Macaulay.

Let X,Y be algebraic curves in P^n over C. We give an effective description of the join J(X,Y)\in P^n of X and Y in terms of local parametrizations of X and Y., 15 pages, LaTeX

Gorenstein liaison seems to be the natural notion to generalize to higher codimension the well-known results about liaison of varieties of codimension~2 in projective space. In this paper we study points in ${\mathbb P}^3$ and curves in ${\mathbb P}^4$ in an attempt to see how far typical codimension~2 results will extend. While the results are satisfactory for small degree, we find in each case examples where we cannot decide the outcome. These examples are candidates for counterexamples to the hoped-for extensions of codimension~2 theorems., Comment: 29 pages

The geometric monodromy of a plane curve singularity is a quasi-finite diffeomorphism. In this paper we locate the reduction curves of the geometric monodromy and the quadratic vanishing cycles of the singularity. An application to the geometric monodromy group is given., 20 pages, 16 figures

The `linear orbit' of a plane curve of degree $d$ is its orbit in $\P^{d(d+3)/2}$ under the natural action of $\PGL(3)$. In this paper we obtain an algorithm computing the degree of the closure of the linear orbit of an arbitrary plane curve, and give explicit formulas for plane curves with irreducible singularities. The main tool is an intersection@-theoretic study of the projective normal cone of a scheme determined by the curve in the projective space $\P^8$ of $3\times 3$ matrices; this expresses the degree of the orbit closure in terms of the degrees of suitable loci related to the limits of the curve. These limits, and the degrees of the corresponding loci, have been established in previous work., Comment: 33 pages, AmS-TeX 2.1

In this paper we study the existence of reduced and irreducible complex or real projective curves contained in an ambient normal projective variety with prescribed singularities and with “low degree”. We consider germs of planar singularities of curves, up to topological or equisingular equivalence. The main result is an existence theorem for plane curves with ordinary singularities which improves previous results by Greuel, Lossen and Shustin.

We study locally Cohen-Macaulay curves in projective three-space which are contained in a double plane 2H, thus completing the classification of curves lying on surfaces of degree two. We describe the irreducible components of the Hilbert schemes of locally Cohen-Macaulay curves in 2H of given degree and arithmetic genus. We show that these Hilbert schemes are connected. We also discuss the Rao modules of these curves, and liaison and biliaison equivalence classes., Comment: 20 pages

We show, among other things, that in most cases the gonality of a subcanonical curve in projective-three space is computable by multisecants., 15 pages, plain tex

Here we prove that the minimal free resolution of a general space curve of large degree (e.g. a general space curve of degree d and genus g with d g+3, except for finitely many pairs (d,g)) is the expected one. A similar result holds even for general curves with special hyperplane section and, roughly, d g/2. The proof uses the so-called methode d'Horace., This paper has been withdrawn due to a crucial error (a key lemma is obviously false). In the meantime I was not able to overcome this error using another (but related) path

A linear series on a curve C in $P^3$ is "primary" when it does not contain the series cut by planes. We provide a lower bound for the degree of these series, in terms of deg(C), g(C) and of the number $s = min{i: h^0(I_C(i))\neq 0}$; as a corollary, we obtain bounds for the gonality of C. Examples show that our bound is sharp. Extensions to the case of general linear series and to the case of curves in higher projective spaces are considered., AMSTeX/AMSppt, 23 pages

Generalizing a classical lemma of Castelnuovo, we characterize rational normal curves (resp. linearly normal elliptic curves) as curves $C\subset \PP^n$ such that the number of linearly independent hypersurfaces $Z\supset C$ of given degree~$m$ is maximal (resp. next to maximal)., Comment: 14 pages, no figures, AMSTeX v 2.1 with amsppt

We define the Seshadri constant of a space curve and consider ways to estimate it. We then show that it governs the gonality of the curve. We use an argument based on Bogomolov's instability theorem on a threefold. The same methods are then applied to the study of the behaviour of a stable vector bundle on P^3 under restriction to curves and surfaces., Comment: 36 pages, amslatex

Let X be a curve non-degenerate in a projective space PN defined over an algebraically closed field of positive characteristic p, consider the Gauss map of order m defined by the osculating m-planes at general points of X, and denote by {b}0≦j≦N the orders of X. We prove that the inseparable degree of the Gauss map of order m is equal to the highest power of p dividing bm+1., Article

We present an algorithm which, given a deformation with trivial section of a reduced plane curve singularity, computes equations for the equisingularity stratum (that is, the mu-constant stratum in characteristic 0) in the parameter space of the deformation. The algorithm works for any, not necessarily reduced, parameter space and for algebroid curve singularities C defined over an algebraically closed field of characteristic 0 (or of characteristic p>ord(C)). It provides at the same time an algorithm for computing the equisingularity ideal of J. Wahl. The algorithms have been implemented in the computer algebra system Singular. We show them at work by considering two non-trivial examples. As the article is also meant for non-specialists in singularity theory, we include a short survey on new methods and results about equisingularity in characteristic 0., 27 pages

For isolated complex hypersurface singularities with real defining equation we show the existence of a monodromy vector field such that complex conjugation intertwines the local monodromy diffeomorphism with its inverse. In particular, it follows that the geometric monodromy is the composition of the involution induced by complex conjugation and another involution. This topological property holds for all isolated complex plane curve singularities. Using real morsifications, we compute the action of complex conjugation and of the other involution on the Milnor fiber of real plane curve singularities. These involutions have nice descriptions in terms of divides for the singularity., Comment: 16 pages, 6 figures

In this paper, we consider the intensity surface of a 2D image, we study the evolution of the symmetry sets (and medial axes) of 1-parameter families of iso-intensity curves. This extends the investigation done on 1-parameter families of smooth plane curves (Bruce and Giblin, Giblin and Kimia, etc.) to the general case when the family of curves includes a singular member, as will happen if the curves are obtained by taking plane sections of a smooth surface, at the moment when the plane becomes tangent to the surface. Looking at those surface sections as isophote curves, of the pixel values of an image embedded in the real plane, this allows us to propose to combine object representation using a skeleton or symmetry set representation and the appearance modelling by representing image information as a collection of medial representations for the level-sets of an image., Comment: 15 pages, 7 figures

A conjecture of Harbourne and Hirschowitz implies that $r \ge 9$ general points of multiplicity $m$ impose independent conditions to the linear system of curves of degree $d$ when $d(d+3) \ge rm(m+1)-2$. In this paper we prove that the conditions are independent provided $d+2\ge (m+1)(\sqrt{r+1.9}+\pi /8) $.

Mark all vertices on a curve evolving under a family of curves obtained by intersecting a smooth surface M with the 1-parameter family of planes parallel to the tangent plane to M at a point p. Those vertices trace out a set, called the vertex set of M through p. We take p to be an isolated umbilic point on M and describe the perestroikas of the vertex set under generic n-parameter small deformations of the surface, as well as the corresponding discriminants. One of the main consequences of our results is that, in some sense (see Theorem 2), generically the study of the discriminants of small n-deformations, with n greater than or equal to 2, simplifies to that of 2-deformations. This work was primarily motivated by the medial representation of shapes in Computer Vision and Image Analysis, where the behaviour of vertices plays a crucial role in the qualitative changes of the skeleton or Blum medial axis of curves. On the other hand, the study of vertices of curves has raised up a great interest in particular in Geometry and Singularity Theory, in line with several problems such as the 4-vertex Theorem, the local geometry of surfaces and Geometry of Caustics. These classical subjects have received a new impulsion due to the development of Contact Geometry, especially in the works of V. Arnold on Legendrian Collapse and Legendrian Sturm Theory showing the relation between vertices of plane curves and Legendrian singularities and Sturm-Hurwitz Theorem., Comment: V2: Latex, 17 pages, 9 figures. Some minor misprints added. Interesting comments from Prof. V.I. Arnold added

We consider pairs of commuting isometries that are annihilated by a polynomial. We show that the polynomial must be inner toral, which is a geometric condition on its zero set. We show that cyclic pairs of commuting isometries are nearly unitarily equivalent if they are annihilated by the same minimal polynomial.