Irreducible Plane Curve Research Articles

Plane Curves With Nodes

A smooth algebraic curve is birationally equivalent to a nodal plane curve. One of the main problems in the theory of plane curves is to describe the situation of nodes of an irreducible nodal plane curve (see [16, Art. 45], [10], [7, Book IV, Chapter I, §5], [12, p. 584], and [3]).Let n denote the degree of a nodal curve and d the number of nodes. The case (AZ, d) — (6,9) has been analyzed by Halphen [10]. It follows from Lemma 3.5 and Proposition 3.6 that this is an exceptional case. The case d ≦n(n + 3)/6, d ≦(n — 1)(n — 2)/2, and (n, d) ≠ (6,9) was investigated by Arbarello and Cornalba [3]. We present a simpler proof (Corollary 3.8).We consider the main case which is particularly important due to its applications to the moduli variety of curves, compare [19, Chapter VIII, Section 4]. Let Vn,d be the variety of irreducible curves of degree n with d nodes and no other singularities such that each curve of Vn,d can be degenerated into n lines in general position (see [17]).

On the variety of plane curves of degree 𝑑 with 𝛿 nodes and 𝜅 cusps

Let P N {{\mathbf {P}}^N} be the projective space which parametrizes all plane curves of degree d d and V ( d , δ , κ ) V(d,\delta ,\kappa ) the subvariety of P N {{\mathbf {P}}^N} consisting of all reduced and irreducible plane curves of degree d d with δ \delta nodes and κ \kappa cusps as their only singularities. In this paper we prove that V ( d , δ , κ ) V(d,\delta ,\kappa ) is irreducible if κ ⩽ 3 \kappa \leqslant 3 , except possibly when κ = 3 \kappa = 3 and d = 5 d = 5 or 6 6 .

On semigroups of irreducible algebroid plane curves

On semigroups of irreducible algebroid plane curves

Topology of the quasi-ordinary surface singularities

IN this paper we are interested in one type of (local) complex surface singularities (nonisolated in general, (2.4.3) ex. 2) the investigation of which is due essentially to Lipman [20]. Let F be the surface in an open set in C3 defined by the vanishing of a holomorphic function in three variables. Assume that the origin 0 is a singular (nonsmooth) point of F. If there is a projection of F to a complex plane so that the discriminant (branch) curve has a smooth point at the image of 0, then F is equisingular at 0 (along the preimage of the discriminant curve) [27]. The surface singularities that we are interested in are those which are next in order of complexity, namely those whose images in some (local) projections are ordinary double points of the discriminant curves. Such singularities are said to be quasi-ordinary or Jungian (2.1). See [21] for an introduction to the quasi-ordinary singularities and a summary of the fundamental work [20]. Quasi-ordinary singularities are amenable to detailed analysis since they can be “parametrized” by fractional power series in two variables [19, p. 2071, [Z], which are called the parametrizations of the singularities (cf. the irreducible plane curve singularities can always be parametrized by fractional power (Puiseux) series in one variable). From a suitably normalized parametrization of a quasi-ordinary singularity one can single out (in the same way one obtains the characteristic pairs of plane curves) some exponents. called the distinguished pairs, which are shown in [20] to depend only on the analytic type ( = isomorphism type of the analytic local ring) of the singularity. Moreover, like the characteristic pairs for plane curves, the distinguished pairs determine quite a lot of geometry of the singularity [21, p. 1631. The main result of this paper is that the distinguished pairs of (F, 0) depend only on the topological type of (C3, F) at 0. In other words, the local topology of a quasi-ordinary singularity determines its distinguished pairs. Since the distinguished pairs determine the local topology (2.2.3), it follows that topo!ogically the quasi-ordinary singulariries are completely classified by rheir disringuished pairs (2.2.4). As a corollary, the question of Zariski about the topological invariance of multiplicity is answered affirmatively for quasi-ordinary singularities (2.2.5). Another consequence of the main result is that the local topology determines the Zariski tangent cone (up to isomorphism) (2.2.5)-ii, this answers a question of L(LTeissier [ 18, p. 107]in the quasi-ordinary case. The following question may be of some interest to topologists: Question (Lipman [ZO, p. 1683). Is there an explicit topological interpretation of the distinguished pairs, as one has via compound torus knots for the characteristic pairs of plane curves? The main result: “local topology determines the distinguished pairs (2.2.1)” will be proved

Semigroup of two irreducible algebroid plane curves

Semigroup of two irreducible algebroid plane curves

On the singularities of polar curves

Let Г be a given equisingularity class of irreducible algebroid plane curves with just one characteristic exponent. We first obtain the equisingularity type of the general polar of a curve of Г with general moduli and we also determine the Newton-Cramer polygon of the general polar of all curves in Г.

A few remarks about the variety of irreducible plane curves of given degree and genus

A few remarks about the variety of irreducible plane curves of given degree and genus

Projective Surfaces Over a Finite Field

Let $k$ be the algebraic closure of a finite field, and let $X$ be an irreducible projective surface over $k$. Let $C$ be a curve on $X$, and let $\Omega$ be a finite set of closed points of $X$ meeting each irreducible component of $X$. We prove that there is an irreducible curve on $X$ whose set-theoretic intersection with $C$ is $\Omega$. Using this theorem we characterize ${\mathbf {P}}_k^2$ as a topological space, and we show that for any two irreducible plane curves $C$, $C’$ there is a homeomorphism from ${\mathbf {P}}_k^2$ onto itself taking $C$ onto $C’$.

Projective surfaces over a finite field

ABsTRAcr. Let k be the algebraic closure of a finite field, and let X be an irreducible projective surface over k. Let C be a curve on X, and let Q be a finite set of closed points of X meeting each irreducible component of X. We prove that there is an irreducible curve on X whose set-theoretic intersection with C is U. Using this theorem we characterize Pk2 as a topological space, and we show that for any two irreducible plane curves C, C' there is a homeomorphism from Pk2 onto itself taking C onto C'.

An arithmetic theory of adjoint plane curves

Introduction. In classical algebraic geometry the adjoint curves to an irreducible plane curve are an essential tool in the study of the geometry on the curve. In this paper we shall give an algebro-arithmetic development of the theory of adjoint curves, and shall extend the classical results to irreducible plane curves with arbitrary singularities defined over arbitrary ground fields. Our definition of the adjoint condition at a given singular point of the curve is stated in terms of the conductor between the local ring of the point and its integral closure. The fundamental properties of the adjoint curves are then derived from corresponding properties of the conductor. The single deepest and most important property of the adjoint curves is that, on a curve of order m, the adjoint curves of order m -3 cut out the complete canonical series. This property is equivalent to the fact that the degree of the fixed component of the adjoint series is twice the number of conditions which the adjoint curves impose on the curves of sufficiently high order(1). We shall give two distinct and independent proofs of this proposition. The first proof is a direct one, based upon a detailed analysis of the singularities of the given curve. This analysis, to which part I is devoted, applies equally well to algebraic number fields, and our treatment will include this case with that of algebraic function fields of one variable. The second proof is more indirect, depending upon the Riemann-Roch theorem and a generalization of the classical representation theorem of the differentials of the first kind. This proof, which will be given in part II, holds only for plane curves whose function field is separably generated over the ground field.