Irreducible Plane Research Articles

Distinct Distances on Algebraic Curves in the Plane

LetSbe a set ofnpoints in${\mathbb R}^{2}$contained in an algebraic curveCof degreed. We prove that the number of distinct distances determined bySis at leastcdn4/3, unlessCcontains a line or a circle.We also prove the lower boundcd′ min{m2/3n2/3,m2,n2} for the number of distinct distances betweenmpoints on one irreducible plane algebraic curve andnpoints on another, unless the two curves are parallel lines, orthogonal lines, or concentric circles. This generalizes a result on distances between lines of Sharir, Sheffer and Solymosi in [19].

Decompositions of the higher order polars of plane branches

Abstract In [1] Casas-Alvero found decompositions of higher order polars of an irreducible plane complex analytic curve generalizing the results of Merle. We improve his result obtaining a finer decomposition where we find out a kind of branches that we call threshold semi-roots. The existence of threshold semi-roots is a new phenomenon observed for the higher order polars. The topological type and the number of these branches is determined by the topological type of the original curve.

Geography of irreducible plane sextics

We complete the equisingular deformation classification of irreducible singular plane sextic curves. As a by-product, we also compute the fundamental groups of the complement of all but a few maximizing sextics.

On fundamental groups of plane curve complements

In this paper we discuss some properties of fundamental groups and Alexander polynomials of plane curves. We discuss the relationship of the non-triviality of Alexander polynomials and the notion of (nearly) freeness for irreducible plane curves. We reprove and restate in modern terms a somewhat forgotten result of Zariski. Finally, we describe some topological properties of curves with abelian fundamental group.

Parametric equations of plane sextic curves with a maximal set of double points

We give explicit parametric equations for all irreducible plane projective sextic curves which have at most double points and whose total Milnor number is maximal (is equal to 19). In each case we find a parametrization over a number field of the minimal possible degree and try to choose coordinates so that the coefficients are as small as we can do.

On classifications of rational sextic curves

Abstract In this paper, we extend the Yang’s list of reduced sextic plane curves to rational irreducible projective plane curves of type (6, 3, 1).

An Algorithm to Compute the Adjoint Ideal of an Affine Plane Algebraic Curve

Let \({\fancyscript{C}}\) be an irreducible affine plane algebraic curve defined by a polynomial \({f \in K[x, y]}\). Assume that f is monic with respect to y and that the function field \({K(\fancyscript{C})}\) of the curve is separable over K(x). In this paper we give a simple algorithm to compute a lexicographic Groebner basis of the adjoint ideal \({{\rm Adj}(\fancyscript{C})}\) of \({\fancyscript{C}}\) from a basis, as a K[x]-module, of the integral closure \({K[\overline{\fancyscript{C}}]}\) of the coordinate ring \({K[\fancyscript{C}]}\).

Galois points for a plane curve in characteristic two

Let C be an irreducible plane curve. A point P in the projective plane is said to be Galois with respect to C if the function field extension induced by the projection from P is Galois. We denote by δ′(C) the number of Galois points contained in P2∖C. In this article we will present two results with respect to determination of δ′(C) in characteristic two. First we determine δ′(C) for smooth plane curves of degree a power of two. In particular, we give a new characterization of the Klein quartic in terms of δ′(C). Second we determine δ′(C) for a generalization of the Klein quartic, which is related to an example of Artin–Schreier curves whose automorphism group exceeds the Hurwitz bound. This curve has many Galois points.

Constructing the set of complete intersection numerical semigroups with a given Frobenius number

Delorme suggested that the set of all complete intersection numerical semigroups can be computed recursively. We have implemented this algorithm, and particularized it to several subfamilies of this class of numerical semigroups: free and telescopic numerical semigroups, and numerical semigroups associated to an irreducible plane curve singularity. The recursive nature of this procedure allows us to give bounds for the embedding dimension and for the minimal generators of a semigroup in any of these families.

ON FINITE BRANCHED UNIFORMIZATIONS OF THE PROJECTIVE PLANE

We give a brief survey of the so-called Fenchel's problem for the projective plane, that is the problem of existence of finite Galois coverings of the complex projective plane branched along a given divisor and prove the following result: Let p, q be two integers greater than 1 and C be an irreducible plane curve. If there is a surjection of the fundamental group of the complement of C into a free product of cyclic groups of orders p and q, then there is a finite Galois covering of the projective plane branched along C with any given branching index.

On the approximate Jacobian Newton diagrams of an irreducible plane curve

We introduce the notion of an approximate jacobian Newton diagram which is the jacobian Newton diagram of the morphism $(f^{(k)},f)$, where $f$ is a branch and $f^{(k)}$ is a characteristic approximate root of $f$. We prove that the set of all approximate jacobian Newton diagrams is a complete topological invariant. This generalizes theorems of Merle and Ephraim about the decomposition of the polar curve of a branch.

CLASSICAL ZARISKI PAIRS

We enumerate and classify up to equisingular deformation all irreducible plane sextics constituting the so called classical Zariski pairs. In most cases we obtain two deformation families, called abundant and non-abundant. Four sets of singularities are realized by abundant sextics only, and one exceptional set of singularities is realized by three families, one abundant and two complex conjugate non-abundant. This exceptional set of singularities has submaximal total Milnor number 18.

Hilbert–Kunz theory for nodal cubics, via sheaves

Suppose B = F [ x , y , z ] / h is the homogeneous coordinate ring of a characteristic p degree 3 irreducible plane curve C with a node. Let J be a homogeneous ( x , y , z ) -primary ideal and n → e n be the Hilbert–Kunz function of B with respect to J. Let q = p n . When J = ( x , y , z ) , it is known that e n = 7 3 q 2 − 1 3 q − R where R = 5 3 if q ≡ 2 ( 3 ) , and is 1 otherwise. We generalize this, showing that e n = μ q 2 + α q − R where R only depends on q mod 3 . We describe α and R in terms of classification data for a vector bundle on C.

Apéry base and polar invariants for real valuations

The main objective of this paper is to study real valuations μ defined on the polynomial ring K[T] with coefficients in a field K such that the restriction of μ to K is a proper valuation on K. For this, we consider the Apery base and the iterated sequence of valuations associated with μ and we show the main properties and the relation between both invariants. We also prove a factorization theorem for elements \({f\in K[T]}\) of the suitable degree such that μ(f) is in the Apery base of μ, obtaining strong properties of their irreducible factors. In particular, some results of M. Merle, R. Ephraim and A. Granja for irreducible algebroid plane curve singularity are a special case of this theorem.

Radical parametrizations of algebraic curves by adjoint curves

We present algorithms for parametrizing by radicals an irreducible curve, not necessarily plane, when the genus is less than or equal to 4 and the curve is defined over an algebraically closed field of characteristic zero. In addition, we also present an algorithm for parametrizing by radicals any irreducible plane curve of degree d having at least a point of multiplicity d−r, with 1≤r≤4 and, as a consequence, every irreducible plane curve of degree d≤5 and every irreducible singular plane curve of degree 6.

Pencils of irreducible rational curves and plane Jacobian conjecture

Pencils of irreducible rational curves and plane Jacobian conjecture

Birational classification of curves on rational surfaces

AbstractIn this paper we consider the birational classification of pairs (S, ℒ), withSa rational surface andℒa linear system onS. We give a classification theorem for such pairs, and we determine, for each irreducible plane curveB, itsCremona minimalmodels, that is, those plane curves which are equivalent toBvia a Cremona transformation and have minimal degree under this condition.

Birational classification of curves on rational surfaces

AbstractIn this paper we consider the birational classification of pairs (S, ℒ), with S a rational surface and ℒ a linear system on S. We give a classification theorem for such pairs, and we determine, for each irreducible plane curve B, its Cremona minimal models, that is, those plane curves which are equivalent to B via a Cremona transformation and have minimal degree under this condition.

Algebraic curves with a large non-tame automorphism group fixing no point

Let K be an algebraically closed field of characteristic p > 0, and let X be a curve over K of genus g ≥ 2. Assume that the automorphism group Aut(X) of X over K fixes no point of X. The following result is proven. If there is a point P on X whose stabilizer in Aut (X) contains a p-subgroup of order greater than gp/(p— 1), then X is birationally equivalent over K to one of the irreducible plane curves (II), (III), (IV), (V) listed in the Introduction.

Geometry of irreducible plane quartics and their quadratic residue conics

Dedicated to Professor Du Plessis on his sixtieth birthday. Abstract. Let D be an irreducible plane curve in P 2 . In this article, we rst introduce a notion of a quadratic residue curve modD, and study quadratic residue conics C mod an irreducible quartic curve Q. As an application, we study a dihedral cover of P 2 with branch locus C + Q and give two examples of Zariski pairs as by-products.