Affine Plane Curves Research Articles

On free and plus-one generated curves arising from free curves by addition–deletion of a line

In a recent paper, after introducing the notion of plus-one generated hyperplane arrangements, Takuro Abe has shown that if we add (respectively, delete) a line to (respectively, from) a free line arrangement, then the resulting line arrangement is either free or plus-one generated. In this paper, we prove that the same properties hold when we replace the line arrangement by a free curve and add (respectively, delete) a line. The proof uses a new version of a key result due originally to H. Schenck, H. Terao and M. Yoshinaga, in which no quasi-homogeneity assumption is needed. Two conjectures about the Tjurina number of a union of two plane curve singularities are also stated. As a geometric application, we show that, under a mild numerical condition, the projective closure of a contractible, irreducible affine plane curve is either free or plus-one generated, using a deep result due to U. Walther.

Frobenius methods for analytic second order linear partial differential equations

The main subject of this text is the study of analytic second order linear partial differential equations. We aim to solve the classical equations and some more, in the real or complex analytical case. This is done by introducing methods inspired by the Frobenius method for second order linear ordinary differential equations. We introduce a notion of Euler type partial differential equation. To such a PDE we associate an indicial conic, which is an affine plane curve of degree two. Then comes the concept of regular singularity and finally convergence theorems, which must necessarily take into account the type of PDE (parabolic, elliptical or hyperbolic) and a nonresonance condition. This condition gives a new geometric interpretation of the original condition between the roots of the original Frobenius theorem for second order ODEs. The interpretation is something like, a certain reticulate has or not vertices on the indexical conic. Finally, we retrieve the solution of all the classical PDEs by this method (heat diffusion, wave propagation and Laplace equation), and also increase the class of those that have explicit algorithmic solution to far beyond those admitting separable variables. The last part of the text is dedicated to the construction of PDE models for the classical ODEs like Airy, Legendre, Laguerre, Hermite and Chebyshev by two different means. One model is based on the requirement that the restriction of the PDE to lines through the origin must be the classical ODE model. The second is based on the idea of having symmetries on the PDE model and imitating the ODE model. We study these PDEs and obtain their solutions, obtaining for the framework of PDEs some of the classical results, like existence of polynomial solutions (Laguerre, Hermite and Chebyshev polynomials).

Hilbert curves of scrolls over threefolds

Let (X,L) be a complex polarized n-fold with the structure of a classical scroll over a smooth projective threefold Y. The Hilbert curve of such a pair (X,L) is a complex affine plane curve of degree n, consisting of n−3 evenly spaced parallel lines plus a cubic. This paper is devoted to a detailed study of this cubic. In particular, existence of triple points, behavior with respect to the line at infinity, and non-reducedness, are analyzed in connection with the structure of (X,L). Special attention is reserved to the case n=4, where various examples are presented and the possibility that the cubic is itself the Hilbert curve of the base threefold Y for a suitable polarization is discussed.

Vanishing of higher order Alexander‐type invariants of plane curves

AbstractThe higher order degrees are Alexander‐type invariants of complements to an affine plane curve. In this paper, we characterize the vanishing of such invariants for a curve C given as a transversal union of plane curves and in terms of the finiteness and the vanishing properties of the invariants of and , and whether or not they are irreducible. As a consequence, we prove that the multivariable Alexander polynomial is a power of , and we characterize when in terms of the defining equations of and . Our results impose obstructions on the class of groups that can be realized as fundamental groups of complements of a transversal union of curves.

Higher order degrees of affine plane curve complements

We study finiteness (and vanishing) properties of the higher order degrees associated to complements of complex affine plane curves with mild singularities at infinity. Our results impose new obstructions on the class of groups that can be realized as fundamental groups of affine plane curve complements. We also clarify the relationship between the higher order degrees and the multivariable Alexander polynomial of a non-irreducible plane curve.

Characterizing some polarized Fano fibrations via Hilbert curves

The Hilbert curve of a complex polarized manifold [Formula: see text] is the complex affine plane curve of degree [Formula: see text] defined by the Hilbert-like polynomial [Formula: see text], where [Formula: see text] is the canonical bundle of [Formula: see text] and [Formula: see text] and [Formula: see text] are regarded as complex variables. A natural expectation is that this curve encodes several properties of the pair [Formula: see text]. In particular, the existence of a fibration of [Formula: see text] over a variety of smaller dimension induced by a suitable adjoint bundle to [Formula: see text] translates into the fact that the Hilbert curve has a quite special shape. Along this line, Hilbert curves of special varieties like Fano manifolds with low coindex, as well as fibrations over low-dimensional varieties having such a manifold as general fiber, endowed with appropriate polarizations, are investigated. In particular, several polarized manifolds relevant for adjunction theory are completely characterized in terms of their Hilbert curves.

Some results on open algebraic surfaces of logarithmic Kodaira dimension zero

Let S be a smooth open algebraic surface of logarithmic Kodaira dimension zero and (V,D) a pair of a smooth projective surface V and a simple normal crossing divisor D on V such that V−D≅S. We prove that ρ(V)−#D≥−2, where #D is the number of all irreducible components of Supp D, and determine the surface S with ρ(V)−#D=−2 and those with ρ(V)−#D=−1 and p‾g(S)=0. Furthermore, by using these results, we determine the affine plane curves whose complements have logarithmic Kodaira dimension zero.

Exceptional isomorphisms between complements of affine plane curves

This article describes the geometry of isomorphisms between complements of geometrically irreducible closed curves in the affine plane $\mathbb{A}^2$, over an arbitrary field, which do not extend to an automorphism of $\mathbb{A}^2$. We show that such isomorphisms are quite exceptional. In particular, they occur only when both curves are isomorphic to open subsets of the affine line $\mathbb{A}^1$, with the same number of complement points, over any field extension of the ground field. Moreover, the isomorphism is uniquely determined by one of the curves, up to left composition with an automorphism of $\mathbb{A}^2$, except in the case where the curve is isomorphic to the affine line $\mathbb{A}^1$ or to the punctured line $\mathbb{A}^1 \setminus \{0\}$. If one curve is isomorphic to $\mathbb{A}^1$, then both curves are in fact equivalent to lines. In addition, for any positive integer $n$, we construct a sequence of $n$ pairwise non-equivalent closed embeddings of $\mathbb{A}^1 \setminus \{0\}$ with isomorphic complements. In characteristic~$0$ we even construct infinite sequences with this property. Finally, we give a geometric construction that produces a large family of examples of non-isomorphic geometrically irreducible closed curves in $\mathbb{A}^2$ that have isomorphic complements, answering negatively the Complement Problem posed by Hanspeter Kraft.. This also gives a negative answer to the holomorphic version of this problem in any dimension $n \geq 2$. The question had been raised by Pierre-Marie Poloni.

A property of Hilbert curves of scrolls over surfaces

ABSTRACTLet (X,L) be a polarized manifold of dimension n. Its Hilbert curve is an affine algebraic plane curve of degree n encoding properties related to fibrations of X, defined by suitable adjoint linear systems to L. In particular, if (X,L) is a scroll over a smooth surface S, its Hilbert curve consists of n−2 parallel lines with a given slope and evenly spaced, plus a conic. Making its equation explicit, we show that this conic turns out to be itself the Hilbert curve of the ℚ-polarized surface , where ℰ is the rank-(n−1) vector bundle obtained by pushing down L via the scroll projection, if and only if ℰ is properly semistable in the sense of Bogomolov.

Algebraic conditions for additive functions over the reals and over finite fields

Let C be an affine plane curve. We consider additive functions $$f{:}\; K\rightarrow K$$ for which $$f(x)f(y)=0$$ , whenever $$(x,y)\in C$$ . We show that if $$K=\mathbb {R}$$ and C is the hyperbola with defining equation $$xy=1$$ , then there exist nonzero additive functions with this property. Moreover, we show that such a nonzero f exists for a field K if and only if K is transcendental over $$\mathbb Q$$ or over $$\mathbb {F}_p$$ , the finite field with p elements. We also consider the general question when K is a finite field. We show that if the degree of the curve C is large enough compared to the characteristic of K, then f must be identically zero.

Exact evaluation of second moments associated with some families of curves over a finite field

Let Fq be the finite field with q elements. Given an N-tuple Q∈FqN, we associate with it an affine plane curve CQ over Fq. We consider the distribution of the quantity q−#Cq,Q where #Cq,Q denotes the number of Fq-points of the affine curve CQ, for families of curves parameterized by Q. Exact formulae for first and second moments are obtained in several cases when Q varies over a subset of FqN. Families of Fermat type curves, Hasse–Davenport curves and Artin–Schreier curves are also considered and results are obtained when Q varies along a straight line.

A generalization of the Bombieri–Pila determinant method

The so-called determinant method was developed by Bombieri and Pila in 1989 for counting integral points of bounded height on affine plane curves. In this paper we give a generalization of that method to varieties of higher dimension, yielding a proof of Heath-Brown's 'Theorem 14' by real-analytic considerations alone.

Rational parametrization of conchoids to algebraic curves

We study the rationality of each of the components of the conchoid to an irreducible algebraic affine plane curve, excluding the trivial cases of the lines through the focus and the circle centered at the focus and radius the distance involved in the conchoid. We prove that conchoids having all their components rational can only be generated by rational curves. Moreover, we show that reducible conchoids to rational curves have always their two components rational. In addition, we prove that the rationality of the conchoid component, to a rational curve, does depend on the base curve and on the focus but not on the distance. As a consequence, we provide an algorithm that analyzes the rationality of all the components of the conchoid and, in the affirmative case, parametrizes them. The algorithm only uses a proper parametrization of the base curve and the focus and, hence, does not require the previous computation of the conchoid. As a corollary, we show that the conchoid to the irreducible conics, with conchoid-focus on the conic, are rational and we give parametrizations. In particular we parametrize the Limaçons of Pascal. We also parametrize the conchoids of Nicomedes. Finally, we show how to find the foci from where the conchoid is rational or with two rational components.

Approximate parametrization of plane algebraic curves by linear systems of curves

It is well known that an irreducible algebraic curve is rational (i.e. parametric) if and only if its genus is zero. In this paper, given a tolerance ϵ > 0 and an ϵ-irreducible algebraic affine plane curve C of proper degree d, we introduce the notion of ϵ-rationality, and we provide an algorithm to parametrize approximately affine ϵ-rational plane curves by means of linear systems of ( d − 2 ) -degree curves. The algorithm outputs a rational parametrization of a rational curve C ¯ of degree d which has the same points at infinity as C . Moreover, although we do not provide a theoretical analysis, our empirical analysis shows that C ¯ and C are close in practice.

Partial degree formulae for plane offset curves

In this paper we present several formulae for computing the partial degrees of the defining polynomial of the offset curve to an irreducible affine plane curve given implicitly, and we see how these formulae particularize to the case of rational curves. In addition, we present a formula for computing the degree w.r.t. the distance variable.

An algebraic analysis of conchoids to algebraic curves

We study the conchoid to an algebraic affine plane curve $${\mathcal C}$$from the perspective of algebraic geometry, analyzing their main algebraic properties. Beside $${\mathcal C}$$, the notion of conchoid involves a point A in the affine plane (the focus) and a non-zero field element d (the distance). We introduce the formal definition of conchoid by means of incidence diagrams. We prove that the conchoid is a 1-dimensional algebraic set having at most two irreducible components. Moreover, with the exception of circles centered at the focus A and taking d as its radius, all components of the corresponding conchoid have dimension 1. In addition, we introduce the notions of special and simple components of a conchoid. Furthermore we state that, with the exception of lines passing through A, the conchoid always has at least one simple component and that, for almost every distance, all the components of the conchoid are simple. We state that, in the reducible case, simple conchoid components are birationally equivalent to $${\mathcal C}$$, and we show how special components can be used to decide whether a given algebraic curve is the conchoid of another curve.

Equi-affine vector fields on manifold with equi-affine structure

The aim is to extend the theory of affine plane curves on R 2 , to the manifolds of 2 and n dimensions. Affine arclength , affine curvatures, and equi-affine vector fields are defined on a manifold of dimension n with equi-affine structure. Classification of equi-affine vector fields on two dimensional equi-affine structure is obtained. Mathematics Subject Classification: 53A15

Degeneration of hyperbolic structures on the figure-eight knot complement and points of finite order on an elliptic curve

In this paper, we consider a hyperbolic structure on a manifold to be a Riemannian metric of constant sectional curvature −1 which is not necessarily complete. A hyperbolic 3-manifold is an orientable 3-manifold with a hyperbolic structure. It is well known that the complement of the figure-eight knot K in the 3-dimensional sphere S admits a complete, finite volume hyperbolic structure σ∞. By Mostow-Prasad rigidity, such a hyperbolic structure on S − K is unique. Incomplete hyperbolic structures are also of interest. Indeed, Thurston [8] analyzed the flexibility of hyperbolic structures on S − K by allowing incomplete hyperbolic structures. In fact, he showed that there are deformations of σ∞ on S3−K that are not complete hyperbolic structures. Such deformations are holomorphically parametrized by points in an open subset U of a complex affine plane curve C. The curve C and this subset U are given in (1) and (2) below. When the parameter of deformation becomes close to the boundary ∂U or enters C−U , degeneration of hyperbolic structures on S3−K occurs. There are cases in which such degeneration results in closed 3-manifolds obtained by Dehn fillings of S −K with other types of geometric structure. In fact, twenty such cases have been identified. In each case, this resultant closed 3-manifold is one of the following types: a Sol-manifold, a PSL2(R)-manifold, a Haken manifold which is decomposed into a PSL2(R)-manifold and a Euclidean manifold along an embedded torus, or a Euclidean orbifold. The genus of the curve C is one, because C is of degree four and has two ordinary double points. In fact, we give an explicit form of a birational map from C to a non-singular plane cubic curve in Weierstrass form E, which is given in (4) below. The curve E is an example of what is called an ‘elliptic curve defined over Q’. It is well known that any elliptic curve is an abelian group under an addition law. In this paper, we see that there is a concrete

Degree formulae for offset curves

In this paper, we present three different formulae for computing the degree of the offset of a real irreducible affine plane curve C given implicitly, and we see how these formulae particularize to the case of rational curves. The first formula is based on an auxiliary curve, called S , that is defined depending on a non-empty Zariski open subset of R 2 . The second formula is based on the resultant of the defining polynomial of C , and the polynomial defining generically S . The third formula expresses the offset degree by means of the degree of C and the multiplicity of intersection of C and the hodograph H to C , at their intersection points.

Braid monodromy and topology of plane curves

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