Cuspidal Curves Research Articles

On Chisini's conjecture. II

We prove that if is a smooth projective surface and is a generic linear projection branched over a cuspidal curve , then is uniquely determined (up to isomorphism) by .

Level Structures on the Weierstrass Family of Cubics

Let W → 𝔸 2 be the universal Weierstrass family of cubic curves over ℂ. For each N ≥ 2, we construct surfaces parameterizing the three standard kinds of level N structures on the smooth fibers of W. We then complete these surfaces to finite covers of 𝔸 2. Since W → 𝔸 2 is the versal deformation space of a cusp singularity, these surfaces convey information about the level structure on any family of curves of genus g degenerating to a cuspidal curve. Our goal in this note is to determine for which values of N these surfaces are smooth over (0, 0). From a topological perspective, the results determine the homeomorphism type of certain branched covers of S 3 with monodromy in SL2 (ℤ/N).

Cuspidal curves contained in a given surface

Cuspidal curves contained in a given surface

Kummer surfaces for the self-product of the cuspidal rational curve

The classical Kummer construction attaches a K3 surface to an abelian surface. As Shioda and Katsura showed, this construction breaks down for supersingular abelian surfaces in characteristic two. Replacing supersingular abelian surfaces by the self-product of the rational cuspidal curve, and the sign involution by suitable infinitesimal group scheme actions, we give the correct Kummer-type construction for this situation. We encounter rational double points of type D 4 D_4 and D 8 D_8 instead of type A 1 A_1 . It turns out that the resulting surfaces are supersingular K3 surfaces with Artin invariant one and two. They lie in a 1-dimensional family obtained by simultaneous resolution, which exists after purely inseparable base change.

Invariants of real symplectic four-manifolds out of reducible and cuspidal curves

Nous construisons des invariants par deformation des varietes symplectiques reelles de dimension quatre. Ces invariants sont obtenus en comptant trois differents types de courbes J-holomorphes rationnelles reelles qui realisent une classe d'homologie donnee et passent par une configuration reelle donnee d'un nombre (adequat) de points. Ces courbes sont des courbes cuspidales, reductibles et des courbes ayant une tangente prescrite en l'un des points de la configuration. Elles sont comptees en fonction d'un signe qui depend de la parite du nombre de leurs points doubles reels isoles et, dans le cas des courbes reductibles, en fonction d'une multiplicite. Dans le cas du plan projectif complexe muni de ses formes symplectiques et structures reelles standards, ces invariants coincident avec ceux precedemment construits dans [11]. Ceci mene a une relation entre le comptage de courbes J-holomorphes rationnelles reelles realise dans [11] et le comptage de courbes J-holomorphes rationnelles reductibles reelles presente ici.

ON RATIONAL CUSPIDAL PROJECTIVE PLANE CURVES

In 2002, L. Nicolaescu and the fourth author formulated a very general conjecture which relates the geometric genus of a Gorenstein surface singularity with rational homology sphere link with the Seiberg--Witten invariant (or one of its candidates) of the link. Recently, the last three authors found some counterexamples using superisolated singularities. The theory of superisolated hypersurface singularities with rational homology sphere link is equivalent with the theory of rational cuspidal projective plane curves. In the case when the corresponding curve has only one singular point one knows no counterexample. In fact, in this case the above Seiberg--Witten conjecture led us to a very interesting and deep set of `compatibility properties' of these curves (generalising the Seiberg--Witten invariant conjecture, but sitting deeply in algebraic geometry) which seems to generalise some other famous conjectures and properties as well (for example, the Noether--Nagata or the log Bogomolov--Miyaoka--Yau inequalities). Namely, we provide a set of `compatibility conditions' which conjecturally is satisfied by a local embedded topological type of a germ of plane curve singularity and an integer $d$ if and only if the germ can be realized as the unique singular point of a rational unicuspidal projective plane curve of degree $d$. The conjectured compatibility properties have a weaker version too, valid for any rational cuspidal curve with more than one singular point. The goal of the present article is to formulate these conjectured properties, and to verify them in all the situations when the logarithmic Kodaira dimension of the complement of the corresponding plane curves is strictly less than 2.

On the number of the cusps of cuspidal plane curves

AbstractIn this paper, we estimate an upper bound of the number of the cusps of a cuspidal plane curve. We prove that a cuspidal plane curve of genus g has no more than (21g +17)/2 cusps. For example, a rational cuspidal plane curve has no more than 8 cusps and an elliptic one has no more than 19 cusps. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Regular homotopy of Hurwitz curves

We prove that any two irreducible cuspidal Hurwitz curves and (or, more generally, two curves with -type singularities) in the Hirzebruch surface with the same homology classes and sets of singularities are regular homotopic. Moreover, they are symplectically regular homotopic if and are symplectic with respect to a compatible symplectic form.

A factorization formula for the full twist of double the number of strings

We give a formula for factorizing the full twist in the braid group in terms of four factorizations of the full twist in. This formula is used to construct a symplectic 4-manifold and two regularly homotopic generic coverings branched along cuspidal Hurwitz curves (without negative nodes) having different braid monodromy factorization types. The class of fundamental groups of complements of affine plane Hurwitz curves is described in terms of generators and defining relations.

On braid monodromy factorizations

We introduce and develop a language of semigroups over the braid groups to study the braid monodromy factorizations (bmf's) of plane algebraic curves and other related objects. As an application, we give a new proof of Orevkov's theorem on the realization of bmf's over a disc by algebraic curves and show that the complexity of such a realization cannot be bounded in terms of the types of factors of the bmf. We also prove that the type of a bmf distinguishes Hurwitz curves with singularities of inseparable type up to -isotopy and -holomorphic cuspidal curves in up to symplectic isotopy.

On rational cuspidal curves

Let \(C\subset\bold{P}^2\) be a rational curve of degree d which has only one analytic branch at each point. Denote by m the maximal multiplicity of singularities of C. It is proven in [MS] that \(d \alpha m\) and this estimate is sharp. The main tool used here, is the logarithmic version of the Bogomolov-Miyaoka-Yau inequality. For curves as above we give an interpretation of this inequality in terms of the number of parameters describing curves of a given degree and the number of conditions imposed by singularity types.

Diffeomorphisms, isotopies, and braid monodromy factorizations of plane cuspidal curves

In this paper we prove that there is an infinite sequence of pairs of plane cuspidal curves, C m,1 and C m,2 , of degree deg( C m,1 )=deg( C m,2 )→∞, such that the pairs ( CP 2,C m,1) and ( CP 2,C m,2) are diffeomorphic, but C m,1 and C m,2 have non-equivalent braid monodromy factorizations. These curves give rise to the negative solutions of “Dif ⇒ Def” and “Dif ⇒ Iso” problems for plane irreducible cuspidal curves. In our examples, C m,1 and C m,2 are complex conjugate.

A focus on focal surfaces

Congruences of lines in P3, i.e. two-dimensional families of lines, and their focal surfaces, have been a popular object of study in classical algebraic geometry. They have been considered recently by several authors as Arrondo, Goldstein, Sols, Verra. Aim of the paper under review is to study from the modern point of view the notions appearing in this context, so to prove in a rigorous way some classical results. More precisely, a congruence is a surface X G(1, 3), its focal locus F, in general a surface, is the branch locus of the natural projection qX from the incidence correspondence IX to P3. It results that a general line of X contains two foci counted with multiplicity, a line L whose points are all foci is called a focal line. In the paper under review the authors compute several invariants of the focal surface of a smooth congruence, e.g. its degree, class and sectional genus, and the degree of its nodal and cuspidal curves (assuming that these are the only 1-dimensional components of its singular locus). Then they study in detail some important examples of congruences, i.e. the bisecant lines of a smooth curve C in P3, the bitangents and the inflectional tangents of a smooth surface . In the first case they find that in general the focal surface F is all formed by focal lines, which are the stationary bisecants. In the other examples they find, among other results, that is a component of the focal surface with high multiplicity, and that at least one component of F is formed by focal lines.Motivated by the examples they state a series of conjectures about congruences whose focal surface is not irreducible or not reduced, and about singularities of congruences of bitangents or inflectionary tangents to not necessarily smooth surfaces of P3.

Defining equations¶of certain rational cuspidal curves. I

In this paper, we describe the defining equations of rational cuspidal plane curves having exactly one cusp such that the tangent line at the cusp intersects only at the cusp.

Rational cuspidal curves of type $(d,d-2)$ with one or two cusps

Rational cuspidal curves of type $(d,d-2)$ with one or two cusps

Braid monodromy factorizations and diffeomorphism types

In this paper we prove that if two cuspidal plane curves and have equivalent braid monodromy factorizations, then and are smoothly isotopic in . As a consequence, we obtain that if , are surfaces of general type embedded in a projective space by means of a multiple canonical class and if the discriminant curves (the branch curves) , of some smooth projections of , to have equivalent braid monodromy factorizations, then and are diffeomorphic (as real four-dimensional manifolds).

Rational Cuspidal Plane Curves of Type (d,d — 3)

In the previous paper [E-print alg-geom/9507004] we classified the rational cuspidal plane curves C with a cusp of multiplicity deg C - 2. In particular, we showed that any such curve can be transformed into a line by Cremona transformations. Here we do the same for the rational cuspidal plane curves C with a cusp of multiplicity deg C - 3.

Base point free pencils of small degree on certain smooth curves

We give sufficient conditions on numbers d and m such that a linear system of degree m on the normalization C of a plane curve \(\overline {C}\) of degree d which is in a certain sense not too singular is in the natural way induced by either a pencil of lines or a pencil of conics in the plane. Those results generalize results on nodal and cuspidal plane curves and seem to complement the recent results of [2]. We present a new approach via the geometry of curves in \({\Bbb P}_1\times {\Bbb P}_2\).

On Chisini's conjecture

Chisini's conjecture asserts that if is a cuspidal curve, then a generic morphism , , of a smooth projective surface to branched along is unique up to isomorphism. In this paper we prove that Chisini's conjecture is true for if is greater than the value of some function depending on the degree, genus and the number of cusps of . This inequality holds for almost all generic morphisms. In particular, on a surface with ample canonical class, it holds for generic morphisms defined by a linear subsystem of the -canonical class, .Moreover, we present examples of pairs (, ) of plane cuspidal curves such that (i) , and these curves have homeomorphic tubular neighbourhoods in , but the pairs and are not homeomorphic;(ii) is the discriminant curve of a generic morphism , , where are surfaces of general type;(iii) the surfaces and are homeomorphic (as four-dimensional real manifolds);(iv) the morphism is defined by a three-dimensional linear subsystem of the -canonical class of .

Examples of non-degenerate and degenerate cuspidal loops in planar systems

In this paper we study some examples of non-degenerate and degenerate cuspidal loops in planar systems. A cuspidal loop is a codimension-three homoclinic orbit given by the intersection of the separatrices of an equilibrium of cusp type. Using a Dulac map analysis and asymptotic expansions we study the stability in the neighbourhood of a cusp point. As a first example we consider an enzyme-catalysed reaction model exhibiting a non-degenerate cuspidal loop. All the codimension-one and-two homoclinic bifurcations present in the unfolding of the corresponding cuspidal loop are found in such a realistic model. Finally, the unfolding of a codimension-five Bogdanov-Takens bifurcation is analysed. A degenerate codimension-four cuspidal loop appearing in this system is located on a non-degenerate cuspidal loop curve, and part of the unfolding of such singularity is shown.