Curve In Complex Plane Research Articles

New Sierpenski Curves in Complex Plane

The Sierpinski triangle also known as Sierpinski gasket is one of the most interesting and the simplest fractal shapes in existence. There are many different and easy ways to generate a Sierpinski triangle. In this paper we have presented a new algorithm for generating the sierpinski gasket using complex variables.

Zeta function of degenerate plane curve singularity

We introduce in this paper a new resolution graph for an isolated complex plane curve singularity and then calculate the monodromy zeta function and the Alexander polynomial for the singularity in terms of this graph.

Algorithms for the implementation of the analytic classification of plane branches

In this paper we present the algorithms used by the second author in Hernandes (2011) for the implementation in MAPLE of the results of Hefez and Hernandes (2011) that allow to perform the analytic classification of irreducible germs of singular complex plane curves.

The Hilbert scheme of a plane curve singularity and the HOMFLY polynomial of its link

The intersection of a complex plane curve with a small three-sphere surrounding one of its singularities is a nontrivial link. The refined punctual Hilbert schemes of the singularity parameterize subschemes supported at the singular point of fixed length and whose defining ideals have a fixed number of generators. We conjecture that the generating function of Euler characteristics of refined punctual Hilbert schemes is the HOMFLY polynomial of the link. The conjecture is verified for irreducible singularities yk=xn whose links are the (k,n) torus knots, and for the singularity y4=x7−x6+4x5y+2x3y2 whose link is the (2,13) cable of the trefoil.

Chebyshev curves, free resolutions and rational curve arrangements

AbstractFirst we construct a free resolution for the Milnor (or Jacobian) algebra M(f) of a complex projective Chebyshev plane curve d : f = 0 of degree d. In particular, this resolution implies that the dimensions of the graded components M(f)k are constant for k ≥ 2d − 3.Then we show that the Milnor algebra of a nodal plane curve C has such a behaviour if and only if all the irreducible components of C are rational.For the Chebyshev curves, all of these components are in addition smooth, hence they are lines or conics and explicit factorizations are given in this case.

On the Enumeration of Complex Plane Curves with Two Singular Points

We study equi-singular strata of plane curves with two singular points of prescribed types. The method of the previous work [Kerner06] is generalized to this case. In particular we consider the enumerative problem for plane curves with two singular points of linear singularity types. First the problem for two ordinary multiple points of fixed multiplicities is solved. Then the enumeration for arbitrary linear types is reduced to the case of ordinary multiple points and to the understanding of "merging" of singular points. Many examples and numerical answers are given.

Symplectic classification of parametric complex plane curves

Based on the discovery that the $\delta $-invariant is the symplectic codimension of a parametric plane curve singularity, we classify the simple and uni-modal singularities of parametric plane curves under symplectic equivalence. A new symplectic deforma

Limits of PGL(3)-translates of plane curves, II

Every complex plane curve C determines a subscheme S of the P 8 of 3×3 matrices, whose projective normal cone (PNC) captures subtle invariants of C . In [P. Aluffi, C. Faber, Limits of PGL(3)-translates of plane curves, I, J. Pure Appl. Algebra 214 (5) (2010) 526–547] we obtain a set-theoretic description of the PNC and thereby we determine all possible limits of families of plane curves whose general element is isomorphic to C . The main result of this article is the determination of the PNC as a cycle; this is an essential ingredient in our computation in [P. Aluffi, C. Faber, Linear orbits of arbitrary plane curves, Michigan Math. J. 48 (2000) 1–37] of the degree of the PGL ( 3 ) -orbit closure of an arbitrary plane curve, an invariant of natural enumerative significance.

Limits of PGL(3)-translates of plane curves, I

We classify all possible limits of families of translates of a fixed, arbitrary complex plane curve. We do this by giving a set-theoretic description of the projective normal cone (PNC) of the base scheme of a natural rational map, determined by the curve, from the P 8 of 3×3 matrices to the P N of plane curves of degree d . In a sequel to this paper we determine the multiplicities of the components of the PNC. The knowledge of the PNC as a cycle is essential in our computation of the degree of the PGL ( 3 ) -orbit closure of an arbitrary plane curve, performed in [P. Aluffi, C. Faber, Linear orbits of arbitrary plane curves, Michigan Math. J. 48 (2000) 1–37].

On the equi-normalizable deformations of singularities of complex plane curves

We study a specific class of deformations of curve singularities: the case when the singular point splits to several ones, such that the total δ invariant is preserved. These are also known as equi-normalizable or equi-generic deformations. We restrict primarily to the deformations of singularities with smooth branches. A natural invariant of the singular type is introduced: the dual graph. It imposes severe restrictions on the possible collisions/deformations. And allows to prove some bounds on the variation of classical invariants in equi-normalizable families. We consider in details deformations of ordinary multiple point, the deformations of a singularity into the collections of ordinary multiple points and deformations of the type x p + y pk into the collections of A k ’s.

The Caporaso–Harris formula and plane relative Gromov-Witten invariants in tropical geometry

Some years ago Caporaso and Harris have found a nice way to compute the numbers N(d, g) of complex plane curves of degree d and genus g through 3d + g − 1 general points with the help of relative Gromov-Witten invariants. Recently, Mikhalkin has found a way to reinterpret the numbers N(d, g) in terms of tropical geometry and to compute them by counting certain lattice paths in integral polytopes. We relate these two results by defining an analogue of the relative Gromov-Witten invariants and rederiving the Caporaso–Harris formula in terms of both tropical geometry and lattice paths.

Vassiliev invariants of quasipositive knots

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button. Quasipositive knots are transverse intersections of complex plane curves with the standard sphere $S^3 \subset {\mathbb C}^2$. It is known that any Alexander polynomial of a knot can be realized by a quasipositive knot. As a consequence, the Alexander polynomial cannot detect quasipositivity. In this paper we prove a similar result about Vassiliev invariants: for any oriented knot $K$ and any natural number $n$ there exists a quasipositive knot $Q$ whose Vassiliev invariants of order less than or equal to $n$ coincide with those of $K$.

Integer points on a curve and the plane Jacobian problem

A polynomial map $F=(P,Q)\in {\mathbb Z} [x,y]^2$ with Jacobian $JF:=P_xQ_y-P_yQ_x\equiv 1$ has a polynomial inverse with integer coefficients if the complex plane curve $P=0$ has infinitely many integer points.

Positive open book decompositions of Stein fillable 3-manifolds along prescribed links

It is known by Loi and Piergallini that a closed, oriented, smooth 3-manifold is Stein fillable if and only if it has a positive open book decomposition. In the present paper we will show that for every link L in a Stein fillable 3-manifold there exists an additional knot L ′ to L such that the link L ∪ L ′ is the binding of a positive open book decomposition of the Stein fillable 3-manifold. To prove the assertion, we will use the divide, which is a generalization of real morsification theory of complex plane curve singularities, and 2-handle attachings along Legendrian curves.

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

Non-triviality of theA–polynomial for knots inS3

The A-polynomial of a knot in S^3 defines a complex plane curve associated to the set of representations of the fundamental group of the knot exterior into SL(2,C). Here, we show that a non-trivial knot in S^3 has a non-trivial A-polynomial. We deduce this from the gauge-theoretic work of Kronheimer and Mrowka on SU_2-representations of Dehn surgeries on knots in S^3. As a corollary, we show that if a conjecture connecting the colored Jones polynomials to the A-polynomial holds, then the colored Jones polynomials distinguish the unknot

Monodromy of real isolated singularities

Complex conjugation on complex space permutes the level sets of a real polynomial function and induces involutions on level sets corresponding to real values. 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.

Plumbing constructions of connected divides and the Milnor fibers of plane curve singularities

A divide is the image of a generic, relative immersion of a finite number of copies of the unit interval or the unit circle into the unit disk. N. A'Campo defined for each connected divide a link in S 3 and proved that the link is fibered. In the present paper we show that the fiber surface of the fibration of a connected divide can be obtained from a disk by a successive plumbing of a finite number of positive Hopf bands. In particular, this gives us a geometric understanding of plumbing constructions of the Milnor fibers of isolated, complex plane curve singularities in terms of certain replacements of the curves of their real morsifications.

Variation of the Milnor Fibration in Pencils of Hypersurface Singularities

Let Φ = (f, g):(Cn+ 1,0) → (C2, 0) be a pair of holomorphic germs with no blowing up in codimension 0. (Two examples are the following: Φ defines an isolated complete intersection singularity; g = lN where ℓ is a generic linear form with respect to f and N> 0.) We study how the Milnor fibrations of the germs ϕ(α:β) = α g-β f are related to each other when (α:β) varies in P1. More precisely, we construct isotopic subfibrations or subfibres of the Milnor fibrations of any two such germs. The proofs are based on the precise study of the subdiscs of complex lines meeting a fixed complex plane curve germ transversally, generalizing Lê's work on the Cerf diagram. 2000 Mathematical Subject Classification: 32S55, 32S15, 32S30.

Symmetry types of hyperelliptic Riemann surfaces

Let $X$ be a compact hyperelliptic Riemann surface which admits anti-analytic involutions (also called symmetries or real structures). For instance, a complex projective plane curve of genus two, defined by an equation with real coefficients, gives rise to such a surface, and complex conjugation is such a symmetry. In this memoir, the real structures $\tau$ of $X$ are classified up to isomorphism (i.e., up to conjugation). This is done as follows: the number of connected components of the set of fixed points of $\tau$ together with the connectedness or disconnectedness of the complementary set in $X$ classifies $\tau$ topologically; they determine the species of $\tau$, which only depends on the conjugacy class of $\tau$ (however, different conjugacy classes may have identical species). On these grounds, for a given genus $g\ge2$, the authors first give a list of all full groups of analytic and anti-analytic automorphisms of genus $g$ compact hyperelliptic Riemann surfaces. For every such group $G$, the authors compute polynomial equations for a surface $X$ having $G$ as full group and then find the number of conjugacy classes containing symmetries; they also compute a representative $\tau$ in every such class. Finally, they compute the species corresponding to such classes. This memoir is an exhaustive piece of work, going through a case-by-case analysis. The problem for general compact Riemann surfaces dates back to 1893, when {\it F. Klein} [Math. Ann. 42, 1--29 (1893)] first studied it. For zero genus, it is easy. For genus one, that is, for elliptic surfaces, it was solved by {\it N. Alling} [Real elliptic curves (1981)]. Partial results for hyperelliptic surfaces of genus two were obtained by {\it E. Bujalance} and {\it D. Singerman} [Proc. Lond. Math. Soc. 51, 501--519 (1985)].