Showing papers on "Plane curve published in 2003"

When Does the Position Vector of a Space Curve Always Lie in Its Rectifying Plane

Abstract: (2003). When Does the Position Vector of a Space Curve Always Lie in Its Rectifying Plane? The American Mathematical Monthly: Vol. 110, No. 2, pp. 147-152.

Welschinger invariant and enumeration of real rational curves

Abstract: Welschinger’s invariant bounds from below the number of real rational curves through a given generic collection of real points in the real projective plane. We estimate this invariant using Mikhalkin’s approach which deals with a corresponding count of tropical curves. In particular, our estimate implies that, for any positive integer d, there exists a real rational curve of degree d through any collection of 3d − 1 real points in the projective plane, and, moreover, asymptotically in the logarithmic scale at least one third of the complex plane rational curves through a generic point collection are real. We also obtain similar results for curves on other toric Del Pezzo surfaces.

Luttinger surgery along Lagrangian tori and non-isotopy for singular symplectic plane curves

Abstract: We discuss the properties of a certain type of Dehn surgery along a Lagrangian torus in a symplectic 4-manifold, known as Luttinger's surgery, and use this construction to provide a purely topological interpretation of a non-isotopy result for symplectic plane curves with cusp and node singularities due to Moishezon [9].

The Alexander polynomial of a plane curve singularity via the ring of functions on it

Abstract: We prove two formulae that express the Alexander polynomial $\Delta\sp C$ of several variables of a plane curve singularity $C$ in terms of the ring $\mathscr {O}\sb C$ of germs of analytic functions on the curve. One of them expresses $\Delta\sp C$ in terms of dimensions of some factors corresponding to a (multi-indexed) filtration on the ring $\mathscr {O}\sb C$. The other one gives the coefficients of the Alexander polynomial $\Delta\sp C$ as Euler characteristics of some explicitly described spaces (complements to arrangements of projective hyperplanes).

Logarithmic Orbifold Euler Numbers of Surfaces with Applications

Abstract: We introduce orbifold Euler numbers for normal surfaces with boundary -divisors. These numbers behave multiplicatively under finite maps and in the log canonical case we prove that they satisfy the Bogomolov–Miyaoka–Yau type inequality. Existence of such a generalization was earlier conjectured by G. Megyesi [Proc. London Math. Soc. (3) 78 (1999) 241–282]. Most of the paper is devoted to properties of local orbifold Euler numbers and to their computation.As a first application we show that our results imply a generalized version of R. Holzapfel's ‘proportionality theorem’ [Ball and surface arithmetics, Aspects of Mathematics E29 (Vieweg, Braunschweig, 1998)]. Then we show a simple proof of a necessary condition for the logarithmic comparison theorem which recovers an earlier result by F. Calderon-Moreno, F. Castro-Jimenez, D. Mond and L. Narvaez-Macarro [Comment. Math. Helv. 77 (2002) 24–38].Then we prove effective versions of Bogomolov's result on boundedness of rational curves in some surfaces of general type (conjectured by G. Tian [Springer Lecture Notes in Mathematics 1646 (1996) 143–185)]. Finally, we give some applications to singularities of plane curves; for example, we improve F. Hirzebruch's bound on the maximal number of cusps of a plane curve.2000 Mathematical Subject Classification: 14J17, 14J29, 14C17.

Integrable Equations Arising from Motions of Plane Curves. II

Abstract: Integrable equations satisfied by the curvature of plane curves or curves on the real line under inextensible motions in some Klein geometries are identified. These include the Euclidean, similarity, and projective geometries on the real line, and restricted conformal, conformal, and projective geometries in the plane. Together with Chou and Qu [Physica D 162 (2002), 9–33], we determine inextensible motions and their associated integrable equations in all Klein geometries in the plane. The relations between several pairs of these geometries provide a natural geometric explanation of the existence of transformations of Miura and Cole-Hopf type.

On the spectrum of curved quantum waveguides

Abstract: The spectrum of the Laplace operator in a curved strip of constant width built along an infinite plane curve, subject to three different types of boundary conditions (Dirichlet, Neumann and a combination of these ones, respectively), is investigated. We prove that the essential spectrum as a set is stable under any curvature of the reference curve which vanishes at infinity and find various sufficient conditions which guarantee the existence of geometrically induced discrete spectrum. Furthermore, we derive a lower bound on the distance between the essential spectrum and the spectral threshold for locally curved strips. The paper is also intended as an overview of some new and old results on spectral properties of curved quantum waveguides.

Symplectic Bifurcations of Plane Curves and Isotropic Liftings

Abstract: We study the classification of varieties in the Marsden–Weinstein reduction and their liftability. In particular the complete symplectic classification of the Bruce–Gaffney plane curve singularites is provided and is applied to obtain naturally the Lagrangian openings.

Closed timelike curves and holography in compact plane waves

Abstract: We discuss plane wave backgrounds of string theory and their relation to G?del-like universes. This involves a twisted compactification along the direction of propagation of the wave, which induces closed timelike curves. We show, however, that no such curves are geodesic. The particle geodesics and the preferred holographic screens we find are qualitatively different from those in the G?del-like universes. Of the two types of preferred screen, only one is suited to dimensional reduction and/or T-duality, and this provides a ``holographic protection'' of chronology. The other type of screen, relevant to an observer localized in all directions, is constructed both for the compact and non-compact plane waves, a result of possible independent interest. We comment on the consistency of field theory in such spaces, in which there are closed timelike (and null) curves but no closed timelike (or null) geodesics.

Realization of finite Abelian groups by nets in P^2

Abstract: In the paper, we study special configurations of lines and points in the complex projective plane, so called k-nets. We describe the role of these configurations in studies of cohomology on arrangement complements. Our most general result is the restriction on k - it can be only 3,4, or 5. The most interesting class of nets is formed by 3-nets that relate to finite geometries, latin squares, loops, etc. All known examples of 3-nets in P^2 realize finite Abelian groups. We study the problem what groups can be so realized. Our main result is that, except for groups with all invariant factors under 10, realizable groups are isomorphic to subgroups of a 2-torus. This follows from the `algebraization' result asserting that in the dual plane, the points dual to lines of a net lie on a plane cubic.

The alexander polynomial of a plane curve singularity and integrals with respect to the euler characteristic

Abstract: It was shown that the Alexander polynomial (of several variables) of a (reducible) plane curve singularity coincides with the (generalized) Poincare polynomial of the multi-indexed filtration defined by the curve on the ring of germs of functions of two variables. The initial proof of the result was rather complicated (it used analytical, topological and combinatorial arguments). Here we give a new proof based on the notion of the integral with respect to the Euler characteristic over the projectivization of the space — the notion similar to (and inspired by) the notion of the motivic integration.

Elementary algebraic geometry

Abstract: Introduction Affine varieties Projective varieties Smooth points and dimension Plane cubic curves Cubic surfaces Introduction to the theory of curves Bibliography Index.

A-polynomial and Bloch invariants of hyperbolic 3-manifolds

Abstract: Let N be a complete, orientable, finite-volume, one-cusped hyperbolic 3-manifold with an ideal triangulation. Using combinatorics of the ideal triangulation of N we construct a plane curve in C×C which contains the squares of eigenvalues of PSL(2,C) representations of the meridian and longitude. We show that the defining polynomial of this curve is related to the PSL(2,C) A-polynomial and has properties similar to the classical A-polynomial. We further show that a factor of this polynomial, A0(l,m), associated to the discrete, faithful representation of π1(N) in PSL(2,C) is independent of the ideal triangulation. The Bloch invariant β(N) of N is related to the volume and Chern-Simons invariant of N . The variation of Bloch invariant is defined to be the change of β(N) under Dehn surgery on N . We relate A0(l,m) to the variation of the Bloch invariant of N . We show that A0(l,m) determines the variation of Bloch invariant in the case when A0(l, m) is a defining equation of a rational curve. We also show that in this case the Bloch invariant reads the symmetry of A0(l,m).

A lecture on the octahedron

Abstract: Algebraic solutions of certain Painlev´ eV I equationsare produced by solving linear systems with monodromy contained in the octahedral subgroup of SO(3). The method uses the algebraic geometry of special plane curves, and makes contact with some classical geometrical problems.

Theory of infinitely near singular points

Abstract: The notion of infinitely near singular points, classical in the case of plane curves, has been generalized to higher dimen- sions in my earlier articles ((5), (6), (7)). There, some basic tech- niques were developed, notably the three technical theorems which were Dierentiation Theorem , Numerical Exponent Theorem and Ambient Reduction Theorem (7). In this paper, using those results, we will prove the Finite Presentation Theorem, which the auther believes is the first of the most important milestones in the gen- eral theory of infinitely near singular points. The presentation is in terms of a finitely generated graded algebra which describes the total aggregate of the trees of infinitely near singular points. The totality is a priori very complex and intricate, including all pos- sible successions of permissible blowing-ups toward the reduction of singularities. The theorem will be proven for singular data on an ambient algebraic shceme, regular and of finite type over any perfect field of any characteristics. Very interesting but not yet apparent connections are expected with many such works as ((1), (8)).

Poincaré series for several plane divisorial valuations

Abstract: We compute the (generalized) Poincare series of the multi-index filtration defined by a finite collection of divisorial valuations on the ring $\mathcal{O}_{\mathbb{C}^2,0}$ of germs of functions of two variables. We use the method initially elaborated by the authors and Campillo for computing the similar Poincare series for the valuations defined by the irreducible components of a plane curve singularity. The method is essentially based on the notions of the so-called extended semigroup and of the integral with respect to the Euler characteristic over the projectivization of the space of germs of functions of two variables. The last notion is similar to (and inspired by) the notion of the motivic integration. AMS 2000 Mathematics subject classification: Primary 14B05; 16W70

Braid monodromy and topology of plane curves

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

Minimal surfaces of rotation in Finsler space with a Randers metric

Abstract: We consider Finsler spaces with a Randers metric F=α+β, on the three dimensional real vector space, where α is the Euclidean metric and β=bdx3 is a 1-form with norm b,0≤b<1. By using the notion of mean curvature for immersions in Finsler spaces introduced by Z. Shen, we get the ordinary differential equation that characterizes the minimal surfaces of rotation around the x3 axis. We prove that for every b,0≤b<1, there exists, up to homothety, a unique forward complete minimal surface of rotation. The surface is embedded, symmetric with respect to a plane perpendicular to the rotation axis and it is generated by a concave plane curve. Moreover, for every \(\) there are non complete minimal surfaces of rotation, which include explicit minimal cones.

Bivariate Hermite Interpolation and Linear Systems of Plane Curves with Base Fat Points

Abstract: . When the multiplicities are all equal,to m say, this problem has been attacked by a number of authors (Lorentz andLorentz, Ciliberto and Miranda, Hirschowitz) and there are a number of goodconjectures (Hirschowitz, Ciliberto and Miranda) on the dimension of these inter-polating spaces. The determination of the dimension has been already solved form ≤ 12 and all d and n by a degeneration technique and some ad hoc geometricarguments. Here this technique is applied up through m = 20; since it fails insome cases, we resort (in these exceptional cases) to the bivariete Hermite interpo-lation with the support of a simple idea suggested by Gr¨obner bases computation.In summary we are able to prove that the dimension of the vector space is theexpected one for 13 ≤ m ≤ 20.

Maximally Inflected Real Rational Curves

Abstract: We introduce and begin the topological study of real rational plane curves, all of whose inflection points are real. The existence of such curves is a corollary of results in the real Schubert calculus, and their study has consequences for the important Shapiro and Shapiro conjecture in the real Schubert calculus. We establish restrictions on the number of real nodes of such curves and construct curves realizing the extreme numbers of real nodes. These constructions imply the existence of real solutions to some problems in the Schubert calculus. We conclude with a discussion of maximally inflected curves of low degree.

Exceptional curves on Del Pezzo surfaces

Abstract: We classify all cases of exceptional curves on Del Pezzo surfaces, which turn out to be the smooth plane curves and some other cases with Clifford dimension 3. Moreover, the property of being exceptional holds for all curves in the complete linear system. We use this study to extend the results of Pareschi [17] on the constancy of the gonality and Clifford index of smooth curves in a complete linear system on Del Pezzo surfaces of degrees ≥ 2 to the case of Del Pezzo surfaces of degree 1, where we explicitly classify the cases where the gonality and Clifford index are not constant.

On affine Sawada–Kotera equation

Abstract: It is shown that motion of plane curves in affine geometry induces naturally the Sawada–Kotera hierarchy. The affine Sawada–Kotera equation is obtained in view of the equivalence of equations for the curvature and graph of plane curves when the curvature satisfies the Sawada–Kotera equation. The affine Sawada–Kotera equation can be viewed as an affine version of the WKI equation since they have similarity properties, such as they have loop-solitons, they are solved by the AKNS-scheme and are obtained by choosing the normal velocity to be the derivative of the curvature with respect to the arc-length. Its symmetry reductions to ordinary differential equations corresponding to an one-dimensional optimal system of its Lie symmetry algebras are discussed.

On the Asymptotic Number of Plane Curves and Alternating Knots

Abstract: We present a conjecture for the power-law exponent in the asymptotic number of types of plane curves as the number of self-intersections goes to infinity. In view of the description of prime alternating links as flype equivalence classes of plane curves, a similar conjecture is made for the asymptotic number of prime alternating knots. The rationale leading to these conjectures is given by quantum field theory. Plane curves are viewed as configurations of loops on a random planar lattices, that are in turn interpreted as a model of 2d quantum gravity with matter. The identification of the universality class of this model yields the conjecture. Since approximate counting or sampling planar curves with more than a few dozens of intersections is an open problem, direct confrontation with numerical data yields no convincing indication on the correctness of our conjectures. However, our physical approach yields a more general conjecture about connected systems of curves. We take advantage of this to design an original and feasible numerical test, based on recent perfect samplers for large planar maps. The numerical datas strongly support our identification with a conformal field theory recently described by Read and Saleur.

On Diophantine Approximations of Dependent Quantities in the p-adic Case

Abstract: In the present paper, we prove an analog of Khinchin's metric theorem in the case of linear Diophantine approximations of plane curves defined over the ring of $$p$$ -adic integers by means of (Mahler) normal functions. We also prove some general assertions needed to generalize this result to the case of spaces of higher dimension.