Showing papers on "Plane curve published in 1996"
Posted Content•
TL;DR: In this paper, the authors obtained a recursive formula answering the following question: How many irreducible, plane curves of degree d and genus g pass through 3d-1+g general points in the plane?
Abstract: We obtain a recursive formula answering the following question: How many irreducible, plane curves of degree d and (geometric) genus g pass through 3d-1+g general points in the plane? The formula is proved by studying suitable degenerations of plane curves.
179 citations
Book•
01 Feb 1996TL;DR: Integral closure plane curves Factorization of ideals The discriminants The ideal class group Projective curves Nonsingular complete curves Zeta-functions The Riemann-Roch Theorem Frobenius morphisms and the RiemANN hypothesis Further topics Appendix Glossary of notation Index Bibliography as mentioned in this paper
Abstract: Integral closure Plane curves Factorization of ideals The discriminants The ideal class group Projective curves Nonsingular complete curves Zeta-functions The Riemann-Roch Theorem Frobenius morphisms and the Riemann hypothesis Further topics Appendix Glossary of notation Index Bibliography.
144 citations
103 citations
TL;DR: In this article, a strategy for computing intersection numbers on smooth varieties with torus actions using a residue formula of Bott is presented, which is consistent with predictions made from mirror symmetry compu- tations.
Abstract: We outline a strategy for computing intersection numbers on smooth varieties with torus actions using a residue formula of Bott. As an example, Gromov-Witten numbers of twisted cubic and elliptic quartic curves on some general complete intersection in projective space are computed. The results are consistent with predictions made from mirror symmetry compu- tations. We also compute degrees of some loci in the linear system of plane curves of degrees less than 10, like those corresponding to sums of powers of linear forms, and curves carrying inscribed polygons. MATHEMATICAL INSTITUTE, UNIVERSITY OF OSLO, P. 0. Box 1053, N-0316 OSLO, NORWAY E-mail address: ellingsr0math.uio .no MATHEMATICAL INSTITUTE, UNIVERSITY OF BERGEN, ALLEG 55, N-5007 BERGEN, NORWAY E-mail address: strommeti i.uib.no This content downloaded from 157.55.39.29 on Tue, 12 Apr 2016 10:23:14 UTC All use subject to http://about.jstor.org/terms
92 citations
TL;DR: A new algorithm computing the topological shape of a real algebraic plane curve whose complexity is better than the best algorithms known and its complexity is dominated by the characterization of the real roots of the discriminant of the polynomial defining the considered curve.
87 citations
Posted Content•
TL;DR: In this article, the Gromov-Witten invariants of the projective plane blown up in r general points are determined by associativity from r+1 intial values, and it is shown that the numbers are enumerative if at least one of the prescribed multiplicities is 1 or 2.
Abstract: We compute the Gromov-Witten invariants of the projective plane blown up in r general points. These are determined by associativity from r+1 intial values. Applications are given to the enumeration of rational plane curves with prescribed multiplicities at fixed general points. We show that the numbers are enumerative if at least one of the prescribed multiplicities is 1 or 2. In particular, all the invariants for r<=8 (the Del Pezzo case) are enumerative.
59 citations
56 citations
31 Jan 1996
54 citations
TL;DR: In this paper, the authors obtained new examples and the complete list of rational cuspidal plane curves with at least three cusps, one of which has multiplicitydegC-2.
Abstract: We obtain new examples and the complete list of the rational cuspidal plane curvesC with at least three cusps, one of which has multiplicitydegC-2. It occurs that these curves are projectively rigid. We also discuss the general problem of projective rigidity of rational cuspidal plane curves.
51 citations
TL;DR: In this article, linear systems and their order sequences for an algebraic curve over a finite field are used to obtain upper bounds on the size of a complete arc in a finite projective plane.
50 citations
TL;DR: In this article, the integral expression of a knot invariant obtained as the second coefficient in the perturbative expansion of Witten's Chern-Simons path integral associated with a knot is studied.
Abstract: We study the integral expression of a knot invariant obtained as the second coefficient in the perturbative expansion of Witten's Chern-Simons path integral associated with a knot. One of the integrals involved turns out to be a generalization of the classical Crofton integral on convex plane curves, and it is related with the invariants of generic plane curves recently defined by Arnold, with deep motivations in symplectic and contact geometry. Quadratic bounds on these plane curve invariants are derived using their relationship with the knot invariant.
01 Jan 1996
TL;DR: In this article, it was shown that principal curves are solutions of a differential equation, and they can be found for uniform densities on rectangles and annuii, and for spherical and elliptical distributions.
Abstract: Principal curves were introduced to formalize the notion of “a curve passing through the middle of a dataset”. Vaguely speaking, a curve is said to pass through the middle of a dataset if every point on the curve is the average of the observations projecting onto it. This idea can be made precise by defining principal curves for probability densities. Principal curves can be regarded as a generalization of linear principal components—if a principal curve happens to be a straight line, then it is a principal component. In this paper we study principal curves in the plane. We show that principal curves are solutions of a differential equation. By solving this differential equation, we find principal curves for uniform densities on rectangles and annuii. There are oscillating solutions besides the obvious straight and circular ones, indicating that principal curves in general will not be unique. If a density has several principal curves, they have to cross, a property somewhat analogous to the orthogonality of principal components. Finally, we study principal curves for spherical and elliptical distributions.
Posted Content•
TL;DR: In this article, enumerative questions about irreducible rational curves on Hirzebruch surfaces were answered by combining an idea of Kontsevich with the study of the geometry of certain natural parameter spaces.
Abstract: We answer some enumerative questions about irreducible rational curves on Hirzebruch surfaces, by combining an idea of Kontsevich with the study of the geometry of certain natural parameter spaces. Our formulas generalize Kontsevich's formula for rational plane curves of given degree.
Book•
01 Aug 1996TL;DR: Plane curves Space curves Surfaces Curvature of a surface Surface patches Composite surfaces as discussed by the authors are the most common surfaces to cover a surface in a 3D space plane trajectory.
Abstract: Plane curves Space curves Surfaces Curvature of a surface Surface patches Composite surfaces.
TL;DR: It is proved that digital plane segments and their least-Squares plane fit are in one-to-one correspondence, which gives a simple representation of a digital plane segment by its base description and coefficients of the least-squares plane Fit, leading to a constant space representation of digital rectangles in space.
Journal Article•
TL;DR: In this paper, the Hurwitz-Kellogg-Tabachnikov theorem of Sturm type is extended to the theory of extatic points of plane curves, which is a generalization of the Sturm theory of caustics.
Abstract: Contents §1. An apologia for Applied Mathematics §2. Theorems on four cusps §3. The Hurwitz-Kellogg-Tabachnikov theorem of Sturm type §4. Trigonometric approximations §5. Lagrangian intersections in symplectic topology §6. Legendrian links in contact topology §7. Lagrangian collapse and the cusps of caustics §8. Legendrian collapse and the cusps of fronts §9. Space curves and their points of flattening §10. Vertices of convex space curves §11. Applications to the theory of extatic points of plane curves §12. Multidimensional generalizations of Sturm theory Bibliography
TL;DR: In this paper, the existence of toric resolution tower for irreducible curve singularities was shown for a smooth affine plane curve from its topology restrictions for its singularity at infinity.
Abstract: We show the existence of toric resolution tower for an irreducible curve singularity which is explicitly described by Tschirnhausen polynomials. We deduce for a smooth affine plane curve from its topology restrictions for its singularity at infinity. For instance, we discribe the singularities at infinity (up to equisingular deformation) for curves of genus 0,1 and 2. From this follows an extension of the Abhyankar-Moh-Suzuki theorem to genus 1 and 2.
01 Jan 1996
TL;DR: In this article, the gonality sequence of a plane curve is computed and a two variable zeta function for curves over a finite field is defined and the rationality and a functional equation are proved.
Abstract: The gonality sequence of a plane curve is computed. A two variable zeta function for curves over a finite field is defined and the rationality and a functional equation are proved.
TL;DR: The numerical approximation schemes for solving the nonlinear initial value problem $\partial_t b(v) = (A v_x)_x+(Bv)-x+Dv+G$ with periodic boundary conditions are presented and the so-called "curve shortening equation" is included.
Abstract: The numerical approximation schemes for solving the nonlinear initial value problem $\partial_t b(v) = (A v_x)_x+(Bv)_x+Dv+G$ with periodic boundary conditions are presented. We assume that the function $b$ is increasing, and asymptotically $b'(s)=0$ and $b'(s)=+\infty$, so that the model describes in a sense both slow and fast diffusions. The solution also may blow up in a finite time. This problem arises from the evolving curves theory, which was used in the construction of models of motion of phase interface in multiphase thermomechanics by Angenent and Gurtin [Arch. Rational Mech. Anal., 108 (1989), pp. 323--391]. The so-called "curve shortening equation" is included in the model. Our approximating solutions converge strongly in $L_2(I,V)$ space to the weak solution. We also derive an "error estimate" for semidiscretization, which implies uniqueness. The numerical experiments in various situations of the "anisotropic curve shortening" are discussed.
TL;DR: The strengths and weaknesses of a number of semi-local methods for describing plane, non-algebraic curves in a projectively invariant fashion are compared on the basis of the same images and edge data.
Abstract: Recently, several methods have been proposed for describing plane, non-algebraic curves in a projectively invariant fashion. These curve representations are invariant under changes in viewpoint and therefore ideally suited for recognition. We report the results of a study where the strengths and weaknesses of a number of semi-local methods are compared on the basis of the same images and edge data. All the methods define a distinguished or canonical projective frame for the curve segment which is used for projective normalisation. In this canonical frame the curve has a viewpoint invariant signature. Measurements on the signature are invariants. All the methods presented are designed to work on real images where extracted data will not be ideal, and parts of curves will be missing because of poor contrast or occlusion. We compare the stability and discrimination of the signatures and invariants over a number of example curves and viewpoints. The paper concludes with a discussion of how the various methods can be integrated within a recognition system.
TL;DR: A dynamic focusing algorithm for the segmentation of plane curves into straight lines and circular arcs is presented, which works well on all types of curves regardless of the degree of curvature smoothness.
Posted Content•
TL;DR: In this article, the authors showed that the Poincar\'e group of irreducible plane curves contains a free group with two generators if the geometric genus of the curve is at least 2 and if a subgroup of the plane curve can be mapped epimorphically onto the fundamental group of the normalization.
Abstract: Let $C \s \pr^2$ be an irreducible plane curve whose dual $C^* \s \pr^{2*}$ is an immersed curve which is neither a conic nor a nodal cubic. The main result states that the Poincar\'e group $\pi_1(\pr^2 \se C)$ contains a free group with two generators. If the geometric genus $g$ of $C$ is at least 2, then a subgroup of $G$ can be mapped epimorphically onto the fundamental group of the normalization of $C$, and the result follows. To handle the cases $g=0,1$, we construct universal families of immersed plane curves and their Picard bundles. This allows us to reduce the consideration to the case of Pl\"ucker curves. Such a curve $C$ can be regarded as a plane section of the corresponding discriminant hypersurface (cf. [Zar, DoLib]). Applying Zariski--Lefschetz type arguments we deduce the result from `the bigness' of the $d$-th braid group $B_{d,g}$ of the Riemann surface of $C$.
Posted Content•
TL;DR: This paper derived recursive equations for the characteristic numbers of rational nodal plane curves with at most one cusp, subject to point conditions, tangent conditions, and flag conditions, developing techniques akin to quantum cohomology on a moduli space of stable lifts.
Abstract: We derive recursive equations for the characteristic numbers of rational nodal plane curves with at most one cusp, subject to point conditions, tangent conditions and flag conditions, developing techniques akin to quantum cohomology on a moduli space of stable lifts.
01 Jan 1996
TL;DR: The Puiseux pairs of the irreducible components and their intersection multiplicities are enough to describe the embedding topological type of C (see as discussed by the authors for a discussion).
Abstract: Let C = f(x,y) = 0 be a germ of a reduced plane curve. As examples of the basic invariants of a plane curve, we have the Milnor number, the number of irreducible components, the resolution complexity, the Puiseux pairs of the irreducible components and their intersection multiplicities. In fact, the Puiseux pairs of the irreducible components and their intersection multiplicities are enough to describe the embedding topological type of C (see [17], [9]).
Journal Article•
TL;DR: In this article, the authors presented a method to construct certain type of Zariski pairs, and made infinite series of examples, and showed that these pairs can be constructed in the projective plane.
Abstract: A couple of complex projective plane curves are said to make a Zariski pair if they have the same degree and the same type of singularities, but their embeddings in the projective plane are topologically different. In this paper, we present a method to construct certain type of Zariski pairs, and make infinite series of examples.
TL;DR: This work characterises the 4-regular plane graphs which contain two orthogonal A-trails, and gives rise to a polynomial algorithm for deciding if two such A-Trails exists.
TL;DR: In this article, the authors show that the dual billiard transformation far from the curve is well approximated by the time 1 transformation of a Hamiltonian flow associated with the curve.
Abstract: Given a strictly convex plane curve, the dual billiard transformation is the transformation of its exterior defined as follows: given a point x outside the curve, draw a support line to it from the point and reflect x at the support point. We show that the dual billiard transformation far from the curve is well approximated by the time 1 transformation of a Hamiltonian flow associated with the curve.
TL;DR: In this paper, an elastic half-plane with an oblique edge crack is considered and the complex stress functions are derived by separating the whole problem into two parts; one is the principal part corresponding to the infinite plane acted on by concentrated forces or dislocations, the other is the holomorphic part, which can be determined by making use of the property of regularity of complex stress function.
Abstract: An elastic half plane with an oblique edge crack is considered in this paper. A pair of concentrated forces or point dislocations is assumed to act at an arbitrary point in the half plane. The half plane with an edge crack is first mapped into a unit circle by a rational mapping function so that the following analysis can be carried out on the mapped plane analytically. Then the complex stress functions are derived by separating the whole problem into two parts; one is the principal part corresponding to the infinite plane acted on by concentrated forces or dislocations, the other is the holomorphic part, which can be determined by making use of the property of regularity of complex stress functions. The stress intensity factors of the crack can be calculated with different inclined angles of the crack, and the displacement and stress components at an arbitrary position in the half plane can be expressed explicitly.