scholarly journals On geometry of the midlocus associated to a smooth curve in plane and space

Filomat ◽  
2018 ◽  
Vol 32 (8) ◽  
pp. 2977-2990
Author(s):  
Azeb Alghanemi ◽  
Peter Giblin
Keyword(s):  
Plane Curve ◽  
Smooth Curve ◽  
Space Curve ◽  
Regular Space ◽  
Geometric Conditions ◽  
Special Cases ◽  
Smooth Plane ◽  
Surface Singularities

The singularities of the midpoint map associated to a smooth plane curve, which is a map from the plane to the plane, are classified. The midlocus associated to a regular space curve is introduced. The geometric conditions for the midlocus of a space curve to have a crosscap or an S?1 singularities are investigated. A more general map, the ?-point map, associated to a space curve is introduced and many known surface singularities are realized as a special cases of this construction.

scholarly journals Higher rank Clifford indices of curves on a K3 surface

2021 ◽  
Vol 27 (3) ◽  
Author(s):  
Soheyla Feyzbakhsh ◽  
Chunyi Li
Keyword(s):  
Vector Bundles ◽  
Plane Curve ◽  
Smooth Curve ◽  
Line Bundles ◽  
K3 Surface ◽  
Clifford Index ◽  
Smooth Plane ◽  
Higher Rank ◽  
Rank Vector ◽  
Semistable Vector Bundle

AbstractLet (X, H) be a polarized K3 surface with $$\mathrm {Pic}(X) = \mathbb {Z}H$$ Pic ( X ) = Z H , and let $$C\in |H|$$ C ∈ | H | be a smooth curve of genus g. We give an upper bound on the dimension of global sections of a semistable vector bundle on C. This allows us to compute the higher rank Clifford indices of C with high genus. In particular, when $$g\ge r^2\ge 4$$ g ≥ r 2 ≥ 4 , the rank r Clifford index of C can be computed by the restriction of Lazarsfeld–Mukai bundles on X corresponding to line bundles on the curve C. This is a generalization of the result by Green and Lazarsfeld for curves on K3 surfaces to higher rank vector bundles. We also apply the same method to the projective plane and show that the rank r Clifford index of a degree $$d(\ge 5)$$ d ( ≥ 5 ) smooth plane curve is $$d-4$$ d - 4 , which is the same as the Clifford index of the curve.

scholarly journals Connected Numbers and the Embedded Topology of Plane Curves

2018 ◽  
Vol 61 (3) ◽  
pp. 650-658 ◽  
Author(s):  
Taketo Shirane
Keyword(s):  
Plane Curve ◽  
Smooth Curve ◽  
Plane Curves ◽  
Irreducible Curve ◽  
Galois Cover ◽  
Splitting Number

AbstractThe splitting number of a plane irreducible curve for a Galois cover is effective in distinguishing the embedded topology of plane curves. In this paper, we define the connected number of a plane curve (possibly reducible) for a Galois cover, which is similar to the splitting number. By using the connected number, we distinguish the embedded topology of Artal arrangements of degree b ≥ 4, where an Artal arrangement of degree b is a plane curve consisting of one smooth curve of degree b and three of its total inflectional tangents.

On the tritangent planes of a quadri-cubic space curve

1935 ◽  
Vol 4 (3) ◽  
pp. 159-169
Author(s):  
H. W. Richmond
Keyword(s):  
Plane Curve ◽  
Space Curve ◽  
Quadric Surface ◽  
Geometrical Aspect ◽  
Quartic Curve ◽  
Abelian Functions

With any twisted curve of order six is associated a system of planes, usually finite in number, which touch the curve at three distinct points. The curve with its system of tritangent planes possesses properties which recall the properties of a plane quartic curve and its system of bitangent lines; and this is specially true of the sextic which is the intersection of a cubic and a quadric surface. But whereas the properties of the plane curve were discovered by geometrical methods, such methods have only recently been applied with success to the space-curve; the earliest properties were obtained by Clebsch from his Theory of Abelian Functions. In the absence of any one place to which reference can conveniently be made, an account of these properties in their geometrical aspect will be useful.

scholarly journals Conjectures and results about parabolic induction of representations of $${\text {GL}}_n(F)$$

2020 ◽  
Vol 222 (3) ◽  
pp. 695-747
Author(s):  
Erez Lapid ◽  
Alberto Mínguez
Keyword(s):  
Linear Group ◽  
Local Field ◽  
General Linear Group ◽  
Irreducible Representations ◽  
General Linear ◽  
Parabolic Induction ◽  
Geometric Conditions ◽  
Irreducible Components ◽  
Special Cases

Abstract In 1980 Zelevinsky introduced certain commuting varieties whose irreducible components classify complex, irreducible representations of the general linear group over a non-archimedean local field with a given supercuspidal support. We formulate geometric conditions for certain triples of such components and conjecture that these conditions are related to irreducibility of parabolic induction. The conditions are in the spirit of the Geiss–Leclerc–Schröer condition that occurs in the conjectural characterization of $$\square $$ □ -irreducible representations. We verify some special cases of the new conjecture and check that the geometric and representation-theoretic conditions are compatible in various ways.

scholarly journals Relative geodesics in bi-invariant Lie groups

2017 ◽  
Vol 473 (2201) ◽  
pp. 20160619
Author(s):  
E. Zhang ◽  
L. Noakes
Keyword(s):  
Order Differential Equation ◽  
Homogeneous Spaces ◽  
Lagrange Equation ◽  
Smooth Curve ◽  
Curve Matching ◽  
Matching Problem ◽  
Smooth Curves ◽  
First Order ◽  
Finite Dimensional ◽  
Special Cases

Motivated by registration problems, this paper deals with a curve matching problem in homogeneous spaces. Let G be a connected finite-dimensional bi-invariant Lie group and K a closed subgroup. A smooth curve g in G is said to be admissible if it can transform two smooth curves f 1 and f 2 in G / K from one to the other. An ( f 1 , f 2 )- relative geodesic (Holm et al. 2013 Proc. R. Soc. A 469 , 20130297. ( doi:10.1098/rspa.2013.0297 )) is defined as a critical point of the total energy E ( g ) as g varies in the set of all ( f 1 , f 2 )-admissible curves. We obtain the Euler–Lagrange equation, a first-order differential equation, satisfied by a relative geodesic. Furthermore, the Euler–Lagrange equation is simplified for the case where G / K is globally symmetric. As a concrete example, relative geodesics are found for special cases where G is SO(3) and K is SO(2). As an application of discrepancy for curves in S 2 , we construct and study a new measure of non-congruency for constant speed curves in Euclidean 3-space. Numerical examples are given to illustrate results.

A classifier for simple space curve singularities

2014 ◽  
Vol 51 (1) ◽  
pp. 92-104
Author(s):  
Faira Janjua ◽  
Gerhard Pfister
Keyword(s):  
Computer Algebra ◽  
Computer Algebra System ◽  
Plane Curve ◽  
Space Curve ◽  
Plane Curve Singularities ◽  
Curve Singularities

The classification of Bruce and Gaffney respectively Gibson and Hobbs for simple plane curve singularities respectively simple space curve singularities is characterized in terms of invariants. This is the basis for the implementation of a classifier in the computer algebra system singular.

scholarly journals Components of the Hilbert scheme of space curves on low-degree smooth surfaces

2015 ◽  
Vol 26 (02) ◽  
pp. 1550017 ◽  
Author(s):  
Jan O. Kleppe ◽  
John C. Ottem
Keyword(s):  
Hilbert Scheme ◽  
Smooth Surface ◽  
Plane Curve ◽  
Picard Number ◽  
Connected Space ◽  
Space Curves ◽  
Cubic Surfaces ◽  
Smooth Component ◽  
Low Degree ◽  
Smooth Plane

We study maximal families W of the Hilbert scheme, H(d, g)sc, of smooth connected space curves whose general curve C lies on a smooth surface S of degree s. We give conditions on C under which W is a generically smooth component of H(d, g)sc and we determine dim W. If s = 4 and W is an irreducible component of H(d, g)sc, then the Picard number of S is at most 2 and we explicitly describe, also for s ≥ 5, non-reduced and generically smooth components in the case Pic (S) is generated by the classes of a line and a smooth plane curve of degree s - 1. For curves on smooth cubic surfaces the first author finds new classes of non-reduced components of H(d, g)sc, thus making progress in proving a conjecture for such families.

Sign in / Sign up

Export Citation Format

Share Document