scholarly journals On Plane Curves with Double and Triple Points

2016 ◽  
Vol 119 (1) ◽  
pp. 60 ◽  
Author(s):  
Nancy Abdallah
Keyword(s):  
Plane Curve ◽  
Plane Curves ◽  
Jacobian Ideal ◽  
Pole Order ◽  
Hodge Filtration ◽  
Triple Points

We describe in simple geometric terms the Hodge filtration on the cohomology $H^*(U)$ of the complement $U=\mathsf{P}^2 \setminus C$ of a plane curve $C$ with ordinary double and triple points. Relations to Milnor algebra, syzygies of the Jacobian ideal and pole order filtration on $H^2(U)$ are given.

scholarly journals On Hodge Theory of Singular Plane Curves

2016 ◽  
Vol 59 (3) ◽  
pp. 449-460 ◽  
Author(s):  
Nancy Abdallah
Keyword(s):  
Numerical Estimate ◽  
Plane Curve ◽  
Hodge Theory ◽  
Plane Curves ◽  
Pole Order ◽  
Hodge Filtration ◽  
The Difference

AbstractThe dimensions of the graded quotients of the cohomology of a plane curve complement U = ℙ2 \ C with respect to the Hodge filtration are described in terms of simple geometrical invariants. The case of curves with ordinary singularities is discussed in detail. We also give a precise numerical estimate for the difference between the Hodge filtration and the pole order filtration on H2(U,ℂ).

The smooth surfaces in ℙ4 with few apparent triple points

2016 ◽  
Vol 18 (01) ◽  
pp. 1550013
Author(s):  
José Carlos Sierra
Keyword(s):  
Plane Curve ◽  
London Mathematical Society ◽  
Plane Curves ◽  
Smooth Surfaces ◽  
Projective Varieties ◽  
Cambridge University ◽  
Projective Character ◽  
Genus Formula ◽  
Triple Points ◽  
Complex Projective

We classify smooth complex projective surfaces in [Formula: see text] with [Formula: see text] apparent triple points, thus recovering and extending the results of Ascione [Sulle superficie immerse in un [Formula: see text], le cui trisecanti costituiscono complessi di [Formula: see text] ordine, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur.[Formula: see text]5[Formula: see text] 6 (1897) 162–169] and Severi [Intorno ai punti doppi impropri di una superficie generale dello spazio a quattro dimensioni, e a’ suoi punti tripli apparenti, Rend. Circ. Mat. Palermo 15 (1901) 33–51] for [Formula: see text], Marletta [Le superficie generali dell’ [Formula: see text] dotate di due punti tripli apparenti, Rend. Circ. Mat. Palermo 34 (1912) 179–186] for [Formula: see text], and Aure [The smooth surfaces in [Formula: see text] without apparent triple points, Duke Math. J. 57 (1988) 423–430] for [Formula: see text]. This is done thanks to a new projective character that can be introduced as a consequence of the main result of [K. Ranestad, On smooth plane curve fibrations in [Formula: see text], in Geometry of Complex Projective Varieties, Sem. Conf., Vol. 9 (Mediterranean, 1993), pp. 243–255; J. C. Sierra and A. L. Tironi, Some remarks on surfaces in [Formula: see text] containing a family of plane curves, J. Pure Appl. Algebra 209 (2007) 361–369; V. Beorchia and G. Sacchiero, Surfaces in [Formula: see text] with a family of plane curves, J. Pure Appl. Algebra 213 (2009) 1750–1755]. Going a bit further, we obtain some bounds on the Euler characteristic [Formula: see text] in terms of the degree [Formula: see text] and the sectional genus [Formula: see text] of a smooth surface in [Formula: see text]. For [Formula: see text], these bounds were first obtained in [A. B. Aure and K. Ranestad, The smooth surfaces of degree [Formula: see text] in [Formula: see text], in Complex Projective Geometry, London Mathematical Society Lecture Note Series, Vol. 179 (Cambridge University Press, Cambridge, 1992), pp. 32–46; K. Ranestad, On smooth surfaces of degree [Formula: see text] in the projective fourspace, Ph.D. thesis, Oslo (1988); S. Popescu, On smooth surfaces of degree [Formula: see text] in the projective fourspace, Dissertation, Saarbrücken (1993)]. Here we give a different argument based on liaison that works also for [Formula: see text] and that allows us to determine the triples [Formula: see text] of the smooth surfaces with [Formula: see text] apparent triple points.

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.

scholarly journals Plane Curves Which are Quantum Homogeneous Spaces

2021 ◽  
Author(s):  
Ken Brown ◽  
Angela Ankomaah Tabiri
Keyword(s):  
Hopf Algebras ◽  
Final Section ◽  
Homogeneous Spaces ◽  
Plane Curve ◽  
Plane Curves ◽  
Closed Field ◽  
Quantum Homogeneous Space ◽  
Central Elements ◽  
Pointed Hopf Algebras ◽  
Future Work

AbstractLet $\mathcal {C}$ C be a decomposable plane curve over an algebraically closed field k of characteristic 0. That is, $\mathcal {C}$ C is defined in k2 by an equation of the form g(x) = f(y), where g and f are polynomials of degree at least two. We use this data to construct three affine pointed Hopf algebras, A(x, a, g), A(y, b, f) and A(g, f), in the first two of which g [resp. f ] are skew primitive central elements, with the third being a factor of the tensor product of the first two. We conjecture that A(g, f) contains the coordinate ring $\mathcal {O}(\mathcal {C})$ O ( C ) of $\mathcal {C}$ C as a quantum homogeneous space, and prove this when each of g and f has degree at most five or is a power of the variable. We obtain many properties of these Hopf algebras, and show that, for small degrees, they are related to previously known algebras. For example, when g has degree three A(x, a, g) is a PBW deformation of the localisation at powers of a generator of the downup algebra A(− 1,− 1,0). The final section of the paper lists some questions for future work.

scholarly journals Isotopies of generic plane curves

1983 ◽  
Vol 24 (2) ◽  
pp. 195-206 ◽  
Author(s):  
J. W. Bruce
Keyword(s):  
Plane Curve ◽  
Gauss Map ◽  
Plane Curves ◽  
Bifurcation Set

The aim of this paper is to explore some facets of the geometry of generic isotopies of plane curves. Our major tool will be the paper of Arnol'd [1] on the evolution of wavefronts. The sort of questions one can ask are: in a generic isotopy of a plane curve how are vertices created and destroyed? How does the dual evolve? How can the Gauss map change? In attempting to answer these questions we are taking advantage of the fact that these phenomena are all naturally associated with singularities of type Ak. Now the bifurcation set of an Ak+1 singularity and the discriminant set of an Ak singularity coincide. So we can apply Arnol'd's results on one parameter families of Legendre (discriminant) singularities (e.g. the duals) to get information on one parameter families of Lagrange (bifurcation) singularities (e.g. the evolutes). For bifurcation sets of functions with singularities other than those of type Ak one runs up against problems with smooth moduli—see [4].

scholarly journals ON RATIONAL CUSPIDAL PROJECTIVE PLANE CURVES

2005 ◽  
Vol 92 (1) ◽  
pp. 99-138 ◽  
Author(s):  
J. FERNÁNDEZ DE BOBADILLA ◽  
I. LUENGO-VELASCO ◽  
A. MELLE-HERNÁNDEZ ◽  
A. NÉMETHI
Keyword(s):  
Singular Point ◽  
Projective Plane ◽  
Plane Curve ◽  
Topological Type ◽  
Homology Sphere ◽  
Plane Curves ◽  
Curve Singularity ◽  
Witten Invariant ◽  
Rational Homology ◽  
Rational Homology Sphere

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.

SHADOWS, DISCRIMINANT AND DIRECT IMAGES OF PLANE CURVE SINGULARITIES

2010 ◽  
Vol 21 (04) ◽  
pp. 453-474
Author(s):  
EDUARDO CASAS-ALVERO
Keyword(s):  
Plane Curve ◽  
Plane Curves ◽  
Plane Curve Singularities ◽  
Curve Singularities

We prove that the images of irreducible germs of plane curves by a germ of analytic morphism φ have a certain contact either with branches of the discriminant of φ or with certain infinitesimal structures (shadows) that arise from the branches of the Jacobian of φ that are mapped to a point (and therefore give rise to no branch of the discriminant).

scholarly journals Freeness versus maximal global Tjurina number for plane curves

2016 ◽  
Vol 163 (1) ◽  
pp. 161-172 ◽  
Author(s):  
ALEXANDRU DIMCA
Keyword(s):  
Plane Curve ◽  
Plane Curves ◽  
Tjurina Number ◽  
Free Plane

AbstractWe give a characterisation of nearly free plane curves in terms of their global Tjurina numbers, similar to the characterisation of free curves as curves with a maximal Tjurina number, given by A. A. du Plessis and C.T.C. Wall. It is also shown that an irreducible plane curve having a 1-dimensional symmetry is nearly free. A new numerical characterisation of free curves and a simple characterisation of nearly free curves in terms of their syzygies conclude this paper.

The aberrancy of plane curves

2004 ◽  
Vol 89 (516) ◽  
pp. 424-436 ◽  
Author(s):  
Russell A. Gordon
Keyword(s):  
Plane Curve ◽  
Plane Curves ◽  
English School ◽  
Real Analysis ◽  
Extended Problem ◽  
Mathematics Class

The aberrancy of a plane curve is a property of the curve that is invariant under both translation and rotation. It provides a numerical measure for the non-circularity of the curve at each point of the curve. (Recall that curvature gives an invariant numerical measure of nonlinearity.) The concept of aberrancy has been around for two centuries, but it has received very little attention. We hope to stir a little interest in the concept by presenting four different derivations of the formula for aberrancy. This may appear to be redundant, but there are some good reasons for doing so. First of all, the only derivation for aberrancy in the literature is a bit confusing and makes a few unjustified assumptions. Secondly, the derivations we present are all significantly different from each other and involve some interesting ideas in elementary real analysis. Finally, although the basic idea behind each derivation is simple, the details can become extremely messy unless a proper path is chosen. We encourage the reader to find one or two approaches of particular interest and use the concept of aberrancy as an extended problem set in an undergraduate analysis course or even a further mathematics class in an English school.

Plane Curves With Nodes

1989 ◽  
Vol 41 (2) ◽  
pp. 193-212 ◽  
Author(s):  
Robert Treger
Keyword(s):  
General Position ◽  
Algebraic Curve ◽  
Plane Curve ◽  
Exceptional Case ◽  
Plane Curves ◽  
Nodal Plane ◽  
Nodal Curve

A smooth algebraic curve is birationally equivalent to a nodal plane curve. One of the main problems in the theory of plane curves is to describe the situation of nodes of an irreducible nodal plane curve (see [16, Art. 45], [10], [7, Book IV, Chapter I, §5], [12, p. 584], and [3]).Let n denote the degree of a nodal curve and d the number of nodes. The case (AZ, d) — (6,9) has been analyzed by Halphen [10]. It follows from Lemma 3.5 and Proposition 3.6 that this is an exceptional case. The case d ≦n(n + 3)/6, d ≦(n — 1)(n — 2)/2, and (n, d) ≠ (6,9) was investigated by Arbarello and Cornalba [3]. We present a simpler proof (Corollary 3.8).We consider the main case which is particularly important due to its applications to the moduli variety of curves, compare [19, Chapter VIII, Section 4]. Let Vn,d be the variety of irreducible curves of degree n with d nodes and no other singularities such that each curve of Vn,d can be degenerated into n lines in general position (see [17]).

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