scholarly journals Nodal curves with a contact-conic and Zariski pairs

2019 ◽  
Vol 19 (4) ◽  
pp. 555-572 ◽  
Author(s):  
Shinzo Bannai ◽  
Taketo Shirane
Keyword(s):  
Plane Curves ◽  
Nodal Plane ◽  
Nodal Curves ◽  
Splitting Type

Abstract To study the splitting of nodal plane curves with respect to contact conics, we define the splitting type of such curves and show that it can be used as an invariant to distinguish the embedded topology of plane curves. We also give a criterion to determine the splitting type in terms of the configuration of the nodes and tangent points. As an application, we construct sextics and contact conics with prescribed splitting types, which give rise to new Zariski-triples.

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]).

On nodal plane curves

1986 ◽  
Vol 86 (3) ◽  
pp. 529-534 ◽  
Author(s):  
Ziv Ran
Keyword(s):  
Plane Curves ◽  
Nodal Plane

scholarly journals Splitting types of bundles of logarithmic vector fields along plane curves

2018 ◽  
Vol 29 (08) ◽  
pp. 1850055 ◽  
Author(s):  
Takuro Abe ◽  
Alexandru Dimca
Keyword(s):  
Vector Fields ◽  
Plane Curve ◽  
Plane Curves ◽  
Tjurina Number ◽  
Arrangements Of Lines ◽  
Splitting Type ◽  
Intersection Points

We give a formula relating the total Tjurina number and the generic splitting type of the bundle of logarithmic vector fields associated to a reduced plane curve. By using it, we give a characterization of nearly free curves in terms of splitting types. Several applications to free and nearly free arrangements of lines are also given, in particular a proof of a form of Terao’s Conjecture for arrangements having a line with at most four intersection points.

scholarly journals On Projections of Smooth and Nodal Plane Curves

2015 ◽  
Vol 15 (1) ◽  
pp. 31-48
Author(s):  
Yu. Burman ◽  
Serge Lvovski
Keyword(s):  
Plane Curves ◽  
Nodal Plane

scholarly journals ON THE POSITION OF NODES OF PLANE CURVES

2020 ◽  
pp. 1-7
Author(s):  
CÉSAR LOZANO HUERTA ◽  
TIM RYAN
Keyword(s):  
Hilbert Scheme ◽  
Plane Curves ◽  
Nodal Curves ◽  
Canonical Class ◽  
Severi Variety

The Severi variety $V_{d,n}$ of plane curves of a given degree $d$ and exactly $n$ nodes admits a map to the Hilbert scheme $\mathbb{P}^{2[n]}$ of zero-dimensional subschemes of $\mathbb{P}^{2}$ of degree $n$ . This map assigns to every curve $C\in V_{d,n}$ its nodes. For some $n$ , we consider the image under this map of many known divisors of the Severi variety and its partial compactification. We compute the divisor classes of such images in $\text{Pic}(\mathbb{P}^{2[n]})$ and provide enumerative numbers of nodal curves. We also answer directly a question of Diaz–Harris [‘Geometry of the Severi variety’, Trans. Amer. Math. Soc.309 (1988), 1–34] about whether the canonical class of the Severi variety is effective.

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