scholarly journals The vanishing cycles of curves in toric surfaces I

2018 ◽  
Vol 154 (8) ◽  
pp. 1659-1697 ◽  
Author(s):  
Rémi Crétois ◽  
Lionel Lang
Keyword(s):  
Linear System ◽  
Simple Closed Curve ◽  
Closed Curve ◽  
Toric Surfaces ◽  
Vanishing Cycles ◽  
Complete Linear System ◽  
Vanishing Cycle

This article is the first in a series of two in which we study the vanishing cycles of curves in toric surfaces. We give a list of possible obstructions to contract vanishing cycles within a given complete linear system. Using tropical means, we show that any non-separating simple closed curve is a vanishing cycle whenever none of the listed obstructions appears.

scholarly journals The vanishing cycles of curves in toric surfaces II

2019 ◽  
Vol 11 (04) ◽  
pp. 909-927 ◽  
Author(s):  
Rémi Crétois ◽  
Lionel Lang
Keyword(s):  
Spin Structure ◽  
Simple Closed Curve ◽  
Isotopy Class ◽  
Closed Curve ◽  
Target Space ◽  
Mapping Class ◽  
Toric Surfaces ◽  
Hyperelliptic Involution ◽  
Vanishing Cycles ◽  
Vanishing Cycle

We resume the study initiated in [R. Crétois and L. Lang, The vanishing cycles of curves in toric surfaces, I, preprint (2017), arXiv:1701.00608]. For a generic curve [Formula: see text] in an ample linear system [Formula: see text] on a toric surface [Formula: see text], a vanishing cycle of [Formula: see text] is an isotopy class of simple closed curve that can be contracted to a point along a degeneration of [Formula: see text] to a nodal curve in [Formula: see text]. The obstructions that prevent a simple closed curve in [Formula: see text] from being a vanishing cycle are encoded by the adjoint line bundle [Formula: see text]. In this paper, we consider the linear systems carrying the two simplest types of obstruction. Geometrically, these obstructions manifest on [Formula: see text] respectively as an hyperelliptic involution and as a spin structure. In both cases, we determine all the vanishing cycles by investigating the associated monodromy maps, whose target space is the mapping class group [Formula: see text]. We show that the image of the monodromy is the subgroup of [Formula: see text] preserving respectively the hyperelliptic involution and the spin structure. The results obtained here support Conjecture [Formula: see text] in [R. Crétois and L. Lang, The vanishing cycles of curves in toric surfaces, I, preprint (2017), arXiv:1701.00608] aiming to describe all the vanishing cycles for any pair [Formula: see text].

scholarly journals Curves with Two Pencils and Associated Maps to Projective Spaces

2006 ◽  
Vol 13 (3) ◽  
pp. 411-417
Author(s):  
Edoardo Ballico
Keyword(s):  
Linear System ◽  
Projective Spaces ◽  
Free Resolution ◽  
Homogeneous Ideal ◽  
Projective Curve ◽  
Minimal Free Resolution ◽  
Complete Linear System

Abstract Let 𝑋 be a smooth and connected projective curve. Assume the existence of spanned 𝐿 ∈ Pic𝑎(𝑋), 𝑅 ∈ Pic𝑏(𝑋) such that ℎ0(𝑋, 𝐿) = ℎ0(𝑋, 𝑅) = 2 and the induced map ϕ 𝐿,𝑅 : 𝑋 → 𝐏1 × 𝐏1 is birational onto its image. Here we study the following question. What can be said about the morphisms β : 𝑋 → 𝐏𝑅 induced by a complete linear system |𝐿⊗𝑢⊗𝑅⊗𝑣| for some positive 𝑢, 𝑣? We study the homogeneous ideal and the minimal free resolution of the curve β(𝑋).

scholarly journals Representing Homology Classes on Surfaces

1976 ◽  
Vol 19 (3) ◽  
pp. 373-374 ◽  
Author(s):  
James A. Schafer
Keyword(s):  
Unit Circle ◽  
Homotopy Class ◽  
Simple Closed Curve ◽  
Closed Curve ◽  
Integral Basis

Let T2 = S1×S1, where S1 is the unit circle, and let {α, β} be the integral basis of H1(T2) induced by the 2 S1-factors. It is well known that 0 ≠ X = pα + qβ is represented by a simple closed curve (i.e. the homotopy class αppq contains a simple closed curve) if and only if gcd(p, q) = 1. It is the purpose of this note to extend this theorem to oriented surfaces of genus g.

A Simple Closed Curve is the Only Homogeneous Bounded Plane Continuum that Contains an Arc

1960 ◽  
Vol 12 ◽  
pp. 209-230 ◽  
Author(s):  
R. H. Bing
Keyword(s):  
Simple Closed Curve ◽  
Closed Curve ◽  
History Of ◽  
Homogeneous Plane ◽  
Plane Continuum

One of the unsolved problems of plane topology is the following:Question. What are the homogeneous bounded plane continua?A search for the answer has been punctuated by some erroneous results. For a history of the problem see (6).The following examples of bounded homogeneous plane continua are known : a point; a simple closed curve; a pseudo arc (2, 12); and a circle of pseudo arcs (6). Are there others?The only one of the above examples that contains an arc is a simple closed curve. In this paper we show that there are no other such examples. We list some previous results that point in this direction. Mazurkiewicz showed (11) that the simple closed curve is the only non-degenerate homogeneous bounded plane continuum that is locally connected. Cohen showed (8) that the simple closed curve is the only homogeneous bounded plane continuum that contains a simple closed curve.

scholarly journals Handel’s fixed point theorem revisited

2012 ◽  
Vol 33 (5) ◽  
pp. 1584-1610
Author(s):  
JULIANA XAVIER
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Unit Disk ◽  
Point Theorem ◽  
Simple Closed Curve ◽  
Closed Curve ◽  
Open Unit ◽  
Open Unit Disk ◽  
Closed Disk ◽  
Oriented Cycle

AbstractMichael Handel proved in [A fixed-point theorem for planar homeomorphisms. Topology38 (1999), 235–264] the existence of a fixed point for an orientation-preserving homeomorphism of the open unit disk that can be extended to the closed disk, provided that it has points whose orbits form an oriented cycle of links at infinity. Later, Patrice Le Calvez gave a different proof of this theorem based only on Brouwer theory and plane topology arguments in [Une nouvelle preuve du théorème de point fixe de Handel. Geom. Topol.10(2006), 2299–2349]. These methods improved the result by proving the existence of a simple closed curve of index 1. We give a new, simpler proof of this improved version of the theorem and generalize it to non-oriented cycles of links at infinity.

A Census of Planar Triangulations

1962 ◽  
Vol 14 ◽  
pp. 21-38 ◽  
Author(s):  
W. T. Tutte
Keyword(s):  
Finite Number ◽  
Interior Point ◽  
Simple Closed Curve ◽  
Closed Curve ◽  
Closed Region ◽  
Straight Segments

Let P be a closed region in the plane bounded by a simple closed curve, and let S be a simplicial dissection of P. We may say that S is a dissection of P into a finite number α of triangles so that no vertex of any one triangle is an interior point of an edge of another. The triangles are ‘'topological” triangles and their edges are closed arcs which need not be straight segments. No two distinct edges of the dissection join the same two vertices, and no two triangles have more than two vertices in common.

scholarly journals AN INTEGRATION APPROACH TO THE TOEPLITZ SQUARE PEG PROBLEM

2017 ◽  
Vol 5 ◽  
Author(s):  
TERENCE TAO
Keyword(s):  
Simple Closed Curve ◽  
Closed Curve ◽  
Finite Sets ◽  
Real Numbers ◽  
Integration Approach ◽  
Sign Patterns ◽  
Lipschitz Constants ◽  
Lipschitz Graphs ◽  
Periodic Version ◽  
Rule Out

The ‘square peg problem’ or ‘inscribed square problem’ of Toeplitz asks if every simple closed curve in the plane inscribes a (nondegenerate) square, in the sense that all four vertices of that square lie on the curve. By a variety of arguments of a ‘homological’ nature, it is known that the answer to this question is positive if the curve is sufficiently regular. The regularity hypotheses are needed to rule out the possibility of arbitrarily small squares that are inscribed or almost inscribed on the curve; because of this, these arguments do not appear to be robust enough to handle arbitrarily rough curves. In this paper, we augment the homological approach by introducing certain integrals associated to the curve. This approach is able to give positive answers to the square peg problem in some new cases, for instance if the curve is the union of two Lipschitz graphs $f$, $g:[t_{0},t_{1}]\rightarrow \mathbb{R}$ that agree at the endpoints, and whose Lipschitz constants are strictly less than one. We also present some simpler variants of the square problem which seem particularly amenable to this integration approach, including a periodic version of the problem that is not subject to the problem of arbitrarily small squares (and remains open even for regular curves), as well as an almost purely combinatorial conjecture regarding the sign patterns of sums $y_{1}+y_{2}+y_{3}$ for $y_{1},y_{2},y_{3}$ ranging in finite sets of real numbers.

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