The vanishing cycles of curves in toric surfaces I

Crétois, Rémi; Lang, Lionel

doi:10.1112/s0010437x18007200

Cited by 13 publications

(39 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Theorem A also addresses a conjecture that was independently formulated by the author in [Sal16] in the case of X = CP 2 , and in full generality by Crétois-Lang [CL17a].…”

Section: Introductionmentioning

confidence: 56%

“…We write ∆(d) := ∆ ∩ dZ 2 , relative to any such embedding. The following is a combination of Propositions 7.13 and 7.16 of [CL17a].…”

Section: Linear Systems In Toric Surfacesmentioning

confidence: 89%

“…The purpose of this section is to give a minimal account of the work of Crétois-Lang in [CL17a]. We do not attempt to give a detailed summary of the theory of toric surfaces; the interested reader is referred to [CL17a] and the references therein.…”

Section: Linear Systems In Toric Surfacesmentioning

confidence: 99%

“…Theorem A is proved using a combination of methods from toric geometry and the theory of the mapping class group. On the toric end of the spectrum, we make essential use of the powerful results developed by Crétois-Lang in [CL17a]. The centerpiece of their theory is a combinatorial model for a curve C 0 ∈ M(L) based around a convex lattice polygon.…”

Section: Introductionmentioning

confidence: 99%

“…Crétois-Lang developed their methods in order to address Question 1.2 and Conjecture 1.4 in the case r ≤ 2, and obtained complete answers in these cases. See [CL17a] for the case r = 1, and [CL17b] for the case r = 2, as well as the case where the general fiber is hyperelliptic.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Monodromy and vanishing cycles in toric surfaces

Salter

2018

Invent. math.

View full text Add to dashboard Cite

Given an ample line bundle on a toric surface, a question of Donaldson asks which simple closed curves can be vanishing cycles for nodal degenerations of smooth curves in the complete linear system. This paper provides a complete answer. This is accomplished by reformulating the problem in terms of the mapping class group-valued monodromy of the linear system, and giving a precise determination of this monodromy group. arXiv:1710.08042v2 [math.AG] 6 Dec 2018Acknowledgements. The author would like to extend his warmest thanks to R. Crétois and L. Lang for helpful discussions of their work. He would also like to acknowledge C. McMullen for some insightful comments on a preliminary draft, and M. Nichols for a productive conversation. A special thanks is due to an anonymous referee for a very careful reading of the preprint and for many useful suggestions, both mathematical and expository. Mapping class groupsThis section collects background material on mapping class groups that will be used throughout the arguments in Sections 3 through 9. Most of the material can be found in [FM12] and so will only be touched on briefly. The exception to this is the D n relation of Section 2.3, which will consequently be dealt with in greater detail.2.1. Basics. The material in this section is almost certainly well-known to a reader conversant in mapping class groups, but is included so as to fix notation and terminology.

show abstract

“…Theorem A also addresses a conjecture that was independently formulated by the author in [Sal16] in the case of X = CP 2 , and in full generality by Crétois-Lang [CL17a].…”

Section: Introductionmentioning

confidence: 56%

“…We write ∆(d) := ∆ ∩ dZ 2 , relative to any such embedding. The following is a combination of Propositions 7.13 and 7.16 of [CL17a].…”

Section: Linear Systems In Toric Surfacesmentioning

confidence: 89%

Section: Linear Systems In Toric Surfacesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Monodromy and vanishing cycles in toric surfaces

Salter

2018

Invent. math.

View full text Add to dashboard Cite

show abstract

Complements to Ample Divisors and Singularities

Libgober

2021

Handbook of Geometry and Topology of Singularities II

View full text Add to dashboard Cite

A note on the Severi problem for toric surfaces

Lang

Tyomkin

2022

Math. Ann.

Self Cite

View full text Add to dashboard Cite

In this note, we make a step towards the classification of toric surfaces admitting reducible Severi varieties. We provide two families of toric surfaces admitting reducible Severi varieties. The first family is general, and is obtained by a quotient construction. The second family is exceptional, and corresponds to certain narrow polygons, which we call kites. We introduce two types of invariants that distinguish between the components of the Severi varieties, and allow us to provide lower bounds on the numbers of the components. The sharpness of the bounds is verified in some cases, and is expected to hold in general for ample enough linear systems. In the appendix, we establish a connection between the Severi problem and the topological classification of univariate polynomials.

show abstract

The vanishing cycles of curves in toric surfaces I

Cited by 13 publications

References 29 publications

Monodromy and vanishing cycles in toric surfaces

Monodromy and vanishing cycles in toric surfaces

Complements to Ample Divisors and Singularities

A note on the Severi problem for toric surfaces

Contact Info

Product

Resources

About