Monodromy and vanishing cycles in toric surfaces

Abstract

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.

Problems, questions, and conjectures about mapping class groups

“…The first tautological class κ 1 , for example, measures the signatures (and first Pontryagin classes) of surface bundles and relates to the Weil-Petersson volume form on Teichmüller space. Salter [138] gives similar characterizations of the other κ i . We would like to have geometric interpretations for the unstable classes.…”

The vanishing cycles of curves in toric surfaces II

“…Our interest in this section is to find generators for the spin mapping class group. To that aim, we introduce the notion of admissible twist, following [Sal17]. Our interest for admissible curves is motivated by the following.…”

“…Few months after appeared the first version of the present work. Finally in [Sal17], Salter completed the description of vanishing cycles of curves in smooth toric surfaces.…”

“…The missing piece is to show that the spin mapping class group is generated by Dehn twists, the so-called admissible twists in [Sal17]. We take care of this in Theorem 5 that we prove in Section 4.…”

Monodromy of rational curves on toric surfaces

“…First, the image of can be thought of as a first approximation of the fundamental group . From this perspective, the study of the map is in line with the work [5] on fundamental groups of complement to discriminant varieties; see [3, 4, 21] for recent developments. The study of also contributes to the Galois theory of enumerative problems in algebraic geometry.…”