"Semitoric systems are a type of four-dimensional integrable system for which one of the integrals generates a global S1-action; these systems were classified by Pelayo and Vu Ngo. c in terms of five symplectic invariants. We introduce and study semitoric families, which are one-parameter families of integrable systems with a fixed S1-action that are semitoric for all but finitely many values of the parameter, with the goal of developing a strategy to find a semitoric system associated to a given partiallist of semitoric invariants. We also enumerate the possible behaviors of such families at the parameter values for which they are not semitoric, providing examples illustrating nearly all possible behaviors, which describes the possible limits of semitoric systems with a fixed S1-action. Furthermore, we introduce natural notions of blowup and blowdown in this context, investigate how semitoric families behave under these operations, and use this to prove that each Hirzebruch surface admits a semitoric family with certain desirable invariants; these families are related to the semitoric minimal model program. Finally, we give several explicit semitoric families on the first and second Hirzebruch surfaces showcasing various possible behaviors of such families which include new semitoric systems"-- Provided by publisher.
1. Introduction
2. Preliminaries on integrable and semitoric systems
3. Fixed-$\mathbb {S}^1$ families
4. Toric type blowups and blowdowns
5. Semitoric transition families on Hirzebruch surfaces
6. Explicit semitoric families on $W_{1}(\alpha ,\beta )$
7. Explicit semitoric families on $W_{2}(\alpha ,\beta )$
A. Comparison of the systems in and using the height invariant
Yohann Le Floch, Universite de Strasbourg et CNRS, France, and Joseph Palmer, University of Illinois at Urbana-Champaign, Illinois, United States, and University of Antwerp, Antwerpen, Belgium
