  We study invariants coming from certain optimisation problems for nef
divisors on surfaces. These optimisation problems arise in work of the author
and collaborators tying obstructions to embeddings between symplectic
4-manifolds to questions of positivity for (possibly singular) algebraic
surfaces. We develop the general framework for these invariants and prove
foundational results on their structure and asymptotics. We describe the
connections these invariants have to Embedded Contact Homology (ECH) and the
Ruelle invariant in symplectic geometry, and to min-max widths in the study of
minimal hypersurfaces. We use the first of these connections to obtain optimal
bounds for the sub-leading asymptotics of ECH capacities for many toric
domains.
