Bridge trisections in rational surfaces

Abstract

We study smooth isotopy classes of complex curves in complex surfaces from the perspective of the theory of bridge trisections, with a special focus on curves in [Formula: see text] and [Formula: see text]. We are especially interested in bridge trisections and trisections that are as simple as possible, which we call efficient. We show that any curve in [Formula: see text] or [Formula: see text] admits an efficient bridge trisection. Because bridge trisections and trisections are nicely related via branched covering operations, we are able to give many examples of complex surfaces that admit efficient trisections. Among these are hypersurfaces in [Formula: see text], the elliptic surfaces [Formula: see text], the Horikawa surfaces [Formula: see text], and complete intersections of hypersurfaces in [Formula: see text]. As a corollary, we observe that, in many cases, manifolds that are homeomorphic but not diffeomorphic have the same trisection genus, which is consistent with the conjecture that trisection genus is additive under connected sum. We give many trisection diagrams to illustrate our examples.