Bridge trisections in rational surfaces

TL;DR

This paper explores the classification of complex curves in rational surfaces using bridge trisections, demonstrating the existence of efficient trisections for various complex surfaces and analyzing their implications for manifold topology.

Contribution

It introduces the concept of efficient bridge trisections for complex curves in rational surfaces and provides explicit examples and diagrams illustrating their existence and properties.

Findings

Every curve in C^2C^1C^1C^1 admits an efficient bridge trisection.

Many complex surfaces, including hypersurfaces and elliptic surfaces, admit efficient trisections.

Trisection genus remains invariant for certain homeomorphic but non-diffeomorphic manifolds.

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 $\mathbb{CP}^2$ and $\mathbb{CP}^1\times\mathbb{CP}^1$. 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 $\mathbb{CP}^2$ or $\mathbb{CP}^1\times\mathbb{CP}^1$ 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 $\mathbb{CP}^3$, the elliptic surfaces $E(n)$, the Horikawa surfaces $H(n)$, and complete intersections of hypersurfaces in $\mathbb{CP}^N$. 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.