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

我们从桥梁三等分理论的角度研究复杂曲面中复杂曲线的光滑同位素类，特别关注曲线${ℂ\mathbb{P}}^2$和${ℂ\mathbb{P}}^1\times {ℂ\mathbb{P}}^1$. 我们对桥三等分和尽可能简单的三等分特别感兴趣，我们称之为高效。我们证明了任何曲线${ℂ\mathbb{P}}^2$或者${ℂ\mathbb{P}}^1\times {ℂ\mathbb{P}}^1$承认有效的桥梁三等分。因为桥三等分和三等分通过分支覆盖操作很好地相关，我们能够给出许多允许有效三等分的复杂表面的例子。其中包括超曲面${ℂ\mathbb{P}}^3$, 椭圆面$乙(n)$, 堀川曲面$H(n)$, 和超曲面的完全交集${ℂ\mathbb{P}}^{ñ}$. 作为推论，我们观察到，在许多情况下，同胚但非微分流形具有相同的三等分属，这与三等分属在连通和下可加的猜想一致。我们给出了许多三等分图来说明我们的例子。