论文标题
图意的多曲线和曲线有理表面
Toric multisections and curves in rational surfaces
论文作者
论文摘要
我们研究了承认有效的圆环动作的4个manifolds中的嵌入式表面的多片。我们表明，当且仅当它承认有效的圆环动作时，一个简单的4个manifold就承认了一个多属。 Orlik和Raymond表明，这些4个manifolds恰恰是$ \ Mathbb {cp}^2 $，$ \ OVERLINE {\ MATHBB {CP}^2} $和$ S^2 \ times S^2 $的连接副本总和。因此，这些4个manifolds中的嵌入式表面可以在一个属的表面上进行示意性编码。我们的主要结果是，$ \ mathbb {cp}^1 \ times \ mathbb {cp}^1 $中的每条平滑，复杂的曲线都可以在有效的桥接位置，相对于一个4段。我们还分析了一个多属属的代数拓扑。
We study multisections of embedded surfaces in 4-manifolds admitting effective torus actions. We show that a simply-connected 4-manifold admits a genus one multisection if and only if it admits an effective torus action. Orlik and Raymond showed that these 4-manifolds are precisely the connected sums of copies of $\mathbb{CP}^2$, $\overline{\mathbb{CP}^2}$, and $S^2\times S^2$. Therefore, embedded surfaces in these 4-manifolds can be encoded diagrammatically on a genus one surface. Our main result is that every smooth, complex curve in $\mathbb{CP}^1\times\mathbb{CP}^1$ can be put in efficient bridge position with respect to a genus one 4-section. We also analyze the algebraic topology of genus one multisections.
