We give characterizations of a finite group $G$ acting symplectically on a rational surface ($\mathbb{C}P^2$ blown up at two or more points). In particular, we obtain a symplectic version of the dichotomy of $G$-conic bundles versus $G$-del Pezzo surfaces for the corresponding $G$-rational surfaces, analogous to a classical result in algebraic geometry. Besides the characterizations of the group $G$ (which is completely determined for the case of $\mathbb{C}P^2\# N\overline{\mathbb{C}P^2}$, $N=2,3,4$), we also investigate the equivariant symplectic minimality and equivariant symplectic cone of a given $G$-rational surface.

中文翻译：

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辛有理 $G$-曲面和等变辛锥

我们给出了一个有限群 $G$ 的表征，它辛地作用在有理曲面上（$\mathbb{C}P^2$ 在两个或多个点上爆炸）。特别是，我们获得了对应的 $G$-有理曲面的 $G$-圆锥丛与 $G$-del Pezzo 曲面的二分法的辛版本，类似于代数几何中的经典结果。除了 $G$ 群的表征（对于 $\mathbb{C}P^2\# N\overline{\mathbb{C}P^2}$ 的情况完全确定，$N=2,3 ,4$)，我们还研究了给定 $G$-有理曲面的等变辛极小和等变辛锥。