Title: Symplectic rational $G$-surfaces and equivariant symplectic cones,
Abstract: 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.