We give characterizations of a finite group G acting symplectically on a rational surface (ℂℙ2 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 ℂℙ2#Nℂℙ2, N = 2, 3, 4), we also investigate the equivariant symplectic minimality and equivariant symplectic cone of a given G-rational surface.
Bibliographical noteFunding Information:
shop and Conference on Holomorphic Curves and Low Dimensional Topology in Stanford, 2012. We are grateful to Igor Dolgachev for inspiring conversations during the FRG conference on symplectic birational geometry 2014 in University of Michigan. This work was partially supported by NSF Focused Research Grants DMS-0244663 and DMS-1065784, under FRG: Collaborative Research: Topology and Invariants of Smooth 4-Manifolds. WW is supported by Simons Collaboration Grant 524427.
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- Cremona group
- Finite group action
- Rational surface
- Symplectomorphism group