In this paper we introduce a new way of displacing Lagrangian fibers in toric
symplectic manifolds, a generalization of McDuff's original method of probes.
Extended probes are formed by deflecting one probe by another auxiliary probe.
Using them, we are able to displace all fibers in Hirzebruch surfaces except
those already known to be nondisplaceable, and can also displace an open dense
set of fibers in the weighted projective space P(1,3,5) after resolving the
singularities. We also investigate the displaceability question in sectors and
their resolutions. There are still many cases in which there is an open set of
fibers whose displaceability status is unknown.Comment: 53 pages, 26 figures; v3: minor corrections and updated references.
To appear in Algebraic & Geometric Topolog
