  In this paper we compute the homotopy groups of the symplectomorphism groups
of the 3-, 4- and 5-point blow-ups of the projective plane (considered as
monotone symplectic Del Pezzo surfaces). Along the way, we need to compute the
homotopy groups of the compactly supported symplectomorphism groups of the
cotangent bundle of $\mathbf{RP}^{2}$ and of $\mathbf{C}^*\times\mathbf{C}$. We
also make progress in the case of the $A_n$-Milnor fibres: here we can show
that the (compactly supported) Hamiltonian group is contractible and that the
symplectic mapping class group embeds in the braid group on $n$-strands.
