  We investigate the existence and location of the surface phase known as the
"Surface-Attached Globule" (SAG) conjectured previously to exist in lattice
models of three-dimensional polymers when they are attached to a wall that has
a short range potential. The bulk phase, where the attractive intra-polymer
interactions are strong enough to cause a collapse of the polymer into a
liquid-like globule and the wall either has weak attractive or repulsive
interactions, is usually denoted Desorbed-Collapsed or DC. Recently this DC
phase was conjectured to harbour two surface phases separated by a boundary
where the bulk free energy is analytic while the surface free energy is
singular. The surface phase for more attractive values of the wall interaction
is the SAG phase. We discuss more fully the properties of this proposed surface
phase and provide Monte Carlo evidence for self-avoiding walks up to length 256
that this surface phase most likely does exist. Importantly, we discuss
alternatives for the surface phase boundary. In particular, we conclude that
this boundary may lie along the zero wall interaction line and the bulk phase
boundaries rather than any new phase boundary curve.
