Models of R-supercompactness

Under various appropriate hypotheses it is shown that there is only one determinacy model of the form L(R, μ) in which μ is a supercompact measure on P!1 (R). In particular, this gives a positive answer to a question asked by W.H. Woodin in 1983. It is also proven that it is relatively consistent that there are different ZF models of the form L(R, μ) in which μ witnesses that !1 is R-supercompact.