It is shown that if there exists a supercompact cardinal then every set of reals, which is an element of L(R), is the projection of a weakly homogeneous tree. As a consequence of this theorem and recent work of Martin and Steel [Martin, D. A. & Steel, J. R. (1988) Proc. Natl. Acad. Sci. USA 85, 6582-6586], it follows that (if there is a supercompact cardinal) every set of reals in L(R) is deter...

Starting from a supercompact cardinal κ, we force and construct a model in which κ is both the least strongly compact and least supercompact cardinal and κ’s strong compactness, but not its supercompactness, is indestructible under arbitrary κ-directed closed forcing.