  In this paper we classify weak Fano varieties that can be obtained by
blowing-up general points in prime Fano varieties. We also classify spherical
blow-ups of Grassmannians in general points, and we compute their effective
cone. These blow-ups are, in particular, Mori dream spaces. Furthermore, we
compute the stable base locus decomposition of the blow-up of a Grassmannian in
one point, and we show how it is determined by linear systems of hyperplanes
containing the osculating spaces of the Grassmannian at the blown-up point, and
by the rational normal curves in the Grassmannian passing through the blown-up
point.
