Smooth Projective Symmetric Varieties with Picard Number Equal to One

We classify the smooth projective symmetric varieties with Picard number equal to one. Moreover we prove a criterion for the smoothness of the simple (normal) symmetric varieties whose closed orbit is complete. In particular we prove that a such variety X is smooth if and only if an appropriate toric variety contained in X is smooth. A Gorenstein normal algebraic variety X over C is called a Fano variety if the anticanonical divisor is ample. The Fano surfaces are classically called Del Pezzo surfaces. The importance of Fano varieties in the theory of higher dimensional varieties is similar to the importance of Del Pezzo surfaces in the theory of surfaces. MoreoverMori’s program predicts that every uniruled variety is birational to a fiberspace whose general fiber is a Fano variety (with terminal