On Fermat varieties

(0.1) x? + x? + + x?+ί = 0 . Throughout this paper, we denote it by Xrm, or by X r m(p), when we need to specify the characteristic p of the base field k; we always assume that m ^ 0 (mod p). The purpose of this paper is to clarify the inductive structure of Fermat varieties of a common degree and of various dimensions, and apply it to the questions concerning the unirationality and algebraic cycles of a Fermat variety. The main results are stated as follows: THEOREM I. For any positive integers r and s, XZf is obtained from the product Xrm x X 8 m by 1) blowing up a subvariety isomorphic to X~ x X'*, 2) taking the quotient of the blown up variety with respect to an action of the cyclic group of order m, and 3) blowing down from the quotient two subvarieties isomorphic to x X'ΰ and Xlr x P