Uniqueness of Generalized Fermat Groups in Positive Characteristic

Let X subset of P-K(m) be a smooth irreducible projective algebraic variety of dimension d >= 1, defined over an algebraically closed field K of characteristic p >= 0. Let n >= d + 1 and k >= 2 be integers. If p > 0, then we also assume k to be relatively prime to p and that k - 1 is not a power of p. We say that X is a generalized Fermat variety of type (d; k, n) if there is a Galois branched covering pi: X -> P-K(d) , with a group of Deck transformations Z(k)(n) congruent to H < Aut(X), whose branch divisor consists of n + 1 hyperplanes in general position (each one of branch order k). In this case, the group H is called a generalized Fermat group of type (d; k, n). We prove that, if either (i) p = 2 or (ii) p not equal 2 and (d; k, n) is not an element of {(2; 2, 5), (2; 4, 3)}, then a generalized Fermat variety of type (d; k, n) has a unique generalized Fermat group of that type.