Surjectivity of the Period Map in the Case of Quartic Surfaces and Sextic Double Planes

The uniqueness has been proved by Piatetskiî-Sapiro and Safarevic. The proof of existence is outlined below. Let o be the closed point of A = Spec C[[t] ]. Let An be the finite covering of A obtained by extracting an nth. root of t; let on be the closed point of An. A family of surfaces over An is a flat, projective map ƒ: Xn —> An such that the generic fiber is smooth, connected, and two dimensional. The special fiber over on will be denoted by Xn. A family of surfaces is said to have ordinary singularities if Xn is reduced and has nonsingular components crossing normally. A modification of a family ƒ: X —> A is a family fn : Xn —> An together with an isomorphism of the generic fiber of fn with the pull-back of the generic fiber of/. Recall that a family of sextic double planes or quartic surfaces, ƒ: X —> A, induces a map of the generic point of A into D\ the map extends to a map n: A —> D if and only if the monodromy group is finite [2]. The existence part of Theorem 1 follows from