A generic K3 surface of degree 4 may be embedded as a nonsingular quartic surface in P3. Let /: X -» Spec C[[t]] be a family of quartic surfaces such that the generic fiber is regular. Let ¿\), 2%, 24 be respectively a nonsingular quadric in P3, a cone in P3 over a nonsingular conic and a rational, ruled surface in P9 which has a section with self intersection —4. We show that there exists a flat, projective morphism /': X' —► Spec C[[i]] and a map p: Spec C[[<]] -» Spec Q[r]] such that (i) the generic fiber of /' and the generic fiber of the pull-back of / via p are isomorphic, (ii) the fiber X¿ of /' over the closed point of Spec C[[/]] has only insignificant limit singularities and (iii) Xg is either a quadric surface or a double cover of 2^,, 2° or 24. The theorem is proved using the geometric invariant theory. The purpose of this paper is to prove projective analog of the Kulikov-PerssonPinkham theorem [7], [11] via the geometric invariant theory in a special case. We recall that a nonsingular, projective surface, V, over C is called a Tí 3 surface if Hl(V, ov) = 0 and the canonical divisor class of the surface is trivial. It is called a AT3 surface of degree n if V carries a line bundle L with L • L = n. V is said to be generic if the rank of its Néron-Severi group is equal to one. If L is a line bundle on a generic ÄT3 surface V such that L ■ L = 4, then, the linear system \L\ has no fixed components and embeds V into P3 as a quartic surface [8]. Conversely, a nonsingular quartic surface is a 7c3 surface of degree 4. Let 5 denote Spec C[[/]]. A family of surfaces over 5 is a flat, projective morphism, f : X -» S such that the generic geometric fiber of f is a nonsingular, connected surface. A family of surfaces, f: A" —» S is called a modification of the family f : X -» S if there exists a map p: S -» 5 such that the generic fiber of f and the generic fiber of the pull-back of f via p are isomorphic. We emphasize that a modification also is a projective morphism. Let 2n = a nonsingular quadric surface in P,, S" = a cone over a nonsingular conic in P3, and S4 = a rational, ruled surface in P9 which has a section whose selfintersection is equal to -4. We prove Theorem 1. Let f: .Y—> S be a family of surfaces such that the generic geometric fiber of f is isomorphic to a quartic surface. Then, there exists a (projective) Received by the editors November 12, 1979. 1980 Mathematics Subject Classification. Primary 14J10, 14J25; Secondary 14C30.
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