Type III degenerations of K3 surfaces (Q1088747)

By results due to Kulikov, Persson, Persson-Pinkham the (birational) study of the semistable degenerations of K3 surfaces is reduced to the study of those degenerations \(\pi: X\to \Delta\) for which the canonical class \(K_ X\) is trivial and for such \(\pi\) the central fiber \(X_ 0\) is: a smooth surface (type I), a chain of elliptic ruled surfaces with rational surfaces on either end (type II), or a union of rational surfaces such that the double curves on each component form a cycle of rational curves and the dual graph of \(X_ 0\) is a triangulation of \(S^ 2\) (type III). Such degenerations are called Kulikov. The Kulikov degenerations of type II are well understood by previous work of the first author. - In this paper the study of type III is achieved. One classifies (up to specific modifications) the degenerations \(\pi\) and the combinatoric given by \(X_ 0=\cup V_ i\) in language of the arithmetic of the monodromy on \(H^ 2(X_ t,{\mathbb{Z}})\), \(X_ t\) being the general fiber. Several applications are also given.