Annals of Mathematics, 165 (2007), 335–395 On the K 2 of degenerations of surfaces and the multiple point formula By A. Calabri, C. Ciliberto, F. Flamini, and R. Miranda* Abstract In this paper we study some properties of reducible surfaces, in particular of unions of planes. When the surface is the central fibre of an embedded flat degeneration of surfaces in a projective space, we deduce some properties of the smooth surface which is the general fibre of the degeneration from some combinatorial properties of the central fibre. In particular, we show that there are strong constraints on the invariants of a smooth surface which degener- ates to configurations of planes with global normal crossings or other mild singularities. Our interest in these problems has been raised by a series of interesting articles by Guido Zappa in the 1950’s. 1. Introduction In this paper we study in detail several properties of flat degenerations of surfaces whose general fibre is a smooth projective algebraic surface and whose central fibre is a reduced, connected surface X ⊂ P r , r  3, which will usually be assumed to be a union of planes. As a first application of this approach, we shall see that there are strong constraints on the invariants of a smooth projective surface which degener- ates to configurations of planes with global normal crossings or other mild singularities (cf. §8). Our results include formulas on the basic invariants of smoothable sur- faces, especially the K 2 (see e.g. Theorem 6.1). These formulas are useful in studying a wide range of open problems, such as what happens in the curve case, where one considers stick curves, i.e. unions of lines with only nodes as singularities. Indeed, as stick curves are used to *The first two authors have been partially supported by E.C. project EAGER, contract n. HPRN-CT-2000-00099. The first three authors are members of G.N.S.A.G.A. at I.N.d.A.M. “Francesco Severi”.