On the $K^2$ of degenerations of surfaces and the multiple point formula


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 degenerates 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.