On degenerations of surfaces

TL;DR

This paper reviews and unifies results on how smooth surfaces can degenerate into unions of planes, highlighting constraints on invariants and providing examples of such degenerations.

Contribution

It offers a comprehensive exposition of degenerations of surfaces to unions of planes, connecting combinatorial features of the central fibre to properties of the smooth general fibre.

Findings

Strong constraints on invariants of degenerating surfaces

Examples of embedded degenerations to unions of planes

Unified exposition of existing results

Abstract

This paper surveys and gives a uniform exposition of results contained in papers published by the team of authors. The subject is degenerations of surfaces, especially to unions of planes. More specifically, we deduce some properties of the smooth surface which is the general fibre of the degeneration from combinatorial features 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. Finally we consider several examples of embedded degenerations of smooth surfaces to unions of planes. Our interest in these problems has been raised by a series of interesting articles by Guido Zappa in 1950's.