On the \(K^2\) of degenerations of surfaces and the multiple point formula (Q2642234)

Inspired by work of Guido Zappa in the 1940-50's, the authors define a Zappatic surface as a projective surface \(X\) which is a union of smooth surfaces such that the singular locus of \(X\) consists of: smooth irreducible curves along which two components of \(X\) meet transversally; a finite number of points \(x\) such that the singularity of \(X\) at \(x\) is analytically equivalent to the cone over a projectively normal stick curve of genus \(\leq 1\). The case of main interest is that of planar Zappatic surfaces, i.e., Zappatic surfaces whose components are planes. It is very hard to decide whether a given Zappatic surface can be obtained as flat limit of smooth projective surfaces. Here, the authors address this problem for a class of Zappatic surfaces, called good Zappatic surfaces, whose singularities belong to certain special types. They are able to compute \(K^2\) of the general element of a \(1\)-parameter family of projective surfaces that specializes to a good Zappatic surface. In addition, they prove an inequality, the Multiple Point Formula, which is satisfied by the combinatorial data attached to a good Zappatic which is the flat limit of a family of smooth surfaces. Using these two basic results, they reobtain and extend results of Zappa, giving restrictions on the type of singularities of a good planar Zappatic surface which is the limit of smooth surfaces of a certain type (scroll, surface of general type, and so on\dots).