10 - Ancillary results

Summary

In this chapter we collect several topics for which good references are either not available or are too scattered in the literature.

In Section 10.1 we prove three well-known results on birational maps of surfaces for excellent 2-dimensional schemes.

General properties of seminormal schemes are studied in Section 10.2. Seminormality plays a key role in the study of lc centers and in many inductive methods involving lc and slc pairs.

In Section 10.3 we gather, mostly without proofs, various vanishing theorems that we use. Section 10.4 contains resolution theorems that are useful for nonnormal schemes and in Section 10.5 we study the action of birational maps on differential forms. The basic theory of cubic hyperresolutions is recalled in Section 10.6.

Assumptions In Section 10.1 we work with excellent surfaces and in Section 10.2 with arbitrary schemes. In later sections characteristic 0 is always assumed.

Birational maps of 2-dimensional schemes

Here we prove the Hodge Index theorem, the Grauert–Riemenschneider vanishing theorem, Castelnuovo's contraction theorem and study rational singularities of surfaces. Instead of the usual setting, we consider these for excellent 2-dimensional schemes.

Theorem 10.1 (Hodge Index theorem) Let X be a 2-dimensional regular scheme, Y an affine scheme and f: X → Y a proper and generically finite morphism with exceptional curves ∪C_i. Then the intersection form (C_i · C_j) is negative-definite.

Proof It is enough to consider all the exceptional curves that lie over a given y ∈ Y.