6 N.M. Katz: 1. Generalities on the Kodaira-Spencer Class and the Gauss-Manin Connection 1.0. The Geometric Setting Throughout this Section 1, we will consider the situation D ~ i ~X, J ~ U=X-D (1.0.1) s T in which T is an arbitrary base scheme, S is a smooth T-scheme (via g), which will play the role of a parameter space, X is a smooth S-scheme (via f), whose fibres over S are "parameterized" by S, and D is (via /) a union of divisors Di in X, each of which is smooth over S (hence also over T), and which have normal crossings relative to S (hence also relative to T). This situation persists after arbitrary change of base T' -~ T. In practice, X is usually proper over S, and should be thought of as a particularly nice compactification of the smooth "open" S-scheme U =X-D, which is psychologically prior to X. We allow D to be the empty divisor, corresponding to U being proper over S. We do
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