Algebraic Deformations of Polarized Varieties
Let V be a projectively embeddable complete non-singular variety of dimension n > 1. Let f be a projective embedding of V, U a non-singular variety, W a non-singular variety and φ a morphism of W onto U such that φ-1(u0) = f(V) for some point u0 of U. Denote by ∑(V) the set of all those complete non-singular fibres φ-1(u), u ∈ U, as we consider all possible (f, U, W). Suppose that we call members of ∑(V) (algebraic) deformations of V and propose to study ∑(V) from the stand point of algebraic geometry, as a generalization of the case of curves. This has been taken up at least locally by Kodaira, Spencer, Kuranishi and others in the case of characteristic 0 from a little more general point of view of complex manifolds (cf. [9] and references given in [16]).
Hyperelliptic and trigonal Fano threefolds
We classify Fano 3-folds with canonical Gorenstein singularities whose anticanonical linear system has no base points but does not give an embedding, and we classify anticanonically embedded Fano 3-folds with canonical Gorenstein singu- larities which are not intersections of quadrics. We also study rationality questions for most of these varieties.
Arithmetic on Algebraic Varieties
D. G. Northcott has recently contributed some interesting new theorems ([4a], [4b]) to a subject which I introduced in my thesis [1] under the above-given title, and which had been further developed by Siegel [2] and myself [3]. It is my purpose here, by making explicit some concepts which had remained implicit in these papers, to supply what seems to be the proper algebraic foundations for that theory, and to give a comprehensive account of its results, including some new ones, up to date.