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]).
Zur algebraischen Geometrie. IX
Es ist prinzipiell wichtig, geometrische Gebilde durch Koordinaten darstellen zu konnen. Ist das namlich fur eine bestimmte Art von Gebilden G einmal geschehen, so hat es einen Sinn, von einer algebraischen Mannigfaltigkeit oder einem algebraischen System von Gebilden G zu sprechen und die gesamte Theorie der algebraischen Mannigfaltigkeiten (Zerlegung in irreduzible, Dimensionsbegriff, Begriff der allgemeinen Elemente einer irreduziblen Mannigfaltigkeit) darauf anzuwenden. Erwunscht ist dabei, das die Gesamtheit aller Gebilde G der betrachteten Art (eventuell nach Hinzufugung von geeigneten Grenzgebilden) eine algebraische Mannigfaltigkeit darstellt, also durch algebraische Gleichungen in den Koordinaten charakterisiert werden kann.
A new class of numerical sequences and its applications to sine and cosine series
First, it is proved that the class of classical quasi-monotone sequences is not comparable to the newly defined class of sequences of ‘rest bounded variation’. Considering this result, we prove three sample theorems for sequences of rest bounded variation, being analogues of the theorems proved earlier for monotone or quasi-monotone sequences. One of them gives a partial answer to a question raised by R. P. Boas Jr.