By I.R. Shafarevich, I.R. Shafarevich, R. Treger, V.I. Danilov, V.A. Iskovskikh

This *EMS* quantity involves elements. the 1st half is dedicated to the exposition of the cohomology idea of algebraic kinds. the second one half offers with algebraic surfaces. The authors have taken pains to provide the fabric carefully and coherently. The publication includes quite a few examples and insights on a variety of themes. This publication might be immensely necessary to mathematicians and graduate scholars operating in algebraic geometry, mathematics algebraic geometry, complicated research and similar fields. The authors are famous specialists within the box and I.R. Shafarevich is additionally identified for being the writer of quantity eleven of the *Encyclopaedia*.

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5 (Saturated=homogeneous+universal). A model M which is K—saturated is K,—homogeneous and κ+—universal. As a partial converse, if λΛ is K—homogeneous and D(ΛΛ) = 5(0), then Λ4 is K—saturated. Proof The tt+— universality of a κ;—saturated model is by the previous corollary while its K—homogeneity is clear from the definition. Now let Λ4 be ft—homogeneous with D(M) = 5(0), A C M of cardinality < K and p £ Sι(A). By the consistency of p there is a model λί containing A such that MA = λίA and p is realized in λί by some element a.

16 2. Constructing Models with Special Properties For uncountable theories the existence and uniqueness of prime and atomic models are all more complicated. The density of the isolated types does not, in general, guarantee the existence of an atomic model. 1), and there may be nonisomorphic prime models of a theory. Many of these issues are discussed in [Kni78]. 5). A model M is minimal ifλί^M = > λί = M. In many of the examples given above the prime model of a theory is also minimal. 8). To find a prime model which is not minimal we need look no further than (Q, <).

2. For T a countable complete theory let n(T) denote the number of countable models of T up to isomorphism. Determine the possible values of n(T), as T ranges over the countable complete theories. 1. For K an infinite cardinal, a theory T is K—categorical (or categorical in K) if T has a unique model of cardinality «, up to isomorphism. Later extensive attention will be given to countable theories which are categorical in some uncountable cardinality. Problem 1 asks for a characterization of NQ—categorical theories.