The author has gone to great trouble to guide the reader, to explain what he is doing at every stage, and finally brings him, slightly dazed perhaps, to the frontier of present-day knowledge in certain branches of algebraic geometry. it must be said that the book is beautifully written, and a remarkable piece of mathematics. It is emphasised that " the main purpose of the book is to present a detailed and connected treatment of the properties of intersection-multiplicities, which is to include all that is necessary and sufficient to legitimise the use made of these multiplicities in classical algebraic geometry, especially of the Italian school ". Andre Weil explains in his preface that his book arose "from the necessity of giving a firm basis to Severi's theory of correspondences on algebraic curves, especially in the case of characteristic p ≠ 0 (in which there is no transcendental method to guarantee the correctness of the results obtained by algebraic means ), this being required for the solution of a long outstanding problem, the proof of the Riemann hypothesis in function-fields ". It has been universally recognised as an important and attractive branch of mathematics, but many mathematicians have been prevented from cultivating it by the feeling that its principles and methods are only fully understood by a small number of people, and that the novice wishing for initiation must undergo a long training in the avoidance of clear-cut (if laborious ) algebraic methods, and in the development of his geometrical intuition. The Mathematical Gazette 31 (297) (1947), 293- 294.Īlgebraic geometry is that branch of mathematics which deals with the geometrical interpretation of algebraic equations.
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