By Solomon Lefschetz

The publication opens with an summary of the consequences required from algebra and proceeds to the basic innovations of the final idea of algebraic kinds: normal element, size, functionality box, rational differences, and correspondences. A targeted bankruptcy on formal energy sequence with functions to algebraic forms follows. an in depth survey of algebraic curves comprises locations, linear sequence, abelian differentials, and algebraic correspondences. The textual content concludes with an exam of structures of curves on a surface.

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C. a power jP in the ideal On of Y is that f contain the points of a canonical set. c for Y to be pure is that the points of a canonical set have the same transcendency r and then dim Y = r. c. for On to be a primary ideal is that a canonical set consist of a single point. 14. Let Y* be the variety resulting from yr through a finite extension of K. Both varieties have the same points and the extension cannot raise the transcendency of any point. Hence dim V* < r =dim V. 4). Thus N still has transcendency r relative to the new groundfield and therefore dim Y* > r.

We are particularly interested in the least value of k for all lines through A. This value will be k itself, when and only when all the (xDa)hf = 0, for h < k, but (xDS'f =F 0. 11) this is equivalent to the property that all the partials off of order < k vanish at A, but that some of order k do not. It is seen at once that k = 1 when and only when A is an ordinary point. If k > 1 the point A is singular and conversely. Such a point is also known as a point of multiplicity k or a k-tuple point off (double, triple, · · · , point for k = 2, 3, · · · ).

In other words we may assume that oc =A(~*). • + · · · + F, = 0, 0 Fi E K[gi, · · · , g:], F, =I= 0. Thus G(X) = F(Xv · · · , Xr; A(X)) E K[X] is such that G(g*) = 0. Since Mis general, we also have G('Y}*) = 0. Since oc('Y}*) = 0, it follows that F,('YJi, · • • , 'YJ:) = 0. Since F,(X) =I= 0, 'YJi, · · · , 'YJ: are not algebraically independent, and hence transc N < r. 1). AL POINTS. DIMENSION 31 Let us discuss certain corollaries of the theorem. 2) Tliere is one and only one irreducible r dimensional variety V'" containing a given point M of transcendency r and M is a general point for V'".