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Extra resources for An Introduction to the Theory of Algebraic Surfaces: Notes by James Cohn, Harvard University, 1957–58
And ~ a normalization of V a. has an affine Then ~ is an affine have the same quotient We note that ~ is detelu~ined only up to a biregular transformation since we only know ~ ~' coordinate ring. Let W W be the center of ~ on V. Clearly is represented by a minimal prime ideal in R. ring of ~ is because Hence the valuation C~W(~/k ). By a set of uniformizin~ coordinates of we m~an any set of uniformizing coordinates of W Let dim W = r-1 ~Or be an r-fold differential of on V. k(V)/k, ~Or # O; let be a prime divisor of the first kind with respect to be uniformizing coordinates of ~ .
Zm) a (~) AijE~. c 8 ~ since R = k~Va], be t h e i n t e g r a l c l o s u r e of be the locus of variety, and Di ~ K/k Va holds. the determinant is be the center of ~ Choose an affine representative re~esentative D i ~ j = 5ij o Hence we can find such This proves (a) of Def. 1. Clearly We want to choose over and let R in its quotient field~ k. and ~ a normalization of V a. has an affine Then ~ is an affine have the same quotient We note that ~ is detelu~ined only up to a biregular transformation since we only know ~ ~' coordinate ring.
We have seen that ~ W k(W)-mcdule. By Prop. 2 and Prop. 3(c), can be identified with the space ~ (W) of W derivations of Let GO q k(W)/k. be a q-fold differential, and assume &)q is regular at W. , Dq)~ ~ . , Dq). , DqgO~W, cosets of DI, Def. , on the traces ~i = Tr~i' i = i, D Q@~ q. (~)q be a q-fold differential which is rega]ar at W. , Dq) q>s, we fix define q-s derivations D~ ' "'" D ,.. , s. We now list some simple properties of the trace of a differential: (2) (3) -- If ~ w ( V / k ) , then Trwd ~ = d~ where ~ = TrW ~ .
An Introduction to the Theory of Algebraic Surfaces: Notes by James Cohn, Harvard University, 1957–58 by Oscar Zariski (auth.)