By Oscar Zariski (auth.)
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Extra resources for An Introduction to the Theory of Algebraic Surfaces: Notes by James Cohn, Harvard University, 1957–58
X r ~ is easily seen to be a set of uniformizing coordinates for W. Prop. ~r) ~ ~"W" a( ) Proof: Assume the ~i ~(~) 3(El and (b)implies = I, are uniformizing coordinates. ~(~) ~(~) Then ~. Hence ac ) is a unit in ~ . , r, AijE K. , z m] contained in field). k(~) = k(g) We call ~ on V (hence R 9 d i m W = r-l). such that W V a . , 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.
Let P' P ~j is regular at P. q Now, let P be any irreducible (r-1)-dimensional subvariety of V such that Pg P. It is immediately seen that the integral closure of the local ring (~r(V/k) is the ring of quotients of to the ultiplicative system are regul~r at ~-. , Dq)~ ~ . 8" ~ is ~ , ~ F P q If ~q is regular at P. is a regular differential of the field K/k, then OJq is regular on V I and conversely, provided V is non- singular. Remark- We shall assume q=l. The proof for larger values of q same (but with more indices ).
S. , s. o,s. of all derivations of and this proves the proposition. If W Prop. 3: is a simple subvariety of (a) ~ W is a free r-dimensional (b) W (c) ~ W / a ~ ~ W Proof: Let I' " " " ~r D 6~W if, sn%d only if, V/k of dimension ~-module ( ~ = s, then ~fw(V/k)~ and is a free s-dimensional ~/~ -module. be uniformizing coordinates of W. , c~ I form an CT -basis of are linearly independent and the module is free because the over k(V). ~, This proves (a). (b) is obvious. Let where the ~I ti be chosen so that ~s+i are uniformizing parameters of W.
An Introduction to the Theory of Algebraic Surfaces: Notes by James Cohn, Harvard University, 1957–58 by Oscar Zariski (auth.)