 By G. Hardy

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Additional info for A Course Of Pure Mathematics

Example text

Ify takes more than two values and x is not GF-function on p2 then we can find h such that lx, h(y)1 does not contain GF-function. PROOF: Remark that any projective line, where y takes at least two values)contains points, where y takes any value from the set y(P2). Indeed for any fixed value y' we have just to take the intersection of the line above with projective line, where y takes value y' and x takes value 1 at some point. Now we fix a line pI with y non constant and I want to find 41 h such that hey) is not GF-function on pl.

Using generality condition for the intersection of this space with Vai, si in pI we obtain that generic value on pH coincides with generic value of x on pl. Then (x) is the point of singular value of x for any line pI containing (x). Thus x is delta-function of (x) on P(L*). It finishes the proof of sublemma. Consider now the case when x is constant on P(L*). Linear envelope of the special 39 strata for flag function is also a special stratum. Thus any subset of points which lie in the union of special strata for codimension one subspaces in L lie also in some codimension two subspace if it fits in some codimension one subspace.

Thus we reduce our lemma to that case. I only have to remark that the union of two subsets isomorphic to valuation ideals will also be like one if k contains exactly two elements. 1, we proved the lemma. Thus we have two functions x, y (we use it instead hey) further) with values 0,1 and the following properties: p2 consists of the three level sets for x, y denoted as Sao, S1O, SOIl respectively, and any projective line lies in the union of no more than two of them. 3 conditions for the case of two valued functions.