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1q-1 arbitrarily and get such a solution, defining -d q . z. (f0 . +f0 . • +(-1) f0 . • lq-1 lr .. lq-1 lo· .. lq-2 and the formal relations would be satisfied. But there is nothing to guarantee that f. •.. x. ). lo••. lq-1 lo lq-1 be checked is that in the Laurent series for fOi. • 1. •. 1. -1+... + <- l)qb . 0 lo ... lq-2 q q What must 54 II. 3. 10 there are no monomials of weight greater than d, so we must choose f0 . , etc. , 11 ... lq-1 so as to get rid of such terms in f0 . . , lq-1 suppose, some iq > i • q- 1 Then the relation shows that there can be in the expansion of f 0 .

Similarly, there is the II. K. THEOREM The set of points at which an algebraic variety is singular is an algebraic subvariety. All this gives a nice interpretation of the behavior of the coherent sheaf OX/Cl::' Consider for example the analytic subset of cr: 2 given by y2 = x3 + x2 31 II. I. 9 2 This will l>e a complex manifold everywhere except at (O, 0). dimcm/ma for a = (O, 0) where the number is 2. = 1, except Note that the tangent space to the variety at (O, 0) contains two linearly independent lines.

A a = 1, while we have seen that this dimension is two. We'll give some more concrete examples of line bundles now. The most important line bundles in algebraic geometry are certain line bundles on projective spaces, both algebraic and holomorphic. Pick homogeneous coordinates (z0 , ... , zn) . n . n+l m JP and (z0 , ... , zn+l) m lP . Let p denote the point (O, O, ... , O, 1) in 1Pn+1, and define a map from lPn+l - {p} to lPn by This map (called the projection from p) is both an algebraic and holomorphic map.

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