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for example r = k[x,y] ... j =
?? so .... ?? k[x,y,a,b]/ay=bx ?? .... ??? or ... k[x,y:0; a,b:1]/ay=bx ... ???....
?? so suppose that we specialize to some point (x,y) not (0,0) ... ????? then we get a linear dependency between a and b ... ???? but not at (0,0) ..... proj of k[a,b:1] as projective line ... ????.....
r graded comm ring .... j graded ideal in r .... as graded r-module .... take symmetric graded r-alg of it .... preserves presentation ... ??? ....
r = k[x,y,z:1] .... j =
?? specialize to (x,y,z) = (0,0,1) .... ???? ???z=1 then trivializes line object (1,0) .... ?? in effect getting k[a,b:1] .... ????? ...
?? so ... should (1,1) (???or something) then be ample here ???? ..... ??or maybe (2,1) ??? ......
ax,ay,az,bx,by,bz .... ay=bx .... ???? ....ax*by=ay*ay ... ax*bz=ay*az ... ay*bz=az*by ... az*by=bz*ay .... hmmm, those last two seem to be the same ... ?? and "close to a consequence of" the two before that?? ?? which might help the dimension arithmetic to work out ... ????.....
d=ax, e=ay, f=az, g=by, h=bz .... dg=ee, dh=ef, eh=fg ... ?? specializing to d=e=g=0 .... ????as giving a projective line, which is supposed to be the blown-up point .... specializing (?? ...) to d invertible .... ????? ??? both a and x being invertible .... ???? ...... ???well, maybe a isn't too important here ... ??? .....
?? well, when d is invertible, then g and h can be eliminated, co-eliminating the first two relations, and making the third one tautological .... ???? ..... ??? so that seems to help make it plausible that this really is the projective plane with a point blown-up .... ???? ......
?? so what about the picard group of this variety ... ????? ...... ?? well, so what about pulling line objects back along the projection ??? ..... ????? ..... possibility of non-ampleness here ... ??? ....
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