A new version is now posted at the usual place (the June 11, 2013 version).   The main reason for this post is to have a reasonably current version on the web.  Thanks to many people for helpful comments — most recently, a number of comments from János Kollár, Jeremy Booher, Shotaro (Macky) Makisumi, Zeyu Guo, Shishir Agrawal, Bjorn Poonen, and Brian Lawrence (as well as many people posting here, whose names I thus needn’t list).

There are a large number of very small improvements, and I’ll list only a few in detail.  I’ve replaced the proof of the Fundamental Theorem of Elimination Theory (Theorem 7.4.7).  (This new proof is much more memorable for me.  It’s also shorter and faster, and generally better.  It was the proof I was trying to remember, which I heard in graduate school.  I couldn’t find it in any of the standard sources, and reproduced it from memory.  But it must be in a standard source, because it is certainly not original with me!)   Jeremy Booher’s comments have led to the completion discussion being improved a lot.

I had earlier called $\rm{Spec} A$, where $A$ is the subring $k[x^3, x^2, xy, y]$ of $k[x,y]$, the “knotted plane”, to suggest the picture that it was a plane, with the origin somehow “knotted” or “pinched”.  János Kollár pointed out that “knotted” is misleading, because it is not in any obvious way knotted.  I’ve changed this to “crumpled plane”, but this isn’t great either.  I’m now thinking about “pinched plane”.  Does anyone have a good suggestion for a name for this important example?

Still to do:  To repeat my comments from the previous post, as usual, the figures, index, and formatting have not yet been thought about.  My to-do list is quite short, so please complain about anything and everything (except figures, index, and formatting).  (Perhaps in the next posted version, I may have “TODO” appearing in the text indicating where there is still work in progress.)

Finally, just for fun, here is a picture of the two rulings on the quadric surface, from the Kobe skyline.  (It is clear why algebraic geometry is so strong in Japan!)

Kobe port (click to enlarge)