The twenty-sixth post is the December 17, 2012 version in the usual place. A large number of small improvements have been made, and the exposition has converged substantially, although there isn’t much big to report.
The (small) changes:
The terminology “local complete intersection” is changed to “regular embedding” (the more usual language, along with “regular immersion” — note that I have gone with “embedding” rather than “immersion” throughout). This was because the notation I was using would cause confusion because of the existence of an importance class of morphisms called “local complete intersection morphisms” (“lci morphisms”).
I am more careful about distinguishing the canonical bundle of a smooth projective variety (the determinant of the cotangent bundle) from the dualizing sheaf, before they are identified in the last chapter, to avoid anyone being confused about what is being invoked when.
A proof of the classification of vector bundles on the projective line (sometimes known as Grothendieck’s Theorem) is given in 19.5.5. (A proof is possible by painful algebra, and another proof is possible using Ext’s. Because of what else is discussed in the notes, I take a middle road: an easy proof without Ext’s, which relies on an easy calculation with 2 by 2 matrices. Note: I have not done the important fact that classifies extensions, which has tied my hands a little.) This required popping out “a first glimpse of Serre duality” into its own section (19.5).
Some discussion relating to Poncelet’s Porism is added in 20.10.7.
Still to do.
The introduction and the Zariski’s Main Theorem / Formal Functions chapter are still to be written. (And of course things like the index, figures, and formatting won’t be dealt with until the end.) The only other substantive things still to be written are a brief discussion of radicial morphisms (for arithmetic folks) and a brief proof and discussion of the Hodge Index Theorem (for geometric folks). Other than that, I expect essentially no other new material to be added.
I’ve caught up with almost all of the corrections and suggestions people have given me, except for a double-digit number of pages from both Peter Johnson and Jason Ferguson. (Thanks in particular to the comments of the reading group at Stanford for useful comments: Macky, Brian, Zeb, Michael, Evan, and Lynnelle!)
My to-do list still has 212 things on it. But that is a drastic improvement; the notes are converging rapidly.
I will not make any further progress until January. I hope to have a draft of the introduction done in January, and the final substantive chapter in February.
Happy holidays everyone!