Wise choice of notation and terminology can help clarify thinking. I would like to make wise choices, constrained somewhat by convention (although I may try to selectively push the envelope), and by lack of ability to hold everything in my head. Here are some conventions I may try to follow (although not religiously).

• A, B, C: rings. I believe using A for a ring comes from french: A is for anneaux. Elements of A might be called some flavor of a, and similarly for B and C.
• i, j, k, l, m, n, p, q: integers. p is often prime.
• p and q may be points too.
• k, K: Different flavors of K are for fields. k is often the ground field. K(A) is the fraction field of an integral domain. K(X) is the function field of an integral scheme X. kappa(p) is the residue field of a point. (Discussion on mathoverflow tentatively convinced me to go with this.)
• L: field
• M is a matrix
• U is an open subset or subscheme
• W, X, Y, Z: schemes. w is a point of W. X = Spec A, Y = Spec B.
• Caligraphic font (\mathcal) is for quasicoherent sheaves.
• Fraktur font is for prime ideals (especially m, n, p, q). m and n are maximal ideals.
• Currently, categories whose names are english words are in bold, but general categories are just in \mathcal. That’s not very consistent. (Perhaps all categories should be in bold?)

Changes I’m considering

Here are some changes I am leaning toward. I likely won’t implement the changes until summer 2011 when more of the notes are posted, so please comment (pro and con, and either here or by email, anonymously or not).

I’m now strongly leaning toward using “open embedding” instead of “open immersion”, and similarly for “closed embedding” and “locally closed embedding”. Enough others have used it that there is sufficient cover. (I’ve most recently seen it in Rob Lazarsfeld’s wonderful “Positivity in Algebraic Geometry I”.) Brian Conrad argues otherwise, and has written to me: “Is there any algebraic geometry reference which uses “open embedding” for “open immersion” in settings where schemes seriously intervene (i.e., not just reduced varieties over a field, which is what I assume comes up for Lazarsfeld’s book) … Please be very careful about changing widely-used terminology unless you’ve checked that the proposed switch is consistent with many references which treat algebraic geometry in a degree of generality comparable to what you are doing in the book; otherwise it can create confusion much as Hartshorne’s textbook has done with the notion of “coherent”.” On the other hand, Dan Abramovich and Georges Elencwajg are I think supportive. (Update some time before March 2012: the change to “embedding” is now made.)

For points of projective space: [0 : 1 : 2] instead of [0 ; 1 ; 2] (and [0,1,2] and (0,1,2) etc.). Ezra Miller made a good argument that the colons suggest ratios. I would have to be cautious of spacing. (Update some time before March 12: I now use [0,1,2].)