2013 Math 632 Daily Update

Guide for Continued Reading. You now have all the tools to go quickly through the classical material of Chapters IV and V of Hartshorne and related material. I suggest for a first read through: • IV §1, • IV §2 First three pages. Cor 2.4 is crucial. Try exercises 2.2, 2.3. Don’t be intimidated by differential forms–everything you need here we did in 631—now it should seem easy. Don’t use Hartshorne’s II §8 (he is just quoting technical stuff from commutative algebra texts anyway). Shafarevich is a better reference because it sticks to the main case where we have a smooth variety over a field, and everything is straightforward—this is the only case you need here. Also, just worry about the separable case on a first read through). If you are interested in char p, you can also continue in that section later. • IV §3 First three pages. Prop 3.1 and its corollaries are important. The theorem that “every curve embeds in P is an interesting application. Exercises: 3.1, 3.2, 3.3, 3.4, 3.6, • IV §4 First few pages. The rest mainly if you like number theory–this one section is a condensed version of a whole book on elliptic curves, so you might find an easier source. Exercises 4.1, 4.2, 4.4, 4.5. More if you like number theory and some catch your eye. • IV §5 Interesting, the idea of the canonical map is crucial. 5.1, 5.2, 5.3, 5.4, 5.5, 5.6 • IV §6 Interesting but not essential on a first read through; Somewhat specialized to a quirky interest of the author. • V Surfaces: All sections are good here. The point is to learn intersection theory on surfaces, which you can do also form Shafarevich. Read in turn and try a few exercises from each section that catch your eye. Other good sources for surfaces: Miles Reid’s ”Chapters on Algebraic Surfaces” is right to point (though some details, such as cohomology, are relegated to a black box, most of which you know from 632). Beauville’s Complex Algebraic Surfaces.