Before moving on to some more exotic topics let’s use these past couple of posts to construct an important example: A proper surface over an algebraically closed field of characteristic 0 that is not projective. This is important because every such curve is projective.
Using the tools we’ve developed, here is the construction. Let be a non-trivial infinitesimal extension of by (the canonical sheaf). How easy was that? Such a non-trivial extension exists because we’ve already shown that iso classes of extensions are in bijective correspondence with .
Here’s how we check it is not projective. We have an exact sequence by the definition of an infinitesimal extension . This induces a sequence (just by the first map being where is the first map of the previous sequence).
This gives a long exact sequence of cohomology: . Now we know the canonical sheaf is really just , so by knowledge of cohomology of projective space and . Also, the Picard group of projective space is generated by .
So if we can show is injective, then , and hence can’t be projective. Since , this just amounts to showing that . We’ll derive a contradiction if it does map to zero. If it does map to zero, then this is the zero map and hence is surjective. i.e. there is some invertible sheaf on such that , which is what it means for a sheaf to map to .
So we examine the exact sequence . It doesn’t matter what is to know , so we get that is surjective and hence we get a map which is the identity on . This gives us a splitting of . Thus . Ack! This means it was the trivial extension, which we assumed it wasn’t. Thus , which means is not projective.
Of course this isn’t the most concrete example of one of these things, but I don’t know of any other examples that are this easy to construct and prove. If you wanted something a little bit more concrete, you could actually pick a non-zero class in and piece together your infinitesimal extension.
I’m not making any promises, but next time I might start trying to figure out what gerbes are for a few posts, and then maybe try to tie them back in with how they relate to deformation theory.