# Classification and Parametrization in Algebraic Geometry and Algebraic Topology - Book recommendation

Logicians, in particular computability theorists, are interested in the distinction between a family of problems being solvable vs. solvable uniformly; in computable analysis and topology we often argue that a function $$f:X\to Y$$ cannot be computable by convincingly topologizing $$X$$ and $$Y$$ and showing that $$f$$ is discontinuous, thus cannot possibly be computable. Here I am thinking for example of the argument that the intermediate value theorem cannot be made constructive because a zero of a function does not depend in a continuous way on the function. Computer scientists deal with the necessary uniformity of parametrization, i.e. in parametric polymorphism, and this is something that comes up a lot in the construction of semantic models for polymorphic type theory. Bart Jacobs, for example, on his book on Categorical Logic and Type Theory, brings in the heavy machinery of Grothendieck fibrations to provide a good setting for the study of parametricity, and I find this to be beautifully done.

I am now convinced that parametrizing a family of things (for instance by maps into another thing) is an important topic manifesting all over mathematics and it would behoove me to learn about it; in particular I am curious about the core examples of this phenomena in geometry and topology. So I am looking for book recommendations which give a broad introduction to some of these ideas. Bear in mind I am aware of my complete ignorance in all that follows here and I would be glad to be set straight, and forgive the abundance of confused questions.

On the topology side we have many instances of the term "classifying space." The n-lab page on this is a great short summary and introduction. We have that the classifying space of a group, $$p: EG\to BG$$, "classifies" principal $$G$$-bundles; there is an explicit construction for $$EG$$ and $$BG$$ given in Hatcher at the end of the first chapter but without any mention of principal $$G$$ bundles, only focusing with the fact that it is $$K(G,1)$$ and unique up to homotopy type. Where can I read about the properties of the classifying space associated to a discrete group, or the classifying space associated to some of the classic topological groups like the linear algebraic groups of introductory Lie theory? (Both a topological and algebraic-geometric perspective on algebraic groups as varieties would be appreciated here.) How does the classifying space of a category generalize this construction? (I have a copy of Ieke Moerdijk's notes on classifying spaces and classifying topoi.)

In algebraic geometry (and topology as well) we have at a fairly elementary level the notion of Grassmannians as generalizing projective space and parametrizing subspaces of a vector space. The demand for, and importance of, spaces that parametrize a family of curves or surfaces - i.e. a family of elliptic curves - seems to lead algebraic geometers to search for bigger categories of spaces that allow us to construct these notions - when varieties no longer suffice we move onto schemes, and from then on to algebraic spaces and stacks. But where does one read about the classical questions that ended up leading to such notions as "classifying stack" or "moduli space?" For instance - Hilbert and Chow varieties (schemes), the Quot scheme, the classical questions of moduli theory. Where does one see a concrete walkthrough, with schemes, of the instances of this problem that can be studied with schemes or varieties, and hints along the way what questions drive us toward generalizations?

Thanks very much. I hope what I'm looking for is clear.

• Geometry of Algebraic Curves Vol II has lots of these things. First chapter constructs the Hilbert scheme, then later introduces the reasons stacks/orbifolds are needed. Obviously it’s focused on the example of moduli of curves, which is IMO the most important example and the original motivation for stacks – Samir Canning Jun 21 '19 at 4:29
• Thanks, this looks great. Looking at the first chapter it looks pretty readable. Can you think of any concepts that are introduced early on that require familiarity with parts of AG outside of Hartshorne? or other supplementary references on basic AG relevant to the material? – Patrick Nicodemus Jun 21 '19 at 8:48
• Volume I is good to have on hand as they reference it in some parts of volume II. Besides that, another useful book to have is Geometric Invariant Theory by Mumford, Fogarty and Kirwan. – Samir Canning Jun 21 '19 at 13:47