# Is there a precise relationship between the goals of moduli theory and the minimal model program?

I want to get into some of the big classification problems in algebraic geometry, but have a very broad question. Ultimately we would like to classify all varieties over some field up to isomorphism, and this is done via moduli theory. I am studying the moduli of elliptic curves at the moment. From the moduli space we construct we are able to obtain information about the relationship between different varieties based on the geometry of the moduli space. For example varieties in the same irreducible component can somehow be deformed to eachother.

On the other hand it seems like the completion of this program is hopelessly difficult for general varieties. Instead it is common to attempt to classify such objects only up to birational equivalence. This is the goal of the minimal model program.

But it seems that these two aren't really running in parallel in the sense that the latter doesn't appear to be a direct stepping stone for the former. I guess what I was hoping for is that classififying up to birational equivalence would somehow be a big step towards classification up to isomorphism. But when we classify curves up to isomorphism we appeal to the genus. But for birational equivalence we just normalize and apply Chow's lemma.

I know this is a broad question, but is there some geometry of the moduli space that would tell us when two varieties are birational equivalent, such as being in the same connected component or some such simple test? Or even more broadly, would the completion of the minimal model program give us any insight into what the moduli space for some family of varieties would look like?

I left this as a soft question since I probably don't even know enough about the subject to ask it in a precise manner. But hopefully some experts understand what I am trying to get at.

• In general, the minimal model program does not tell much about isomorphism classes. Just to give an elementary examples, all Hirzebruch surfaces $\mathbb{F}_n$ ($n \neq 1)$ are minimal models of a rational surface. They are all birational, but pairwise non-isomorphic. Apr 13, 2020 at 7:23
• When $X$ is not covered by rational curves and the dimension is $\leq 2$, there is a unique minimal model in each birational class, so birational equivalence and isomorphism coincide. When the dimension is at least $3$ the minimal model is not unique (and you must also allow some mild singularities), but two distinct minimal models are isomorphic outside subsets of codimension at least 2, and more precisely they are related by a sequence of flops. Apr 13, 2020 at 7:25
• This is a far reaching question so I kept my answer short. There are also approaches to construct nice moduli spaces of Fano varieties (and of vars of intermediate Kodaira dimension) insprired by the work of Donaldson and others. A very important step towards this is Birkar's work on the boundedness of Fanos. Apr 13, 2020 at 19:18

Koll\'ar's paper "Moduli of varieties of general type" https://arxiv.org/abs/1008.0621 provides an excellent introduction. We now know that moduli spaces of canonical models of varieties of general type exist and once you fix the volume $$K_X^{dim X}$$, they can be compactified by adding stable models, obtaining a projective moduli space. Note that the volume is the higher dimensional analog of the genus as the volume of a curve is $$2g-2$$.
BTW, if $$X$$ is not a canonical model, then the volume is computed by $$\lim \frac{h^0(mK_X)}{m^d/d!}$$ where $$d=dim X$$. Since any variety of general type is birational to its canonical model (and hence shares many properties), this is considered a satisfactory answer. For example, canonical models determine the fundamental group and the cohomology groups $$h^i(\mathcal O _X)$$.
There are likely other issues such as the boundedness. In dimension 2, if we fix the volume and the Picard number $$\rho$$ of a surface $$X$$ then this surface is obtained from its canonical model by blowing up at most $$\rho$$ times. Thus we expect that these surfaces belong to a bounded family. However, in dimension $$\geq 3$$ if you fix the volume, the Picard number, the topological type, it is still unclear if you can bound the corresponding families (some results exist in dimension 3).