# Methods to compute the Kodaira dimension of moduli spaces

It is known that the moduli space $$\bar{M_g}$$ of genus $$g$$ stable curves over $$\mathbb C$$ is of general type for $$g \geq 24$$ with Kodaira dimension $$3g-3=\dim \bar{M_g}$$.

The idea is that one can compute the canonical divisor by deformation theory (using moduli interpretation) and Grothendieck-Riemann-Roch formula, and then show it's numerically equivalent to the sum of an ample and an effective divisor, see e.g https://arxiv.org/abs/1610.09589. It seems some careful analysis of the boundary is needed. Beyond $$\bar{M_g}$$, what do we know about variants of it e.g $$M_{g,n}$$?

And it seems people also use Borcherds modular forms to show some moduli spaces of abelian varieties (with some levels) are of general type. What's the idea of this method? What type Shimura varieties can we apply this method?

Moreover, are there other ways to compute the Kodaira dimension of interesting moduli spaces? How about moduli of vector bundles / stable sheaves (with other extra structures)? Moduli of other type varieties? Moduli of rings and ideals?

• I think you mean g >= 24? Also, the Kodaira dimensions of $M_g$ and $\bar{M}_g$ are the same (Kodaira dimension is a birational property). – cgodfrey Oct 9 at 2:45
• @cgodfrey This is a typo, thanks. You're right, and the formula of the canonical class has a simple form on $M_g$. However, it's important to use the boundary for construction. – loos Oct 9 at 4:22