# Applications of GIT to Moduli of Varieties (Thursday, December 01)

:::{.remark}
Last time: moduli of sheaves on a fixed variety, a linear case where GIT works very well by reducing to a computation on a Grassmannian.

- Successes: 
  - $\Mgbar$ (compactification of moduli of genus $g$ curves)
  - Moduli of varieties with nef $K_X$, e.g. K3 surfaces, CYs, varieties of general type, all with (very) mild singularities.
    However, GIT does not give a compactification here.
- Failures:
  - Compactifications for higher-dimensional varieties analogous to $\Mgbar$.
  Computations are infeasible most of the time, and unreasonable when computable.

Recall $\Mg$ is the moduli of smooth projective genus $g$ curves, and for $g\geq 2$ it is known that $\dim \Mg = 3g-3$ which is locally a quotient of a smooth variety and is thus a smooth orbifold/stack.
It is quasiprojective but not projective and not complete.
One would like an inclusion $\Mg\injects \Mgbar$ a projective variety with mild singularities such that points in $\bd\Mgbar$ correspond to curves with certain singularities.
This is constructed by Deligne-Mumford, locally $U/G$ where $U$ is smooth and $G\in\Fin\Grp$.
Moreover $\bd \Mgbar = \Union_i D_i/G$ with $D_i$ smooth and SNC, and points in $\bd \Mgbar$ correspond to *DM-stable curves*:
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:::{.definition title="?"}
Let $C = \Union_i C_i$ be a connected reduced projective curve, then $C$ is **DM-stable** iff

1. (Singularities) $C$ has at worst double crossing points, so analytically-locally of the form $V(xy)$.
2. (Numerical) Any of the following equivalent conditions:
  a. $\size \Aut(C) < \infty$,
  b. For any $C_i\cong \PP^1$, $\size( C_i \intersect (C\sm C_i))\geq 3$,
  and for any $C_i$ which is rational nodal (elliptic), $\size(C_i \intersect (C\sm C_i)) \geq 1$.
  c. The dualizing sheaf $\omega_C$ is ample.

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:::{.remark}
Why these three conditions are the same: recall $\Aut \PP^1 = \PGL_2$ which has dimension 3.
Note that $\Aut E\cong E \times G$ for $G\in \Fin\Grp$, usually $G\cong C_2$, and $\dim \Aut E = 1$.
Consider the nodal curve $E$, equivalent to $\PP^1/ 0\sim\infty$, so $\Aut(C) = \CCstar \semidirect C_2$ which again has dimension 1.
If $g(C) \geq 2$ then $\size \Aut(C) < \infty$.
The 3 in condition b is due to the need to fix 3 points, to drop $\dim \PGL_2$ from dimension 3 to zero.

If $X$ is Gorenstein then $\omega_X\in \Pic(X)$ and can be written $\omega_X = \OO(K_X)$ where $K_X$ is defined to be the canonical class.
This holds if e.g. $X$ has hypersurface singularities.
Note that $\omega_X$  is ample iff $\deg \ro{ \omega_X }{C_i} > 0$ for each $C_i$, and by adjunction one has
\[
\deg \ro{\omega_X}{C_i} = \deg \omega_{C_i} + \size{ C_i \intersect (C\sm C_i) } = 2p_a(C_i) - 2
.\]
Thus if $p_a(C_i) \geq 2$ this is always positive; if $p_a(C_i) = 0$ then $C_i = \PP_1$, otherwise if $p_a(C_i) = 1$ and we get the curves appearing in condition b.
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:::{.remark}
Consider a family $\mcc \to \Delta\interior$ of smooth projective curves.
By the semistable reduction theorem, after a finite base change $\Delta'\to\Delta$ any family can be completed to $X'\to \Delta'$ such that $X'$ is smooth and $X_0'$ is SNC:

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Note that $\deg \ro{\omega_X}{X_i} < 0 \iff C_i = \PP^1$ and $\size\qty{C_i \intersect (C\sm C_i)} = 1$, or just $\size\qty{C_i \intersect (C\sm C_i)} = 2$.
If $C\cdot C_i = 0$ since $C$ can be replaced with a disjoint fiber $F$.
Writing $0 = C\cdot C_i = C_i^2 + (C-C_i)C_i$, where get $C_i^2 = -1$ in the first case and $C_i^2 = -2$ in the second case.
In the first case, $C_i$ can be contracted to yields $X\to X_1$ (Castelnuovo's lemma) with $X_1$ smooth, so we can get rid of $-1$ curves in stages to get a new surface with (potentially) only $-2$ curves, which is a *minimal model* and is smooth.
Contracting all $-2$ curves yields the *canonical model*, which may be singular but has only *canonical singularities*.
Note that $X$ has canonical singularities iff $(X, X_0)$ has slc singularities iff $X_0$ has slc singularities, which is a form of log adjunction for degenerating pairs.

> See the BCHM paper.

So it's clear how to degenerate in one parameter families, but how does one organize these various limits into a compactification?
The idea is to construct $\Mg,\Mgbar$ using GIT to realize them as $H/\PGL_n$ where $H$ is a moduli of curves with additional data.
The HM criterion gives stable and semistable points, and one hopes these coincide with the above notions.
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:::{.remark}
What is $H$? Two answers: the Chow variety, or the Hilbert scheme.
Start with $C$ a smooth curve of genus $g\geq 2$ with $\deg K_C = 2g-2 \geq 2$ so that $K_C$ is ample.
Then $nK_C$ is very ample for any $n\geq 2$, and there is an embedding $C \injectsvia{\abs{nK_C}} \PP^N$.
Such an embedding is given by a choice of basis of $H^0(C; \OO(nK_C))$ where two bases differ by a $\PGL\dash$action.
Note that $N = n(2g-2) - (g-1) - 1$ by Riemann-Roch.

The Chow variety $\CH(d_1, d_2, \PP^N)$ parameterizes cycles of dimension $d_1$ and degree $d_2$ in $\PP^N$.
For example, $\CH(1, n(2g-2), \PP^N) \embeds \PP H^0(\Gr, \OO(k))$ for some $k$ which sends a cycle to a certain hypersurface.
Since $\PGL$ acts on the latter, it acts on the former, and there is a notion of *Chow stability*.
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:::{.theorem title="Mumford"}
For $n\geq 4$, DM curves are Chow stable
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:::{.remark}
The Hilbert scheme is preferable since Chow doesn't have an immediate deformation theory.
We can take a scheme parameterizing closed embeddings $Z\embeds \PP^N$ with a given Hilbert polynomial; recalling $p_X = \chi(\OO_Z(X))$ and setting $p_Z(x) = n(2g-2)x + (1-g)$, consider $\Hilb(\PP^n, p_Z)$.
Constructing this scheme: for $n\gg 0$, there is a surjection $H^0(\OO_{\PP^N}(n) ) \surjects H^0(\OO_Z(n))$ defines a point $g_n\in \Gr$ as the codomain varies.
Mumford proves there exists an $N$ such that $(g_N, g_{N+1})$ defines $Z$ uniquely, although $N$ is not canonically defined.
Thus $\Hilb$ embeds into a product of Grassmannians, and there is a notion of asymptotic Hilbert stability in terms of growth in $N$, and one takes the leading term.
One shows that for $n\geq 4$, HM-stable curves are asymptotically Hilbert stable in this since.
This almost completely fails for surfaces.
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:::{.remark}
The generalization: algebraic spaces, $H/G$ or more generally $H/R$ for $R\subset H\times H$ an equivalence relation.
By Artin, these exist, and they are natural to consider for non-polynomial equations like $y = \sqrt{x^3+x+1}$.
The construction of moduli spaces as algebraic spaces is easy, one then tries to prove they are projective.
See KSB varieties and KSBA pairs.
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