# Thursday, October 06

## Projective GIT Quotients

:::{.remark}
Mumford's approach for e.g. $G = \SL_n$: $G\actson R = R(X, L) + \bigoplus _{d\geq 0} H^0(X; L^d)$ where e.g. $L = \OO(1)$ for $X \subseteq \PP^n$.
This yields $R^G$ a graded ring and $X\modmod G = \Proj R^G$.
Setting $Y = CX = \spec R$, we can consider $Y\modmod G = \spec R^G$.
We have $Y \contains Y^s = \ts{g\in G\st G.y \text{ is closed}, G_y \text{ is finite}}$, the open subset of stable points.
For each equivalence class of orbits under the orbit-closure equivalence, there is a unique closed orbit.

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:::{.definition title="Stable"}
\envlist

- $X^s$: $x\in X$ is **stable** iff for all $[y] = x$ the point $y\in Y$ is stable:
- $X^\ss$: $x$ is **semistable** iff $0\not\in \cl(G.y)$ for all $y$ with $[y] = x$.
  Note stable implies semistable.
- $x$ is **unstable** iff $0\in \cl(G.y)$.

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:::{.theorem title="?"}
\envlist

- Points of $X\modmod G$ biject with closure-equivalence classes of $G\dash$orbits on $X^\ss$, each containing a unique closed orbit.
- There is a **geometric quotient** $X^s/G \subseteq X\modmod G$  whose points biject with $G\dash$obits on $X$.

\begin{tikzcd}
	{X^\ss} && {X\modmod G} \\
	\\
	{X^s} && {X^s/G}
	\arrow[hook, from=3-1, to=3-3]
	\arrow[hook, from=1-1, to=1-3]
	\arrow["{\text{open}}", hook, from=3-1, to=1-1]
	\arrow[hook, from=3-3, to=1-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJYXlxcc3MiXSxbMiwwLCJYXFxtb2Rtb2QgRyJdLFswLDIsIlhecyJdLFsyLDIsIlhecy9HIl0sWzIsMywiIiwwLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XSxbMCwxLCIiLDAseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJob29rIiwic2lkZSI6InRvcCJ9fX1dLFsyLDAsIlxcdGV4dHtvcGVufSIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzMsMSwiIiwxLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XV0=)

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:::{.remark}
One can show that unstable points are a closed condition, so $Y^s$ is open in $Y^\ss$.
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:::{.remark}
Mukai's approach: for $\GG = \GL_n$ define $Y\modmod \GG = X\modmod G$.
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:::{.example title="?"}
Consider $V_{d, n}$ the space of degree $d$ hypersurfaces in $\CC^n$, which is isomorphic to $\CC^N$ where $N = {d+n\choose n}$.
We can also consider $\PP V_{d, n} = \ts{p_d(x_0,\cdots, x_n) = 0 \text{ homogeneous}}$ the space of hypersurfaces of degree $d$ in $\PP^n$, up to the action of $\PGL_{n+1} = \Aut(\PP^n)$.
We have $\PP V_{d, n}\modmod \PGL_{n+1} = \PP V_{d, n} \modmod \SL_{n+1} = V_{d, n}\modmod \GL_{n+1}$, and $\SL_{n+1}\actson \AA(V_{d, n}) = S(V_{d, n})$, so we want to describe the ring $S(V)^{ \SL_{n+1} }$ since its proj is $\PP V_{d, n} \modmod \PGL_n$.
Recall $S(V) = \bigoplus_{k\geq 0} \Sym^k(V)$.
Describing this ring is equivalent to describing the variety and yields a solution to moduli problems involving hypersurfaces.
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:::{.remark}
Consider sextic curves in $\PP^2$, given by polynomials $p_6(x_0, x_1, x_2)$.
Note that smooth curves are stable since their orbits are closed.
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:::{.theorem title="Shah, Ann. Math, Moduli space of K3 surfaces of degree 2"}
He lists the unique closed semistable orbits.
One is a cube of a quadratic, one is three quadrics tangent at two points, one is a double line with a smooth quartic:

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![](figures/2022-10-06_13-45-48.png)

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:::{.remark}
The case of interest: a moduli space $\Mg$ parameterizing smooth curves.
One choose a linear system $\phi_{\abs{2K_C}}: C\injects\PP^n$, then one shows that the smooth curves $\ts{C\injects \PP^n} \subseteq \Hilb, \CH$ are stable, but one also picks up singular stable curves and quotienting yields a compactification $\Mgbar$.

> See Sylvester, a first good American mathematician!
He proved some important theorems in invariant theory.

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:::{.example title="?"}
The easiest case: $\ts{f(x, y) = 0} \subseteq \PP^1$ with coordinates $x,y$.
There is an action of $\SL_2$ given by $\matt abcd \cvec xy = \cvec{ax+by}{cx+dy}$.
Write $V_{d, 1} = \gens{x^d, x^{d-1}y,\cdots}$ and $\SL_2$ acts by variable substitution.
The algebra is $R = S(V_{d, 1})$, and we want to find $R^G = S(V_{d, 1})^{\SL_2}$.
This is a moduli problem for configurations of distinct points (with multiplicity) on $\PP^1$, up to reparameterizing by $\PGL_2$.
It turns out $f$ is

- Semistable if each point's multiplicity is $\leq d/2$.
- Stable if $< d/2$.

Note that for $d$ odd, $X^s = X^\ss$.

- For $d=1$ we have $R^{G} = \CC$.
  Here $\SL_2\actson \gens{x, y}$ and $S(\gens{x, y}) = \CC[x, y]$.
  Note that $\CC[x,y] = \CC \oplus \gens{x,y} \oplus \gens{x^2, xy, y^2} \oplus \cdots$, each corresponding to an irreducible $\SL_2$ representation.
  Since $\SL_2$ is completely reducible, the higher graded pieces have no invariants.
- For $d = 2,3$ we similarly get $R^G = \CC$ using Mobius transformations.
- For $d=4$, one can fix $0,1,\infty$ and there is a free parameter $\lambda$.
  One can take a double cover $y^2 = x(x-1)(x- \lambda)$, i.e. an elliptic curve.
  The projective is 1-dimensional, since the $j\dash$invariant is a moduli space, so the affine version is dimension 2 and turns out to be $R^G = \CC[g_2, g_3], \abs{g_2} = 4, \abs{g_3} = 6$.
  So $\Proj R^G = \PP(4, 6)$.
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