# Tuesday, October 11

:::{.remark}
Setup: $X \subseteq \PP^n$ with $G\actson X, \PP^n$, $G$ linearly reductive, which is linearized so that $G\actson \AA^{n+1}$ acting on projective coordinates is a linear action.
Thus each $g\in G$ induces $\rho_g$ which is linear in the coordinates $x_0, x_1,\cdots, x_n$.
We have

\begin{tikzcd}
	X \\
	{X^\ss} && {X^\ss\modmod G} & {\text{GIT quotient, projective}} \\
	{X^s} && {X^s/G} & {\text{geometric quotient}}
	\arrow["{\subseteq, \text{open}\qquad }", hook, from=3-1, to=2-1]
	\arrow["{\subseteq, \text{open}\qquad }", hook, from=2-1, to=1-1]
	\arrow[two heads, from=2-1, to=2-3]
	\arrow[two heads, from=3-1, to=3-3]
\end{tikzcd}

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

Define $X^u = X\sm X^s$ to be the unstable points; our main problem is to describe $X^u, X^s, X^\ss$.
:::

:::{.theorem title="Mumford-Hilbert criterion"}
For $x\in X$, $x\in X^\ss \iff$ the following holds:
Let $\lambda: \GG_m\to G$ be nonconstant, and note the action $\GG_m\actson \AA^{n+1}$ corresponds to a grading and thus some system of linear coordinates $x_0,\cdots, x_n$ with weights $\omega_0, \cdots, \omega_n \in \ZZ$ where $t.x_i = t^{w_i} x_i$.
Then for all such $\lambda$, there should exist a coordinate $x_i$ with $x_i(p) \neq 0$ and $w_i \leq 0$.

Similarly, $x\in X^s \iff \exists \lambda, x_i$ with $x_i(p) \neq 0$ and $w_i < 0$ is strictly negative.
:::

:::{.remark}
Note that replacing $\lambda$ with $-\lambda$, one can replace the above conditions with $w_i \geq 0$ and $w_i > 0$ respectively.
Most papers on GIT start with this theorem, and finding the unstable locus is a computation.
:::

:::{.corollary title="?"}
$x\in X^u \iff$ there exists a 1-parameter subgroup $\lambda: \GG_m\to G$ such that $w_i > 0$ for all $i$ with $x_i(p) \neq 0$.
:::

:::{.example title="?"}
Consider binary degree $d$ forms, corresponding to degree $d$ cycles/subschemes in $\PP^1$.
Each point corresponds to a homogeneous polynomial $f_d(x,y)$ of degree $d$.
Recall that $V_d = \gens{x^d, x^{d-1}y,\cdots, y^d}$, the irrep of $\SL_2$ where $\SL_2\actson \gens{x,y}$ by matrix multiplication:
\[
\matt abcd \cvec xy \da \cvec{ax+by}{cx+dy}
.\]

We have a 1-parameter subgroup corresponding to $\diag(t, t\inv) \actson [x,y] = [tx, t\inv y]$ which gives $x$ weight $1$ and $y$ weight 2.
Call this $\lambda^\std$.
:::

:::{.claim}
All 1-parameter subgroups of $\SL_2$ are powers $(\lambda^\std)^n$ for some $n$, up to a linear change of coordinates for $x,y$.
:::

:::{.proof title="?"}
The general theory: if $G$ is semisimple (e.g. $G=\SL_n$) then $G\contains T$ a maximal torus, and any two such are conjugate. 
For $\SL_n$ these tori are diagonal matrices $M$ with $\det M = 1$.
Moreover all 1-parameter subgroup is contained in a maximal torus.
Powers can be ignored here, since they correspond to multiplying weights by a positive integer.
By the theorem, a point is unstable iff the monomials that appear in the binary form are all of negative degree for some choice of coordinates $x,y$.
For $d=3$, we have

<!-- Xournal file: /home/zack/SparkleShare/github.com/Notes/Class_Notes/2022/Fall/Moduli/sections/figures/2022-10-18_13-12.xoj -->

![](figures/2022-10-18_13-14-06.png)

So $f$ is unstable iff $f(x,y) = axy^2 + by^3 = y^2(ax + by)$, i.e. in some coordinates $y^2\divides f$, so $f$ has a double root.
:::

:::{.proposition title="?"}
A binary form of degree $d$ is

- Unstable iff there exists a root of multiplicity $m > d/2$.
- Semistable iff there exists a root of multiplicity $m \leq d/2$.
- Stable iff there exists a root of multiplicity $m < d/2$.

:::

:::{.remark}
Note that for odd $d$, stable = semistable, and for even $d$ they are different.
:::

:::{.remark}
For $d=4$, consider the double cover $z^2 = f(x,y)$:

\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2022/Fall/Moduli/sections/figures}{2022-10-18_13-17.pdf_tex} };
\end{tikzpicture}

So 

- Smooth curves are stable, corresponding to $(1,1,1,1)$ or $z^2 = (x - a)(x-b)(x-c)(x-d)$
- Nodal curves are semistable, corresponding to $(1,1,2)$ ($z^2 = (x-a)(x-b)(x-c)^2$) or $(2, 2)$ ($z^2 = (x-a)^2(x-b)^2$).
- Tacnodes are unstable, corresponding to $(4)$, so $z^2 = (x-a)^4$,
- Cusps are unstable, corresponding to $(1,3)$, so $z^2 = (x-a)(x-b)^3$

Thus the $j\dash$line $\AA^1$ corresponds to smooth/stable curves, and compactifies to $\PP^1 = \PP(2,3)$ by adding nodal curves.
:::

:::{.remark}
For $\SL_3\actson X$ for $X$ the space of cubic curves in $\PP^2$, we have several possibilities for curves $f_3(x,y) = 0$:

\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2022/Fall/Moduli/sections/figures}{2022-10-18_13-38.pdf_tex} };
\end{tikzpicture}

We have $f_3\in \PP^{{3+2\choose 2} - 1} = \PP^9$, with 10 coordinates:
\[
x^3,y^3,z^3, 
x^2 y, x^2 z,
y^2 x, y^2 z,
z^2 x, z^2 y,
xyz
.\]
Each curve $f = 0$ to a closed subscheme of $\PP^2$ whose ideal is $\gens{f}$.
There is an action of $\SL_3 \actson \tv{x,y,z}$ on coordinates, and a maximal torus $T = \diag(t_1, t_2, t_1\inv t_2\inv)$.
Choosing this torus in a diagonal form is equivalent to choosing a coordinate system.
One can then look at $\GG_m \injects T$ and consider its action to define weights.
We get the following triangle of monomials:

<!-- Xournal file: /home/zack/SparkleShare/github.com/Notes/Class_Notes/2022/Fall/Moduli/sections/figures/2022-10-18_13-46.xoj -->

![](figures/2022-10-18_13-49-30.png)

Take this and project to get weights:


![](figures/2022-10-18_13-51-58.png)

This gives $w(x) = (1,0), w(y) = (0, 1), w(z) = (-1, -1)$
Those with the right weights are $x^2 z, xz^2, z^3, yz^2$, all containing a factor of $z$.
So any polynomial of the form $f(x,y,z) = (axz + bxz^2 + cz^2 + dyz)$ is unstable.

Thus the following are unstable:

![](figures/2022-10-18_13-54-49.png)

The game: take a line through the center point $xyz$, rotate, take monomials on the positive side, and check for instability, since we need $w(xyz) = 0$.
It turns out that smooth cubics are stable, simple nodes are semistable, and anything worse than $xyz = 0$ is unstable.
:::