# Thursday, September 29

## Projective Quotients

:::{.example title="?"}
Let $G = \GG_m\actson \AA^n$ by \( \lambda. \tv{x_0,\cdots, x_n} = \tv{ \lambda x_0, \cdots, \lambda x_n} \).
What are the stable points?

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Note that the orbits are either $\ts{0}$ or lines $L\smz$.
The former has stabilizer $\GG_m$ and the latter orbits are open, so there are no stable points.
The affine quotient is $\mspec \kxn^G \cong \mspec k$, a point.
However, the projective quotient will be $\AA^{n+1} \modmod_{\proj} \GG_m \cong \PP^{n} = {\AA^{n+1} \smz\over \GG_m}$, not a point.
:::

:::{.remark}
Write 
\[
R = \kxn = \bigoplus_d R_d = k \oplus \gens{x_0, \cdots, x_n}_k[1] \oplus \gens{x_0^2, x_0x_1,\cdots}[2] \oplus \cdots
.\]
We'll say $R_d$ are *semi-invariants* of degree $d$, where \( \lambda.f = \lambda^d f \).
More generally, for a character \( \chi: G\to \GG_m \) given by \( \lambda\mapsto \chi( \lambda) \), for $\lambda \in G$ we can act by $\lambda.f = \chi( \lambda) f$.
:::

:::{.definition title="?"}
Given a character $\chi: G\to \GG_m$ with $G\actson R$ a ring, define the $k\dash$vector space of **semi-invariants**
\[
R_\chi^G = \ts{f \st \lambda .f = \chi( \lambda) f} \subseteq R
.\]
Say this action is of **ray type** with respect to $\chi$ iff for all $d < 0$, the semi-invariants $R_{d\chi}^G$ vanish and $R_0 = k$.
We can then define the projective quotient in the direction of a character as 
\[
X \modmod_\chi G \da \mproj \oplus _{d\in \ZZ} R^G_{d\chi}
.\]

:::

:::{.remark}
Note that $R^G = R_{\triv}$, and \( \lambda.(fg) = (\chi_1 + \chi_2)( \lambda) fg \).
:::

:::{.example title="?"}
Let $G = \GG_m$ and take $\chi = \id_{\GG_m}$, then \( \bigoplus _{d\in \ZZ} R^G_{d\chi} = R \) and $\AA^{n+1}\modmod_\chi \GG_m = \PP^n$.
:::

:::{.remark}
Constructing Proj: write \( \bigoplus _{d\in \ZZ} R^G_{d\chi} = k[f_1,\cdots, f_n] \) with $f_i$ homogeneous of degree $d$ in $R^G_{d\chi }$.
Note that $\prod f_i^{m_i}/\prod f_i^{n_i}$ of total degree zero yield regular functions on $D(\prod f_i^{n_i}) = V(\prod f_i^{n_i})^c$.
:::

:::{.example title="?"}
Let $X = \Mat_{n\times n}(\CC) \cong \AA^n \in \Aff\Var\slice \CC$, and let $G \da \GL_n(\CC) \actson X$ by $g.A = g\inv A g$.
Recall $G = \mspec \CC[a_{ij}, b] / \gens{b\det(a_{ij}) - 1 }$, so the action is algebraic since $g\inv = g^{\adj}/\det(g)$.
Considering that characters $\chi: \GL_n\to \GG_m$ must be multiplicative, it turns out that every character is a power of $\det: \GL_n\to \GG_m$.
What is $\Mat_{n\times n}(\CC)\modmod_{\det} \GL_n(\CC)$?
Identify $R = \CC[c_{ij}]$ as the coordinate ring of $X$, and recall 
\[
\charpoly(gAg\inv, x) = \charpoly(A, x) = \det(A-xI) = (-1)^n(x^n - \Trace(A)x^{n-1} + \cdots \pm \det(A))
.\]
Note that the affine quotient is $\mspec \CC[\Trace(A),\cdots, \det(A)]$.
:::

:::{.exercise title="?"}
Take $G = \GL_2(\CC) \actson V_d = k[x,y]_d\cong \CC^{d-1}$ where 
\[
\matt abcd .\cvec xy \da \cvec{ax+by}{cx+dy}
.\]
The ring is $R = \Symalg V_d$.
Let $\chi: G\to \GG_m$ and $\det: \GL_2\to \CC\units$.

Find polynomials such that $A.p(x,y) = \det(A)^N p(x,y)$ for some power $N$.
:::

:::{.remark}
This is Mukai's POV, an alternative POV is described in Mumford's GIT.
Start with $G\actson Y = \mproj R$ and $L\in \Pic(Y)$ ample.

1. Linearize the action.
  E.g. for $G\actson \PP^n =\mproj k[x_0,\cdots, x_n]$, there is not necessarily an action on $\AA^{n+1}$, and a linearization is a lift to $G\actson k[x_0,\cdots, x_n]$.
  Taking $L\smz$ yields $X\smz$, and $X\smz\to Y$ is a $\GG_m\dash$torsor, and gluing the zero section yields an $\AA^1\dash$bundle.
  So lifting $G\actson R(Y, L)$ is the same as lifting to the affine cone $G\actson X$.
  Then define the projective quotient $\Proj R(X, L)^G \da Y\modmod G$.
:::

:::{.remark}
Basic examples:

- Mumford: $\SL_n, \PGL_n\actson Y$ a projective variety with affine cone $X = CY$.
  Take $R^G$ to get a graded ring, take $\Proj$.
- Mukai: $\GL_n\actson X$ which is already affine.
  Take semi-invariants $R^G_\chi$ to get a graded ring and take $\Proj$.


Note 

- $\SL_n \injects \GL_n \surjectsvia{\det} \GG_m$ 
- $\mu_n \injects \SL_n \surjects \PGL_n$, and 
- $R^{\PGL_n}_{\det} = R^{\SL_n}$.
:::

:::{.remark}
Consider $\PGL_n \actson \PP^n$, corresponding to ${\SL_{n+1} \over \mu_n} = {\GL_{n+1}\over \GG_m}\actson {\AA^{n+1}\smz\over \GG_m}$.
The linearization is an action $\PGL_{n+1}\actson \AA{n+1} = \ts{\tv{x_0, \cdots, x_n}}$, which does not exist.
However, there are natural actions $\GL_{n+1}\actson \AA^{n+1}$ and thus $\SL_{n+1} \actson \AA^{n+1}$ by restriction.
Thus $\OO_{\PP^n}(1)$ is not linearisable for the $\PGL_{n+1}$ action, but is for $\SL_n$.
One solution: work with $\SL_n$, which has trivial $\pi_1$, while $\PGL_n$ has nontrivial solution.
Mumford's solution: the power $\OO_{\PP^n}(n+1)$ is linearisable for $\PGL_{n+1}$, since $\mu_{n+1}$ acts trivially.
This amounts to replacing $L$ by $L^{n+1}$ and $R(X, L)$ by the subring $R(X, L^{n+1})$ whose powers are divisible by $n+1$, but their Projs are equal.
The difference here is that $\SL_n$ has no characters and $\GL_n$ has only one character.
Mukai's approach is easier when there are many characters, e.g. when $G$ is a torus with characters $\Hom(T, \GG_m)\cong \ZZ^n \da M$.
:::