# Thursday, August 18


:::{.remark}
Some examples of moduli spaces:
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## Picard varieties

:::{.example title="The Picard group and Picard variety"}
Consider $X=E$ an elliptic curve, which can be defined as:

- A 1-dimensional abelian variety,
- A Weierstrass equation $y^2 = x^3 + ax + b$,
- A nonsingular genus 1 algebraic curve with a fixed origin, so $\CC/\Gamma$ for $\Gamma\cong \ZZ^2$ a lattice.

Recall that the group is 
\[
\Pic(X) \da \ts{\text{Invertible } \OO_X\dash\text{sheaves}}/\sim \cong \ts{\text{Line bundles over } X}/\sim
.\]
There is a homomorphism $\Pic(X) \mapsvia{\deg}  \ZZ\to 0$ with $\Pic^0(X) \da \ker \deg$.
A priori $\Pic^0(X)$ is a group, but in fact has the structure of a variety -- there exists a **Jacobian** variety $\Jac(X)$ such that $\Pic^0(X) \cong \Jac(C)(k)$, the $k\dash$points of the Jacobian.
Thus $\Jac(X)$ is a moduli space of invertible sheaves of degree zero.
:::

:::{.fact}
For $X=E$ an elliptic curve, $\Jac(E) \cong E$.
:::

:::{.fact}
There are distinct varieties with the same $k\dash$points: take for example the cuspidal curve $X = V(y^2-x^3)$ and $\AA^1$ -- there is a map 
\[
\AA^1 &\to X \\
t &\mapsto \tv{t^2, t^3}
.\]
with inverse $t=y/x$:

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\end{tikzpicture}

Note that these have the same $k\dash$points over *any* field $k$.
Thus we need to consider not just objects, but families of objects.
:::

## Elliptic curves

:::{.example title="?"}
The moduli space of elliptic curves 
\[
\mcm_{1} = \ts{\text{Elliptic curves over }\kbar }\slice{\cong}
.\]
As an algebraic variety, $\mcm_1 \cong \AA^1_j$ (the $j\dash$line) coming from taking the $j\dash$invariant
\[
j(X) = j(a, b) =_? {2a\over 4a^3 + 27b^2}
.\]

Then if $X\to S$ is a family of genus 1 algebraic curves, there exists a unique map $S\to \AA^1_j$ where $s\in S$ maps to $j(X_s)$.
How would you prove this?
See Hartshorne's treatment using the Weierstrass $\wp\dash$function.
Alternatively, factor to get $y=x(x-1)(x-\lambda)$ for $\lambda\not\in\ts{0, 1}$ and quotient by $S_3$ acting by permuting $\ts{0,1,\lambda}$.
One can then form $M_1 = \AA^1_\lambda/S_3$ and construct $j(\lambda)$ invariant under this action.
Note that when $X = \spec R$ is affine and $G$ is finite, there is an isomorphism $\spec R/G \cong \spec R^G$ to the GIT quotient.
If $X$ is not affine but $G$ is finite, one can still patch together quotients locally.
:::

## Vector bundles

:::{.example title="?"}
Moduli of sheaves or vector bundles (locally free $\OO_X\dash$modules of rank $n$) on a fixed base variety $X$, e.g. a curve.
One might fix invariants like a rank $r$, degree $d$, etc in order to impose a finiteness/boundedness condition on the moduli space.
For $X = \PP^1$, a vector bundle $F\to X$ decomposes as $F = \bigoplus _{i=1}^r \OO(d_i)$ where $\deg F = \sum d_i$ by Grothendieck's theorem.
Since some $d_i$ can be negative, the moduli come in a countably infinite set.
To impose boundedness one can additionally add **stability conditions** such as semistability, which here ensures only finitely many degrees appear and the existence of a moduli space $\mcm_{r, d}(X)$.
To do this, twist by a large integer and take global sections to get $H^0(X; F(n))$ for $n\gg 0$.
Understanding $\bigoplus_{n\geq 0} H^0(X; F(n))$ as a module over $R = \bigoplus _{n\geq 0} \OO(n)$ allows one to reconstruct $F$.
Thus one can construct $\mcm_{r, d}(X) = ? / \PGL_N$ corresponding to choosing a basis for $H^0$.
Here we remove some "unstable" locus before taking the quotient -- note that points correspond to orbits, except that some orbits become identified.

> This is an "easy" moduli problem, since vector bundles are somehow linear. See Ramanujan, ?, Mumford, 40+ years ago.

:::

## Nonlinear examples: moduli of curves/varieties

:::{.example title="?"}
Let $\mcm_2$ be the moduli of curves $C$ with $g(C) = 2$.
All such curves are hyperelliptic, so similar to the $g=1$ theory.
In the $g=1$ case, curves can be realized as ramified covers of $\PP^1$:

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\end{tikzpicture}

In the $g=2$ case, they can similarly be realized as 2-to-1 maps ramified at 6 points:

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\end{tikzpicture}


One can realize $\AA^3\contains U \da\ts{\tv{ \lambda_1, \lambda_2, \lambda_3} \st \lambda_i\neq 0,1,\infty, \lambda_i \neq \lambda_j}$ and $M_2 = U/S_6$.

For $g=3$, one has $g=(1/2)(d-1)(d-2)$ by the adjunction formula, so $g=3$ corresponds to $d=4$ and one obtains

- Hyperelliptic: degree 4 curves in $\PP^2$ (the generic case), or

- Non-hyperelliptic: 2-to-1 covers of $\PP^1$ ramified at 8 points.

There is no analog of the Weierstrass equation for degree 4 polynomials, so write $f_4(x_1, x_2, x_3) = \sum a_m x^m$ where $x^m \da x_1^{m_1}x_2^{m_2}x_3^{m_3}$.
How many such polynomials are there?
Count points in the triangle:

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![](figures/2022-08-18_13-48-50.png)

This yields $5+4+3+2+1 = 15$ such monomials, and one can write 
\[
\AA^{15}\smz / k\units = \PP^{14} \contains U = \PP^{14}\sm \Delta
\]
where $\Delta$ is the discriminant locus.
This is an affine variety, since $\Delta$ is a high degree hypersurface.
Then form $U/\PGL_3$, noting that $\dim \PGL_3 = 3^2-1 = 8$, so $\dim \PP^{14}\sm\Delta = 14-8 = 6$.
:::

:::{.remark}
Ways of forming moduli spaces:

- GIT,
- Hodge theory over $\CC$,
- Stacks (e.g. Artin's method).

These rarely produce compact/complete spaces, so we'll discuss compactification.
Why compactify?
Computing things, projectivizing, intersection theory.
See Bailey-Borel and toroidal compactifications.
:::

:::{.remark}
A note on Hodge theory: for an elliptic curve, one can write $E = \CC/\gens{1, \tau}$ with $\Im(\tau) > 0$ (so $\tau\in \HH$), one can form $\mcm_1 = \dcosetr{\HH}{\SL_2(\ZZ)}$.
This is Hodge theory: $\tau$ is a **period**, and we quotient a bounded symmetric domain by an arithmetic group.
Similarly, for PPAVs one can write $\mca_g = \dcosetr{H_g}{\Sp_{2g}(\ZZ)}$, and for K3 surfaces one has $F_{2d} = \DD_{2g}/\Gamma_{2g}$ where $\omega_X \in \DD$.
One can determine things like Jacobians using Torelli theorems.
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:::{.remark}
Todo: how much do you know, and what are you trying to get out of the course?
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