# Thursday, August 25

:::{.remark}
Goal: showing $\Mg$ exists as a quasiprojective complex variety, and can in fact be defined over any field $k$ or even over $\ZZ$.
Here quasiprojective over $\ZZ$ means $X \subseteq \PP^n\slice{\ZZ}$ is a closed subset given as $X = V(f_i)$ for homogeneous integral polynomials $f_i$.
Note that $\Mg(\kbar) = \ts{\text{smooth projective curves of genus } g} = X(\kbar) \sm Z(\kbar) \subseteq \PP^n\slice {\kbar}\modiso$ where $Z = V(f_i, g_i)$ -- this says $\Mg$ satisfies exactly the equations $f_i$ and no more.
Anytime objects have isomorphisms, one only gets a coarse moduli space instead of a fine moduli space, which we'll later describe.
Families $\mcx \to S$ yield to maps $S\to \Mg$ over $\spec \kbar$, and this will be a bijection when $\Mg$ is a fine moduli space and $\mcx$ is the pullback of a universal family $\mce\to\Mg$.
Since we only have a coarse moduli space, a family yields a map to $\Mg$, but these are not in bijection.

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:::{.remark}
We'll want projective varieties in order to do intersection theory.
The most fundamental compactification: the Deligne-Mumford compactification $\Mgbar$ of $\Mg$, i.e. the moduli of stable curves of genus $g$.
This is a projective moduli space containing $\Mg$ as an open dense subset, and is obtained by adding degenerate curves "at infinity".
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:::{.example title="?"}
Consider $x_0 x_2 = t^n x_1^2$ in $\PP^2_{x_0, x_1, x_2}$ and take the 1-parameter degeneration $t\to 0$.
This is smooth for $t\neq 0$, since this is a full rank conic.
In affine coordinates this is $xy=t^n$, which degenerates to the simple node (double point) $xy=0$.
Part of this degeneration data can be recovered from a tropical curve, which is a metric graph whose points are singularities and lengths correspond to the $n$ in $t^n$:

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![](figures/2022-08-25_13-14-26.png)

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:::{.definition title="Stable curves"}
A **stable curve** of genus $g$ is a connected reduced (possibly reducible) projective curve $C$ such that 

- (Mild singularities) $C$ has at worst nodes, locally of the form $xy=0$.
- (Numerical) The dualizing sheaf $\omega_X$ is ample.

Note that $g=h^0(\omega_X) = h^1(\OO_X)$.
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:::{.remark}
Writing a multi-component curve as $X = \Union X_i$, the numerical condition requires that for every $X_i\cong \PP^1$, one has $\abs{X_i \intersect (X\sm X_i)} \geq 3$, and for all $X_i$ of the following form (or $X_i\cong E$ an elliptic curve), $\abs{X_i \intersect (X\sm X_i)} \geq 1$:

\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}{2022-08-25_13-20.pdf_tex} };
\end{tikzpicture}

This is equivalent to $\size \Aut X < \infty$.
For $g\geq 2$ and $C_g$ smooth of genus $g$, one has $\size \Aut C_g < \infty$, and for $g=1$ enforces $\dim \Aut C_g = 1$.
For $g=0$, note $\Aut \PP^1 = \PGL_2$ which has dimension 3, so fixing at least 3 points cuts this down to a finite automorphism group.
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:::{.remark}
The dualizing sheaf $\omega_X$ is invertible if $X$ has only nodes.
The adjunction formula yields a twist $\ro{\omega_X}{X_i} = \omega_{X_i}(X\sm X_i)$
Then $\omega_X$ is ample iff $\deg\ro{\omega_X}{X_i} > 0$.
One can compute $\deg \omega_{X_i}(X\sm X_i) = 2g_i - 2 + \abs{X_i \intersect (X\sm X_i)}$, hence the lower bound on the number of intersection points.
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:::{.proposition title="?"}
Without the numerical condition, the limit is not unique.

To see this, take a trivial family over $\PP^1$, so a surface, and blow up a point on the central fiber.
This yields a multi-component curve, which we allow, and we can continue blowing up such points:

\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}{2022-08-25_13-31.pdf_tex} };
\end{tikzpicture}

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:::{.remark}
If $\omega_{X_t}$ is ample for all $t$, then $\omega_{\mcx}/S$ is relatively ample, which implies $\mcx/S$ is the canonical model.
One can contract $(-1)$ curves to get a minimal model, and $(-2)$ curves to get canonical models.
See degenerations of elliptic curves to wheels of copies of $\PP^1$:

\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}{2022-08-25_13-36.pdf_tex} };
\end{tikzpicture}


See Kodaira's elliptic fibers -- classified by extended Dynkin diagrams $\tilde A_n, \tilde D_n, \tilde E_n$, and special types $\tilde A_i^*$ for $i=0,1,2$:

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![](figures/2022-08-25_13-47-01.png)

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:::{.definition title="Stable maps"}
For $V$ a projective variety, a **stable curve** is a map $f:C\to V$ satisfying

- (Mild singularities) $C$ has at worst nodes.
- (Numerical condition) $\omega_C$ is very ample, or equivalently has positive degree on components which map to points.

So for example, we can ignore vertical curves:

\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}{2022-08-25_13-49.pdf_tex} };
\end{tikzpicture}

One can define a moduli space of stable curves passing through $n$ marked points, $\bar{\mcm}_{g, n}(V)$:

\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}{2022-08-25_13-53.pdf_tex} };
\end{tikzpicture}

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:::{.remark}
Defined to formulate Gromov-Witten invariants.
Motivated by physics, but originally non-algebraic and used almost complex structures.
The second condition yields unique limits since it will yield a relative canonical model, which exist and are unique.
This moduli space can be generalized to higher dimensions, see KSBA compactifications.
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:::{.remark}
Constructing $\Mg$ and $\Mgbar$:

- Step 1: Parameterize embedded curves $C_g \embeds \PP^N$ by the picking a basis of the linear system $\abs{2K_X}$, where $N = 2(2g-2) - (g-1) - 1 = 3g-4$ and $\deg C_g = 2(2g-2)$.
  Use either the Chow variety $\Ch_{d, N}$, parameterizing cycles/subvarieties of $\PP^N$ with degree $d$, or the Hilbert scheme $\Hilb_h$ parameterizing closed subschemes $X \embeds \PP^N$ with a fixed Hilbert polynomial $h$.
  The latter may not yield reduced curves, but closed subschemes are easier than varieties since they are just defined by equations.

- Step 2: Divide by $\PGL_{N+1}$, using GIT (next week) to produce a space $X/G$ whose points (ideally) correspond to $G\dash$orbits.

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