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created: 2023-03-29T11:30
updated: 2023-03-29T11:30
---
# Questions 

-   Why is $\Fix(\sigma)$​ a disjoint union of curves and points?
    
-  Hodge index theorem implies $\Fix(σ)$​ for $(X,σ)$​ a nonsymplectic automorphism has at most one curve with $g\geq 2$ -- why?
  
- What are primitive bundles?

- Issue of separatedness for quasipolarized K3s:
	- There exist families $(X, L)$ and $(Y, M)$ over $\spec R$ for $R$ a DVR with isomorphic generic fibers 
	- But the isomorphism does not extend over the closed fibers to an isomorphism of the families
	- But special fibers are isomorphic
	  
- Easier instances of toric or semitoric compactifications?
- How much to study Baily-Borel compactifications if they are generalized by semitoroidal?
- Is the ADE contraction moduli stack separated?
- Recognizable divisors?

# Notes on moduli

## Stacks

- Defining the moduli functor: or a Noetherian base $S$, $\mcm_{2d}: \opcat{\Sch\slice S^\ft}\to \Grpd$ sends $T$ to $( f:X\to T, L)$ with $f$ smooth proper and $L\in \Pic_{X/T}(T)$ such that
	- Every geometric base change yields $X_k$ a K3 with $L_{X_k}$ primitive ample and $L_{X_k}^2 = 2d$
	- Thus the fibers are polarized K3s of degree $2d$.
	- Note $\Pic_{X/T}(A) \da \Pic(X_T)/q^* \Pic(T)$ where $q: X_T\to T$.
- **Definition:** a **fine moduli space** is $M_{2d} \in \Sch\slice S$ with a natural transformation $h_{M_{2d}}\iso \mcm_{2d}$ for $h$ the functor of points.
- **Definition:** a **coarse moduli space** is $M_{2d}$ with a natural transformation $\mcm_{2d}\to h_{M_{2d}}$ which is
	- A set bijection on geometric points
	- Initial: if $\mcm_{2d}\to h_N$ then it factors through $h_{M_{2d}}\to h_N$.
- **Hard result:** over $\spec \CC$ there is a coarse space $M_{2d}$ which is quasiprojective (proof: Hilbert schemes or GIT)
- **Easier result:** over any $S$, there is a coarse **separated** algebraic space $M_{2d}\in \Sch\slice S^{\lft}$.
- Regard as a groupoid over $\Sch\slice S$: take the category of pairs $\mcm_{2d} \da (f:X\to T, L)$ and define the projection $\mcm_{2d}\to \Sch\slice S$ by $(X\to T, L)\mapsto T$.
	- Morphisms are Cartesian squares where the upper arrow induces $\bar g^* L \cong L'$: ![](attachments/2023-02-14-morphism.png)
- **Theorem:** $\mcm_{2d}$ is a separated DM stack of finite type.
	- Becomes smooth over $\ZZ\invert{2d}$, but is not smooth.
	- **Theorem:** the coarse space $M_{2d}$ over $S = \spec \CC$ is not far from smooth: it is etale locally the quotient of a smooth scheme by a finite group.
- Can partially compactify by adding **quasi-polarized** K3s: replace $L$ ample with big and neg.
	- **Issue**: not separated!! See question above.
- Workaround: replace quasipolarized K3s with singular K3s
	- Let $L$ be big and nef, then $L^3$ is BPF 
		- Note $L$ ample implies $L^3$ very ample by Saint-Donat.
	- $\phi_{\abs{L^3}}: X \surjects \bar{X}$ contracts only ADE curves, so $\bar{X}$ has only RDPs
	- Yields a smooth DM stack with quasiprojective coarse space
	- Can describe via periods.

## Hilbert schemes

- For $(X, L)$ polarized, by RR, the Hilbert polynomial of $X$ is $p(t) = dt^2 + 2$.
- Embed in $\PP^N$ for $N \da p(3) - 1$; the Hilbert polynomial wrt $\OO_{\PP^N}(1)$ is $p(3t)$.
- Take $\Hilb \da \Hilb_{\PP^N}^{p(3t)}$, a projective scheme $\Hilb: \Sch\slice k\to \Set$ sending $T$ to $T\dash$flat closed subschemes $Z \subset T\fiberprod{k}\PP^N\slice k$ where all geometric fibers have Hilbert polyomial $p(3t)$.
	- Has a universal family $\mcz$ with a flat projection.
- **Theorem**: there is a subscheme $H \leq \Hilb$ such that:
	- Universal property: $T\to \Hilb$ factors through $Z$ iff the universal pullback $f: \mcz_T\to T$ satisfies:
	- $f$ is smooth, all fibers are K3s
	- For $p:\mcz_T \to \PP^N$ the projection, $p^* \OO_{\PP^N}(1) \cong L^3\tensor f^*L_0$  for some $L\in \Pic(\mcz_t)$ and $L_0\in \Pic(T)$.
	- $L$ is primitive on geometric fibers
	- All fibers $\mcz_s$ of $f$, restriction induces isos $H^0(\PP^N\slice{k(s)}; \OO(1))\iso H^0(\mcz_s; L_s^3)$. 
- **Propostion**: there is an right-action $\PGL_{N+1}\actson \Hilb$ which preserves $H$, and $h_H \to \mcm_d$ which forgets the embedding is equivariant, so get a natural transformation $\Theta: h_H / \PGL_{N+1}\to \mcm_d$ which is injective and locally surjective.
- $\Theta$ induces a bijection on $k\dash$points, so one just needs a categorical quotient (points parameterize orbits); if so it is a coarse moduli space.
	- **Theorem (Viehweg)**: over $k=\CC$ one $Q$ exists as a coarse quasiprojective scheme, and $[H/\PGL_{N+1}](k) \iso Q(k)$.
	- Avoids checking GIT stability, which only works for low-dim, but doesn't generalize to positive characteristic.
	- Open problem: existence of quasiprojective coarse spaces in positive characteristic
- **Definition**: $x\in H(k)$ is GIT stable if $\Stab_G(x) < \infty$ and there is a $G\dash$invariant section $s\in H^0(H; \mcl^n)$ for $\mcl\in \Pic(H)$ $\SL_n\dash$linearized where $s(x) \neq 0$, $H_s\da H\sm Z(s)$ is affine, and $\PGL\actson H_s$ is closed
- **Theorem (Keel-Mori)**: it's easy to show a coarse algebraic space exists. If $G$ is linear algebraic, $G\actson H$ properly for $H\in \Sch^\ft\slice S$ for $S$ Noetherian, the categorical quotient $\pi: H\to Q\da H/G$ exists as a separated algebraic space and for $k=\kbar$ induces an isomorphism $H(k)/G(k)\iso Q(k)$.
	- One needs to show the action is proper, ie $\PGL_n\times H\to H$ where $(g,x)\mapsto (g,gx)$ is a proper morphism (separated ft univ closed).
	- Reduces to showing closed orbits and finite stabilizers.