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created: 2024-05-03T00:14
updated: 2024-05-15T17:53
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Misc

- How can you tell if $D\in \Div_T(X_\Sigma)^\amp$ for $X_\Sigma$ toric?
- If $(X, L)\in \Ag$, $d = 1 \implies L$ is a principal polarization
- By Riemann-Roch, $d = h^0(L) = L^g/g!$, and if $g=2$ then $L^2 = 2d$.
- See Delauney tilings
- See Kulikov 1979 for Type III degenerations
- K3s are a bit like spheres: $h^2(\OO_X) = h^0(\OO_X) = 1$, so similar to $H^2(S^2; \ZZ)$.
- How to formulate mirror symmetry in terms of nef line bundles?
- Type A:
	- Symplectic
	- Nef line bundles
	- $\omega$ a symplectic form
	- Moment maps $\mu: (X, \omega) \to S^2$
	- $\IAS^2$ with 24 singularities
	- **$L$ a nef line bundle**
- Type B:
	- AG
	- 1-parameter degenerations
	- Kulikov models $\mcx$ with $\mcx_0 = \union (V_i, D_i)$ a Kulikov surface (broken K3), almost toric, described by a Symington polytope, $\mu\inv(p) = (S^1)^2$ almost everywhere
		- ![[2023-08-01 Old Notes 2023-08-25 11.00.30.excalidraw]]
	- $K_{V_i} + D_i \sim 0$, so all pairs are ACPs
	- **Monodromy invariant $\lambda$**
- All yield Lagrangian torus fibrations, if $p$ is an $I_1$ singularity then the fiber $\mu\inv(p)$ is a pinched torus
	- ![[2023-08-01 Old Notes 2023-08-25 11.06.13.excalidraw]]
- See Oda and Schenck for more info on moment maps

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- For a toric variety $X_P$ coming from a polytope $P$, one has $\vol P = L^{\dim X} = L^2$ for toric surfaces.
	- Note that we take the **lattice** volume here, which is $\vol P \da (\dim X)! \cdot m(P)$ for the standard Lebesgue measure.
	- So the standard simplex $P$ for $(\PP^2, \OO(1))$ has $m(P) = {1\over 2}$ so $\vol P = 2! \cdot {1\over 2} = 1$ and thus $\OO(1)^2 = 1$.
	- Similarly $3P$ has $m(3P) = {9\over 2}$ so $\vol P = 9$, so this represents $(\PP^2, \OO(3))$.
	- Since $\Pic(\PP^2) \cong \ZZ$, all $\ZZ\dash$line bundles correspond to integer dilations. Taking $\QQ$ or $\RR$ coefficients for the dilations yields $\QQ$ or $\RR$ line bundles
- To go from $P$ to $\Sigma$: take the **inward** pointing normal fan.
- When subdividing a fan, a vector $(a, b)$ corresponds to a blowup at $(x^a, y^b)$.
- How does one compute $\Pic(X_\sigma)$? The intersection matrix of $T\dash$invariant curves?
- How does one describe $\Pic(X_\sigma)^\amp$ and $\Div(X_\Sigma)^\amp$?
- Fano: $-K_X$ is ample.
- When is a toric variety Fano?
	- Classified for $\dim X = 2$, due to Batyrev? There are very few.
	- For $\dim X = 3$, I've written that there are 11, but that doesn't seem right.
- What is a del Pezzo?
- For surfaces, log terminal is equivalent to quotients by finite $G \leq \GL_n$.
	- Quotients always imply log terminal, but the converse only holds for surfaces.
- Look at Gorenstein surface singularities.
- One can classify resolutions of surfaces by looking at the graphs that occur.

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- A Kulikov degeneration is described by a number of polytopes glued together in a honeycomb.
	- Most give toric varieties by taking the inward-pointing normal fans
	- Check by adjunction $\ro{K_X}{V_i} = K_{V_i} + D_i = 0$.
	- All have charges $Q(V_i, D_i)\in \NN$ with $Q = 0 \iff$toric.
	- Each $V_j$ is glued to some union of other surfaces along an anticanonical cycle $D_j = \sum D_{ij}$.
- A corner blowup inserts a new edge that replaces a corner. The result is again toric.
  ![[2023-08-01 Old Notes 2023-08-25 11.31.06.excalidraw]]
- At the level of curves, we get the following:
  ![[2023-08-01 Old Notes 2023-08-25 11.35.49.excalidraw]]
- Compute $K_{V_1} = \pi^* K_{V} + E$ and $D' = \pi^*  - 2E$ to get $K_V + D' + E = \pi^*(K_V + D) = \pi^*(0)  = 0$, so the result is still anticanonical.
- Internal blowups: not at a corner, introduces a $-1$ curve $E$:
  ![[2023-08-01 Old Notes 2023-08-25 11.38.29.excalidraw]]
	- Still anticanonical: let $(V', D')$ be the result of the blowup of $(\bar V, \bar D)$, then $K_{V'} = \pi^* K_{\bar V}$ and $D' = \pi^* D - E$, where the coefficient of $E$ is the multiplicity of the curve, and so $K_{V'} + D' = \pi^*(K_{\bar V} + \bar D) = 0$.
- A theorem: given $(V, D)$, there exists a span:
  [Quiver diagram](https://q.uiver.app/#q=WzAsMyxbMiwwLCIoVicsIEQnKSJdLFs0LDIsIihcXGJhciBWLCBcXGJhciBEKSJdLFswLDIsIihWLCBEKSJdLFswLDIsIlxcdGV4dHtDb3JuZXIgYmxvd3Vwc30iLDJdLFswLDEsIlxcdGV4dHtJbnRlcm5hbCBibG93dXBzfSJdXQ==)
  ![[Pasted image 20230825114600.png]]
  where the left leg is a composition of corner blowups (thus toric) and the right is a composition of internal blowups (nontoric).