---
title: New Periodic Note Template
created: 2023-03-26T11:59
updated: 2024-05-11T12:05
---

[📅 Google Calendar](https://calendar.google.com/calendar/u/0/r)

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# May 5th


- When a semitoroidal compactification with semifan $\mcf$ is isomorphic to a KSBA compactification, any semifan refining $\mcf$ yields a semitoroidal compactification which corresponds to a family of stable pairs.
	- In particular, if the Coxeter fan refines $\mcf$, there is an associated family of stable pairs.
- A generic element of $F_2$ is a branched cover of a sextic in $\PP^2$.
- $\Ag$ analogue: for $M \cong \ZZ^{2g}$ a lattice, the cone $\mcc$ is that of positive definite symmetric bilinear forms $Q: M\tensor M\to \ZZ \to \RR$. Any choice of a form $Q$ defines a Voronoi and Delaunay decomposition of $\mcc^{\mathrm{rc}}$ in $M_\RR$.
	- Write a form as $Q\leadsto f_Q: M\to N_\RR$, then the infinite toric variety is $f_Q(\Vor Q)/f_Q(M)$ up to a shift.
	- The quotient $N_\RR/f_Q(M)$ is an integral-affine torus $(S^1)^g$.
- Prove that toric blowups preserve charge and nontoric blowups increase the charge by one.
- Describe $\Gamma(\mcx_0)$ as a simplicial complex
- A subdiagram is called elliptic if the restriction of the quadratic form of $N$ to $\mathbb{R} V$ is negative definite. It is called parabolic if it negative semi-definite. Maximal parabolic means maximal by inclusion among the parabolic diagrams.
- Background on Coxeter groups and hyperbolic geometry: <https://www.cambridge.org/core/services/aop-cambridge-core/content/view/A117FD150E108CACCACF9EF00C61A23C/S1461157015000273a.pdf/coxiter_computing_invariants_of_hyperbolic_coxeter_groups.pdf#page=2&zoom=160,-279,661>
- What is $(r, a, \delta)$ for $F_2$, where $T = A_1 \oplus U \oplus E_8^2$? This seems to be $(19, 1, 1)_1$.
	- ![](2024-05-11.png)
- Coxeter conventions: ![](2024-05-11-1.png)
- Cusp correspondences:
	- $F_{\En, 2}\to F_{\En}$: ![](2024-05-11-2.png)
	- $F_{\En, 2} \to F_{(2,2,0)}$: ![](2024-05-11-3.png) 
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