---
date: 2022-09-10 22:14
modification date: Saturday 10th September 2022 22:14:41
title: "2022-09-10"
aliases: [2022-09-10]
---

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---

- Tags: 
	- #unsorted #AG/toric
- Refs:
	- #todo/add-references
- Links:
	- #todo/create-links 

---

# 2022-09-10

![](attachments/Pasted%20image%2020220910221621.png)

When Weil are Cartier
![](attachments/Pasted%20image%2020220910221638.png)
![](attachments/Pasted%20image%2020220910221649.png)

Cartier data:
![](attachments/Pasted%20image%2020220910221927.png)

Theorems:
![](attachments/Pasted%20image%2020220910222053.png)

![](attachments/Pasted%20image%2020220912132252.png)
![](attachments/Pasted%20image%2020220912133131.png)
![](attachments/Pasted%20image%2020220912133645.png)
![](attachments/Pasted%20image%2020220912133827.png)
![](attachments/Pasted%20image%2020220912133946.png)
![](attachments/Pasted%20image%2020220912135600.png)
![](attachments/Pasted%20image%2020220912135656.png)
![](attachments/Pasted%20image%2020220912135738.png)

$$
\left(D_{\Delta}\right)^n=n ! \operatorname{vol}(\Delta)
$$
![](attachments/Pasted%20image%2020220912140158.png)
![](attachments/Pasted%20image%2020220912140248.png)
![](attachments/Pasted%20image%2020220912140326.png)

![](attachments/Pasted%20image%2020220912140337.png)
![](attachments/Pasted%20image%2020220913144520.png)
![](attachments/Pasted%20image%2020220913151810.png)
![](attachments/Pasted%20image%2020220913151930.png)
![](attachments/Pasted%20image%2020220913155158.png)
![](attachments/Pasted%20image%2020220913155227.png)

See [[cone multiplicity]].

![](attachments/Pasted%20image%2020220913155615.png)

![](attachments/Pasted%20image%2020220913160808.png)
![](attachments/Pasted%20image%2020220913160858.png)

![](attachments/Pasted%20image%2020220913161109.png)
![](attachments/Pasted%20image%2020220913161200.png)
![](attachments/Pasted%20image%2020220913161459.png)
![](attachments/Pasted%20image%2020220913161633.png)
![](attachments/Pasted%20image%2020220913161935.png)

See [Riemann-Roch](Unsorted/Riemann-Roch.md).

![](attachments/Pasted%20image%2020220913162606.png)
![](attachments/Pasted%20image%2020220913162652.png)
![](attachments/Pasted%20image%2020220913162718.png)
![](attachments/Pasted%20image%2020220913164109.png)

![](attachments/Pasted%20image%2020220913164220.png)
![](attachments/Pasted%20image%2020220913164306.png)

![](attachments/Pasted%20image%2020220913164844.png)
![](attachments/Pasted%20image%2020220913164855.png)
![](attachments/Pasted%20image%2020220913164932.png)
![](attachments/Pasted%20image%2020220913165030.png)


Example: cohomology for a 3-dimensional toric variety
![](attachments/Pasted%20image%2020220913165135.png)
![](attachments/Pasted%20image%2020220913165143.png)

See [log forms](Unsorted/log%20differential%20forms.md).

Denominator in definition of [Chow](Unsorted/Chow%20ring.md), i.e. what is rational equivalence?

![](attachments/Pasted%20image%2020220913171237.png)

![](attachments/Pasted%20image%2020220913171216.png)

![](attachments/Pasted%20image%2020220913171315.png)

Intersecting a [Cartier divisor](Unsorted/Weil%20divisor.md) with a subvariety:
![](attachments/Pasted%20image%2020220913171611.png)

Doing this torically:
![](attachments/Pasted%20image%2020220913171809.png)
![](attachments/Pasted%20image%2020220913171845.png)
![](attachments/Pasted%20image%2020220913171909.png)

![](attachments/Pasted%20image%2020220913172225.png)
![](attachments/Pasted%20image%2020220913172257.png)

![](attachments/Pasted%20image%2020220913172454.png)
![](attachments/Pasted%20image%2020220913172530.png)
![](attachments/Pasted%20image%2020220913172603.png)

Betti numbers:
![](attachments/Pasted%20image%2020220913172657.png)

![](attachments/Pasted%20image%2020220913172810.png)