---
date: 2022-02-11 10:53
modification date: Friday 11th February 2022 10:53:03
title: very ample divisor
aliases: [ample, very ample, ample divisor, ample bundle, globally generated, "semi-ample", globally generated, basepoint free]

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Last modified date: <%+ tp.file.last_modified_date() %>

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

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# ample

![](attachments/Pasted%20image%2020220921204354.png)

![](attachments/Pasted%20image%2020220905150344.png)

Idea: a notion of "positivity" for e.g. line bundles, related to having lots of global sections.

- $\mcl$ is **basepoint free** iff $\mcl$ admits enough sections to define a morphism to some $\PP^N$.
	- Equivalently, $\mcl$ is **globally generated**, i.e. $\mcl \in \mods{\OO_X}$ admits a set of global sections $\ts{s_i \in \globsec{X; \mcl}}_{i\in I}$  such that $\bigoplus_{i\in I} \OO_X \surjects \mcl$.
- $\mcl$ is **semi-ample** iff some positive power $\mcl\tensorpowerk{n}$ is basepoint free (idea: non-negative).
- $\mcl$ is **very ample** iff $\mcl$ admits enough sections to define a [closed immersion](Unsorted/closed%20immersion.md) into some $\PP^N$. 
	- Equivalently, $\mcl$ is basepoint free and the associated morphism $f: X\to \PP^N$ is a closed immersion.
	- Equivalently, $X$ can be embedded into some $\PP^N$ such that $\mcl = \ro{ \OO(1) }{X}$.
- $\mcl$ is **ample** iff some positive power $\mcl\tensorpowerk{n}$ is very ample.

![](attachments/Pasted%20image%2020220928225756.png)

Kleiman's criteria:
![](attachments/Pasted%20image%2020220601125451.png)

Pushforwards of ample divisors are [nef](Unsorted/nef%20divisor.md):
![](attachments/Pasted%20image%2020220601125527.png)