Let $X$ be an algebraic variety (separated quasi-compact scheme of finite type) over a field $k$.

One of the possible definitions of an ample line bundle goes as follows:

Def 1: A line bundle $\mathcal{L}$ on $X$ is said to be ample iff some tensor power of it $\mathcal{L}^{\otimes k}$ admits $n+1$-generating sections (for some $n$) s.t. the associated morphism $X \to \mathbb{P}^n$ is a closed embedding.

I always found this definition rather subtle and mysterious. The classical story goes through showing that this definition is equivalent to the following one (which is manifestly much more useful in practice and much less easy to check):

Def 2: A line bundle $\mathcal{L}$ on $X$ is said to be ample if for every coherent sheaf $\mathcal{F}$ there exists some $n>0$ (depending on $\mathcal{F}$) such that for all $m>n$ the sheaf $\mathcal{L}^{\otimes m} \otimes_{\mathcal{O}_X} \mathcal{F}$ is generated by global sections (i.e. is a quotient of a trivial vector bundle).

The proof I know of this equivalence is subtle and goes through a reduction argument to the projective case and using serre vanishing (notice that may be why the relation between $k$ in the first definition and $n$('s) in the second is highly indirect).

Let $QCoh(X)$ denote the derived (stable $\infty$-)category of sheaves of quasi-coherent $\mathcal{O}_X$-modules with symmetric monoidal structure given by $\otimes_{\mathcal{O}_X}$.

Given this structure we can easily to detect (shifted) line bundles inside $QCoh(X)$ as those are given by the $\otimes$-invertible objects. Here's the question:

Questions: Given a (shifted-)line bundle $\mathcal{L}$ in $QCoh(X)$ can we... (increasing level of difficulty).

1. Detect whether $\mathcal{L}$ is ample (in the classical sense above) without "leaving" the derived category $QCoh(X)$? (using $\otimes$-structure).

2. Detect whether $\mathcal{L}$ is ample by considering $QCoh(X)$ without the $\otimes$-structure, but remembering the action of $Pic(X)$ (the $\infty$-picard groupoid of line bundles) on it.

3. Detect whether $\mathcal{L}$ is very ample without using the $\otimes$-structure at all?. ֿ

Let $X$ be an algebraic variety (separated quasi-compact scheme of finite type) over a field $k$.

One of the possible definitions of an ample line bundle goes as follows:

Def 1: A line bundle $\mathcal{L}$ on $X$ is said to be ample iff some tensor power of it $\mathcal{L}^{\otimes k}$ admits $n+1$-generating sections (for some $n$) s.t. the associated morphism $X \to \mathbb{P}^n$ is a closed embedding (then $\mathcal{L}^{\otimes k}$ is said to be very ample).

I always found this definition rather subtle and mysterious. The classical story goes through showing that this definition is equivalent to the following one (which is manifestly much more useful in practice and much less easy to check):

Def 2: A line bundle $\mathcal{L}$ on $X$ is said to be ample if for every coherent sheaf $\mathcal{F}$ there exists some $n>0$ (depending on $\mathcal{F}$) such that for all $m>n$ the sheaf $\mathcal{L}^{\otimes m} \otimes_{\mathcal{O}_X} \mathcal{F}$ is generated by global sections (i.e. is a quotient of a trivial vector bundle).

The proof I know of this equivalence is subtle and goes through a reduction argument to the projective case and using serre vanishing (notice that may be why the relation between $k$ in the first definition and $n$('s) in the second is highly indirect).

Let $QCoh(X)$ denote the derived (stable $\infty$-)category of sheaves of quasi-coherent $\mathcal{O}_X$-modules with symmetric monoidal structure given by $\otimes_{\mathcal{O}_X}$.

Given this structure we can easily to detect (shifted) line bundles inside $QCoh(X)$ as those are given by the $\otimes$-invertible objects. Here's the question:

Questions: Given a (shifted-)line bundle $\mathcal{L}$ in $QCoh(X)$ can we... (increasing level of difficulty).

1. Detect whether $\mathcal{L}$ is ample (in the classical sense above) without "leaving" the derived category $QCoh(X)$? (using $\otimes$-structure).

2. Detect whether $\mathcal{L}$ is ample by considering $QCoh(X)$ without the $\otimes$-structure, but remembering the action of $Pic(X)$ (the $\infty$-picard groupoid of line bundles) on it.

3. Detect whether $\mathcal{L}$ is very ample without using the $\otimes$-structure at all?. ֿ

Let $X$ be an algebraic variety (separated quasi-compact scheme of finite type) over a field $k$.

One of the possible definitions of an ample line bundle goes as follows:

Def 1: A line bundle $\mathcal{L}$ on $X$ is said to be ample iff some tensor power of it $\mathcal{L}^{\otimes k}$ admits $n+1$-generating sections (for some $n$) s.t. the associated morphism $X \to \mathbb{P}^n$ is a closed embedding.

I always found this definition rather subtle and mysterious. The classical story goes through showing that this definition is equivalent to the following one (which is manifestly much more useful in practice and much less easy to check):

Def 2: A line bundle $\mathcal{L}$ on $X$ is said to be ample if for every coherent sheaf $\mathcal{F}$ there exists some $n>0$ (depending on $\mathcal{F}$) such that for all $m>n$ the sheaf $\mathcal{L}^{\otimes m} \otimes_{\mathcal{O}_X} \mathcal{F}$ is generated by global sections (i.e. is a quotient of a trivial vector bundle).

The proof I know of this equivalence is subtle and goes through a reduction argument to the projective case and using serre vanishing (notice that may be why the relation between $k$ in the first definition and $n$('s) in the second is highly indirect).

Let $QCoh(X)$ denote the derived (stable $\infty$-)category of sheaves of quasi-coherent $\mathcal{O}_X$-modules with symmetric monoidal structure given by $\otimes_{\mathcal{O}_X}$.

Given this structure we can easily to detect (shifted) line bundles inside $QCoh(X)$ as those are given by the $\otimes$-invertible objects. Here's the question:

Questions: Given a (shifted-)line bundle $\mathcal{L}$ in $QCoh(X)$ can we... (increasing level of difficulty).

1. Detect whether $\mathcal{L}$ is ample (in the classical sense above) without "leaving" the derived category $QCoh(X)$? (using $\otimes$-structure).

2. Detect whether $\mathcal{L}$ is ample by considering $QCoh(X)$ without the $\otimes$-structure, but remembering the action of $Pic(X)$ (the $\infty$-picard groupoid of line bundles) on it.

3. Detect whether $\mathcal{L}$ is very ample without using the $\otimes$-structure at all?. ֿ

I think the interesting information is probably contained in the action of $Pic(X)$ on $QCoh(X)$. Perhaps through studying this action one can identify the ampleness condition naturally. However I haven't been able to make serious progress about this. Is this trivial/well known somehow?

Let $X$ be an algebraic variety (separated quasi-compact scheme of finite type) over a field $k$.

One of the possible definitions of an ample line bundle goes as follows:

Def 1: A line bundle $\mathcal{L}$ on $X$ is said to be ample iff some tensor power of it $\mathcal{L}^{\otimes k}$ admits $n+1$-generating sections (for some $n$) s.t. the associated morphism $X \to \mathbb{P}^n$ is a closed embedding.

I always found this definition rather subtle and mysterious. The classical story goes through showing that this definition is equivalent to the following one (which is manifestly much more useful in practice and much less easy to check):

Def 2: A line bundle $\mathcal{L}$ on $X$ is said to be ample if for every coherent sheaf $\mathcal{F}$ there exists some $n>0$ (depending on $\mathcal{F}$) such that for all $m>n$ the sheaf $\mathcal{L}^{\otimes m} \otimes_{\mathcal{O}_X} \mathcal{F}$ is generated by global sections (i.e. is a quotient of a trivial vector bundle).

The proof I know of this equivalence is subtle and goes through a reduction argument to the projective case and using serre vanishing (notice that may be why the relation between $k$ in the first definition and $n$('s) in the second is highly indirect).

Let $QCoh(X)$ denote the derived (stable $\infty$-)category of sheaves of quasi-coherent $\mathcal{O}_X$-modules with symmetric monoidal structure given by $\otimes_{\mathcal{O}_X}$.

Given this structure we can easily to detect (shifted) line bundles inside $QCoh(X)$ as those are given by the $\otimes$-invertible objects. Here's the question:

Questions: Given a (shifted-)line bundle $\mathcal{L}$ in $QCoh(X)$ can we... (increasing level of difficulty).

1. Detect whether $\mathcal{L}$ is ample (in the classical sense above) without "leaving" the derived category $QCoh(X)$? (using $\otimes$-structure).

2. Detect whether $\mathcal{L}$ is ample by considering $QCoh(X)$ without the $\otimes$-structure, but remembering the action of $Pic(X)$ (the $\infty$-picard groupoid of line bundles) on it.

3. Detect whether $\mathcal{L}$ is very ample without using the $\otimes$-structure at all?. ֿ

Let $X$ be an algebraic variety (separated quasi-compact scheme of finite type) over a field $k$.

One of the possible definitions of an ample line bundle goes as follows:

Def 1: A line bundle $\mathcal{L}$ on $X$ is said to be ample iff some tensor power of it $\mathcal{L}^{\otimes k}$ admits $n+1$-generating sections (for some $n$) s.t. the associated morphism $X \to \mathbb{P}^n$ is a closed embedding.

I always found this definition rather subtle and mysterious. The classical story goes through showing that this definition is equivalent to the following one (which is manifestly much more useful in practice and much less easy to check):

Def 2: A line bundle $\mathcal{L}$ on $X$ is said to be ample if for every coherent sheaf $\mathcal{F}$ there exists some $n>0$ (depending on $\mathcal{F}$) such that for all $m>n$ the sheaf $\mathcal{L}^{\otimes m} \otimes_{\mathcal{O}_X} \mathcal{F}$ is generated by global sections (i.e. is a quotient of a trivial vector bundle).

The proof I know of this equivalence is subtle and goes through a reduction argument to the projective case and using serre vanishing (notice that may be why the relation between $k$ in the first definition and $n$('s) in the second is highly indirect).

Let $QCoh(X)$ denote the derived category of sheaves of quasi-coherent $\mathcal{O}_X$-modules with symmetric monoidal structure given by $\otimes_{\mathcal{O}_X}$.

Given this structure we can easily to detect (shifted) line bundles inside $QCoh(X)$ as those are given by the $\otimes$-invertible objects. Here's the question:

Question: Given a (shifted-)line bundle $\mathcal{L}$ in $QCoh(X)$ can we detect whether it is ample (in the classical sense above) without "leaving" the derived category $QCoh(X)$? (Bonus: Can we detect whether $\mathcal{L}$ is very ample without using the $\mathcal{otimes}$-structure?).

I think the interesting information is probably contained in the action of $Pic(X)$ on $QCoh(X)$. Perhaps through studying this action one can identify the ampleness condition naturally. However I haven't been able to make serious progress about this. Is this trivial/well known somehow?

Let $X$ be an algebraic variety (separated quasi-compact scheme of finite type) over a field $k$.

One of the possible definitions of an ample line bundle goes as follows:

Def 1: A line bundle $\mathcal{L}$ on $X$ is said to be ample iff some tensor power of it $\mathcal{L}^{\otimes k}$ admits $n+1$-generating sections (for some $n$) s.t. the associated morphism $X \to \mathbb{P}^n$ is a closed embedding.

I always found this definition rather subtle and mysterious. The classical story goes through showing that this definition is equivalent to the following one (which is manifestly much more useful in practice and much less easy to check):

Def 2: A line bundle $\mathcal{L}$ on $X$ is said to be ample if for every coherent sheaf $\mathcal{F}$ there exists some $n>0$ (depending on $\mathcal{F}$) such that for all $m>n$ the sheaf $\mathcal{L}^{\otimes m} \otimes_{\mathcal{O}_X} \mathcal{F}$ is generated by global sections (i.e. is a quotient of a trivial vector bundle).

The proof I know of this equivalence is subtle and goes through a reduction argument to the projective case and using serre vanishing (notice that may be why the relation between $k$ in the first definition and $n$('s) in the second is highly indirect).

Let $QCoh(X)$ denote the derived (stable $\infty$-)category of sheaves of quasi-coherent $\mathcal{O}_X$-modules with symmetric monoidal structure given by $\otimes_{\mathcal{O}_X}$.

Given this structure we can easily to detect (shifted) line bundles inside $QCoh(X)$ as those are given by the $\otimes$-invertible objects. Here's the question:

Questions: Given a (shifted-)line bundle $\mathcal{L}$ in $QCoh(X)$ can we... (increasing level of difficulty).

1. Detect whether $\mathcal{L}$ is ample (in the classical sense above) without "leaving" the derived category $QCoh(X)$? (using $\otimes$-structure).

2. Detect whether $\mathcal{L}$ is ample by considering $QCoh(X)$ without the $\otimes$-structure, but remembering the action of $Pic(X)$ (the $\infty$-picard groupoid of line bundles) on it.

3. Detect whether $\mathcal{L}$ is very ample without using the $\otimes$-structure at all?. ֿ

I think the interesting information is probably contained in the action of $Pic(X)$ on $QCoh(X)$. Perhaps through studying this action one can identify the ampleness condition naturally. However I haven't been able to make serious progress about this. Is this trivial/well known somehow?