8
votes

Accepted

### Pushforward of a very ample line bundle on a curve to $\mathbb{P}^1$

No, not in general. Take $C=\mathbb{P}^1$, $L=\mathcal{O}(1)$, $p$ to be map $x\mapsto x^2$ in affine coordinates. Then $p_*L$ has rank $2$, but
$$2=h^0(L)=h^0(p_*L)=h^0(\mathcal{O}(e_1))+h^0(\mathcal{...

7
votes

Accepted

### Does a projective variety have only finitely many associated Hilbert polynomials?

Yes for the first question, by Riemann--Roch.
No for the second --- even in the simplest case of a projective line, the polynomial $td + 1$ is the Hilbert polynomial (with respect to $L = O(d)$).

7
votes

Accepted

### On a proposition in Hartshorne's paper "Ample vector bundles on curves"

The assertion (X) is false in any characteristic $p > 0$ for any $Y$ with genus at least 2. (It is true and easy for $Y$ of genus 1, and true and easy and uninteresting for $Y$ of genus 0.)
To ...

6
votes

Accepted

### Does normalization of projective varieties preserve very ampleness

1) There are certainly many cases where $f^*L$ is still very ample: e.g. if $X$ is an irreducible curve, any line bundle on $X$ of degree $\geq 2g(\tilde{X} )+1$ will have this property.
2) A ...

6
votes

### Varieties with an ample vector bundle mapping to their tangent bundle

This is not a complete answer, but it's way too long for a comment.
You can define ampleness for arbitrary coherent sheaves. This is done for example in V. Ancona, "Faisceaux amples sur les espaces ...

5
votes

Accepted

### Does ampleness descend along finite maps?

There is a non-quasi-affine variety $X$ with quasi-affine normalization. See Tag 0271. Then $\mathcal{O}_X$ is a counter example.

5
votes

Accepted

### Classifying ample line bundles over the flag manifold $G/B$

As Donu said, line bundles on $G/B$ correspond to elements of the weight lattice. The ample bundles correspond to the dominant regular weights. This means that the pairing of the weight $\lambda$ with ...

4
votes

Accepted

### Ample vector bundles and embeddings

The answer to your first question is yes. Gieseker (https://projecteuclid.org/download/pdf_1/euclid.nmj/1118798367) calls such bundles as strongly ample. ( see p 92 of above paper). To see that such ...

4
votes

Accepted

### The ample cone of a surface with an algebraic $\mathbb C^*$ action

This is true, and follows from Corollary 3.29 of https://arxiv.org/pdf/0811.0517.pdf

4
votes

Accepted

### Varieties with an ample vector bundle mapping to their tangent bundle

The answer to this question is now known to be yes -- it is Corollary 1.2 of this paper of Jie Liu.

4
votes

### Global section of very ample line bundles and its value on stalks

It seems to me that the $m$ is a red herring. Since you already assume that $\mathcal L$ is very ample, why do you need a power? I'll ignore $m$ here.
For the questions: In case the base field is ...

4
votes

### Weak Fano and Log fano varieties

As pointed out by the other answers, every smooth weak Fano is certainly log Fano. But you also asked for an example of a log Fano variety that is not weakly Fano.
Take any toric variety for which $-...

4
votes

### On a proposition in Hartshorne's paper "Ample vector bundles on curves"

Yes, this is classic. A counterexample was found by Serre, and more classes of counterexamples are in Fulton's paper Ample Vector Bundles, Chern Classes,
and Numerical Criteria (Inventiones, 1976).

3
votes

Accepted

### An ample line bundle on a K3 surface

The ampleness follows from Nakai--Moichezon criterion. The degree is twice the degree of $O(1,1)$ on $P^1\times P^1$, so is $4$.

2
votes

Accepted

### Making a vector bundle ample by twisting with ample line bundle

Because $L$ is ample, $E\otimes L^n$ is generated by global sections for $n\gg 0$, i.e., there is a surjective morphism $\mathscr O_X^{\oplus r} \to E\otimes L^n$, which implies that there is a ...

2
votes

Accepted

### Ample divisors on $\mathbb{P}^n$ blown-up at $k$ general points

If $k\leq n+3$ and if $n = 3, k = 7$, or $n = 4, k = 8$ then $X$ is a Mori Dream Space. You can find this here http://arxiv.org/abs/math/0505337.
In particular this implies that the cone of curves $...

2
votes

### When is the determinant of the push-forward of an ample line bundle ample

Actually, I think that the phenomenon Jason is describing is very much due to using a finite morphism, so there is something intelligent one could say about this.
Of course, as stated there is no ...

2
votes

Accepted

### Weak Fano and Log fano varieties

That is true. Basically it is a consequence of the following fact:
Let $D$ be a nef and big divisor on an irreducible projective variety $X$. Then there exist an effective divisor $E$ and a rational ...

2
votes

### Weak Fano and Log fano varieties

I believe that all weak Fano varieties are log Fano. Basically, you can find an effective divisor $D$ such that $-K_X-\epsilon D$ is ample for arbitrarily small $\epsilon$. For a reference, see the ...

2
votes

### Global section of very ample line bundles and its value on stalks

Maybe it's worth saying something more about question (2) (like Sándor, I'll stay in the algebraically closed case). This gets very close to the realm of Seshadri constants, see Positivity in ...

2
votes

Accepted

### Fiber product of projective varieties and ample line bundles

Let $D_X$ and $D_Y$ be the Chern classes of $L_X$ and $L_Y$. Assume $X$ and $Y$ have dimension at least one. Thus $pr_{1*}pr_1^*D_X=0$ and $pr_{2*}pr_2^*D_Y=0$. Applying the projection formula we ...

1
vote

Accepted

### Pencils in very ample linear systems without curve in its base locus

Here is a more precise formulation of my suggestion above. First of all, I do not believe that you can find such a surface in $\mathbb{P}^3$. The base locus curves must be coordinate lines. In ...

Community wiki

1
vote

### Global section of very ample line bundles and its value on stalks

I propose that this is what you "really" mean to say by 2): "what is the maximum length $t(L)$ such that for any $0$-dimensional scheme $Z$ of length $t(L)$, the evaluation morphism $H^0(L) \otimes \...

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