11
votes
Accepted
One-point compactification of ample line bundle
This is EGA 2, Prop. 8.8.2. It basically says that if $L$ is ample then one can contract the zero section of the geometric realization $\mathbb V(L)$ of $L$ to a point. The result is called the affine ...
- 15.5k
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{...
- 35.2k
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 $L = O(d)$).
- 39.3k
6
votes
Accepted
Ampleness verifiable over faithfully flat cover
According to stacks project tag 0B5V, if $S$ is Noetherian, $g$ is finite and surjective, and $X$ is proper over $S$ (which automatically implies properness of $X'$), then $\mathcal L$ is ample if and ...
- 148k
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 ...
- 6,829
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.
- 66
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.
- 23k
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
- 56
3
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 ...
- 42.9k
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 ...
- 569
1
vote
Bertini type theorem for very ample line bundle
1. General hypersurface of a normal variety is normal;
see ``The hyperplane sections of normal varieties'', Seidenberg, A. (1950), Transactions of the American Mathematical Society, 69(2), 357-386. (...
- 85
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