All Questions
7 questions
1
vote
0
answers
143
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Existence and injectivity of a map from $\mathrm{Num}(X) \otimes_\mathbb{Z} \mathbb{C}$ to $H^1(X,\Omega^1_X)$
I am currently reading a paper by Mustata and Popa on the Van de Ven theorem, you can read the paper here.
On page 51 there is the following map
$$\alpha_\mathbb{C} : \mathrm{Num}(X) \otimes_\mathbb{Z}...
- 201
5
votes
1
answer
463
views
Square root of a line bundle up to a finite surjective morphism
Given a projective variety $X$ over a field of any characteristic, consider a line bundle $\mathcal{L}$ over $X$.
The existence of a line bundle $\mathcal{L}^\prime$ with an isomorphism ${\mathcal{L}^...
- 100
2
votes
1
answer
482
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Divisors on projective bundles
Let $\pi:X = \mathbb{P}(\mathcal{E})\rightarrow\mathbb{P}^n$ be a projective bundle, where $\mathcal{E}$ is a rank two vector bundle over $\mathbb{P}^n$.
If $n = 0$ then $X = \mathbb{P}^1$, and for $n ...
- 8,998
9
votes
1
answer
443
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Set theoretic equation for Veronese varieties
Consider the embedding $f:\mathbb{P}^n\rightarrow\mathbb{P}^N$ induced by the complete linear system of degree $d$ hypersurfaces of $\mathbb{P}^n$. Its image $V_{n,\,d}$ is degree $d$ Veronese variety ...
user125056
6
votes
2
answers
719
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Intersection numbers in $\mathbb{P}^1$-bundles
Let $\mathcal{E}$ be a rank two vector bundle on $\mathbb{P}^2$ fitting in the following exact sequence
$$0\rightarrow \mathcal{O}_{\mathbb{P}^2}\rightarrow \mathcal{E}\rightarrow \mathcal{I}_p(-1)\...
user101890
1
vote
2
answers
985
views
Top self-intersection of the tautological line bundle
Let $\mathcal E$ be a rank $n$ vector bundle over a curve $Y$ and let $X=\mathbb P(\mathcal E)$ and let $\pi: X \to Y$ be the projection. I would like to compute the value of the top self-intersection ...
- 427
0
votes
0
answers
161
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sections of vector bundles transversal to a divisor
Let $X$ a smooth projective curve over $\mathbb{C}$, $S$ a finite subscheme of $X$.
$E$ a vector bundle over $X$ with a divisor $D$.
We look at the sections $A:=H^{0}(X,E)$ with $\deg E$ big enough.
...
- 3,472