All Questions
7 questions
1
vote
1
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249
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Higher cohomology of line bundles and small modifications
I am a PhD student in algebraic geometry and I am blocked on a question that I asked to myself for my research. I am not yet very comfortable with the traditional tools. Also, I apologize in advance ...
- 113
5
votes
2
answers
527
views
Divisors whose restriction is big
Let $f:X\rightarrow Y$ be a flat morphism of smooth projective varieties, and $\mathcal{L}$ an effective and ample line bundle on $Y$. For a divisor $A\in H^0(Y,\mathcal{L})$ set $X_A := f^{-1}(A)$.
...
user329228
6
votes
2
answers
524
views
Global sections of multiples of a divisor
Let $D$ be an integral divisor on a smooth projective variety $X$. Consider the multiples $mD$ of $D$ for $m\geq 0$. Clearly, $h^0(X,mD) = 1$ for $m = 0$.
Is there any example where $h^0(X,mD) = 0$ ...
- 8,998
2
votes
1
answer
1k
views
Cohomology of tangent sheaf of a hypersurface
Let $X\subset\mathbb{P}^n$ be an irreducible and reduced hypersurface of degree $d$. How can one explicitly compute the dimension of the vector spaces $H^0(X,T_X),H^1(X,T_X),H^2(X,T_X)$? Here $T_X$ is ...
user125056
5
votes
2
answers
676
views
Log canonical counterexample to Kawamata-Viehweg vanishing
I found in the literature that, in characteristic 0, Kodaira vanishing holds for log-canonical pairs. On the other hand, the usual statement for Kawamata-Viehweg vanishing talks about a klt pair $(X,\...
- 625
2
votes
0
answers
263
views
Global section of line bundle on anti-canonical rational surface
Let $X$ be an anti-canonical rational surface(i.e. $-K_X$ is effective) such that $K_X^2\geq 1$. Let $D$ be a $r$-class divisor ($D^2=r, D^2+D.K_X=-2$, the latter condition can be re-interpreted as $\...
- 1,982
3
votes
1
answer
430
views
What is $h^0(\mathcal O_F)$ where $F$ is a fiber of a normal surface over a smooth curve?
Lately I am studying the bend-and-break, and I follow the proof in the following note written by Olivier Debarre:
http://www.math.ens.fr/~debarre/M2.pdf
There is a detail that I just cannot go ...
- 71