All Questions
Tagged with sheaf-cohomology projective-geometry
12 questions
1
vote
0
answers
295
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Exterior product of Euler Exact Sequence
Consider the Euler exact sequence:
$ 0\longrightarrow \mathcal{O}_{\mathbb{P}^n} \longrightarrow \mathcal{O}_{\mathbb{P}^n}(1)^{n+1}\longrightarrow \mathcal{T}_{\mathbb{P}^n} \longrightarrow 0 $
This ...
- 131
4
votes
1
answer
649
views
Cohomology of divisors on Hirzebruch surfaces
Consider the Hirzebruch surface $\mathbb{F}_n = \mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus \mathcal{O}_{\mathbb{P}^1}(n))\rightarrow\mathbb{P}^1$. The Picard group of $\mathbb{F}_n$ is generated by ...
user114666
5
votes
2
answers
527
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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
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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
6
votes
2
answers
632
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Geometric meaning of coherent sheaves $\mathcal{F} \otimes \mathcal{O}_{\mathbb{P}^n}(d)$ over $\mathbb{P}^n$
Maybe it sounds like a silly question but I'm not able to figure out in my head the geometric meaning of "twisting" vector bundles (or more generaly coherent sheaf) over $\mathbb{P}^n$.
I ...
- 1,343
2
votes
1
answer
279
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Help about "Varieties with small Dual Varieties" by L.Ein
I'm studying the paper "Varieties with small Dual Varieties" by L.Ein and in the construction he gives about the $10-$dimensional spinor variety $S_4 \subset \mathbb{P}^{15}$ I'm finding ...
- 1,343
1
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0
answers
152
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What is the smallest number $d$ such that $H^1(X,\pi^*\mathcal{O}_{\mathbb{P}_k^1}(d))$ vanishes?
Let $X$ be a reduced projective scheme of pure dimension 1 over the field $k$. Let $\pi: X \to \mathbb{P}_k^1$ be a finite, flat and surjective morphism onto the projective line.
What is the ...
- 435
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
3
votes
0
answers
285
views
Reference for the Koszul--Malgrange Theorem
The Koszul--Malgrange theorem, roughly, identifies holomorphic vectors bundles over a complex manifold, as those finitely generated projective modules admitting a flat $(0,1)$-connection. The ...
- 31
6
votes
1
answer
753
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Sections of the conormal bundle
Let $X\subset\mathbb{P}^N$ be a quadratic manifold. That is $I(X)$ is generated by quadratic polynomials $Q_1,...,Q_m$.
Let $\mathcal{I}_X$ be the ideal sheaf of $X$ and $\mathcal{I}_X/\mathcal{I}_X^...
user82886
1
vote
2
answers
276
views
Sections of a sheaf of differentials on a weighted complete intersection
Let $X\subset\mathbb{P}(a_0,...,a_N)$ be a smooth $n$-dimensional weighted complete intersection in a weighted projective space $\mathbb{P}(a_0,...,a_N)$.
Is it true that if $q\geq 1$ then $H^0(X,\...
user75933
6
votes
1
answer
261
views
Do general sheaves on P^2 have cohomology governed by their Euler characteristic?
Suppose $\xi$ is chern character on $\mathbb P^2$. Then there is a moduli space $M(\xi)$ of semistable sheaves of chern character $\xi$.
If $\xi$ has Euler characteristic 0, then apparently there is ...
- 1,509