All Questions
Tagged with sheaf-cohomology algebraic-curves
10 questions
5
votes
1
answer
299
views
First cohomology of tangent sheaf of rational curve
Let $C$ be a reduced, connected, projective and purely one-dimensional scheme of finite type over a field $k$.
Suppose that $C$ is rational, i.e. that the normalisation of $C$ is a disjoint union of ...
- 984
2
votes
1
answer
385
views
Very weak Riemann-Roch on curves (by J. Kollar)
I have a question on an unequality used in the proof of the Very weak Riemann-Roch on curves in Janos Kollar's Lecture on Resolution of Singularities (page 14):
1.13 (Very weak Riemann-Roch on curves)...
- 5,966
2
votes
0
answers
405
views
Cohomology of a family of twisted cubic curves (Hartshorne III, 12.9.2)
I'm trying to understand following Example in Hartshorne (Chapter III, Example 9.8.2
& Example 12.9.2):
Let $X_1 \subset \mathbb{P}^{3}$ be a twisted cubic curve not containing the point
$(0:...:1)...
- 5,966
1
vote
1
answer
452
views
Restriction of sheaves on curves
Let $C$ be a scheme of pure dimension $1$. Let $C_1$ be a closed subscheme of $C$ of pure dimension $1$. Denote by $i:C_1 \hookrightarrow C$ a closed immersion. Given a sheaf $\mathcal{F}$ on $C$, ...
- 2,323
1
vote
0
answers
152
views
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
1
vote
0
answers
80
views
Sections of nodal curves
We work over an algebraically closed field. Suppose $X\subset \mathbf{P}^n$ is an integral projective curve and $\pi:X\to Y$ is a linear projection that identifies two distinct points $p,q\in X$ to a ...
- 141
1
vote
0
answers
445
views
Pull-back of globally generated sheaves
Let $X$ be a smooth projective surface in $\mathbb{P}^3$, $D=\sum_i n_iD_i$ an effective Cartier divisor. Let $C$ be a smooth irreducible curve on $X$. Denote by $i:C \hookrightarrow X$ is the closed ...
- 1,981
0
votes
1
answer
193
views
Is the square of a special line bundle also special?
Suppose $C$ is a smooth projective curve over, say, $\mathbb{C}$. I'm interested in knowing whether the following is true.
Let $\mathcal{L} \in Pic^d(C)$ be a special line bundle, i.e. its $H^1 \neq 0$...
- 129
0
votes
1
answer
660
views
Example of non-vanishing of first cohomology of a torsion coherent sheaf on a curve
By a curve we mean a projective scheme of pure dimension one. Can some one give an example of a curve $C$ and a torsion coherent sheaf on $C$ such that its first cohomology group does not vanish?
...
- 2,032
0
votes
0
answers
57
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Lifting of quadrics containing hyperplane section for projectively normal curves
Let $C \subset \mathbb{P}^r$ be a projective curve (over $k=\mathbb{C}$), smooth, irreducible and nondegenerate of degree $d$, ie the embedding line bundle $\mathcal{O}_C(1)=(\mathcal{O}_{\mathbb{P}^r}...
- 5,966