All Questions
Tagged with divisors ag.algebraic-geometry
313 questions
16
votes
3
answers
4k
views
Contracting divisors to a point
This is quite possibly a stupid question, but it is pretty far from what I normally do, so I wouldn't even know where to look it up.
If $X$ is a projective variety over an algebraically closed field ...
- 4,450
1
vote
3
answers
913
views
Terminology issue: meaning of 'ample class' ?
What is meant by an "ample class" in general? Motivation: In the document I am reading, the phrase in question is "fix an ample class $\alpha\in H^1(X,\Omega^1_X)$." I know what ampleness of a line ...
- 545
17
votes
2
answers
1k
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Line bundles vs. Cartier divisors on a non-integral scheme
It is well-known that if $X$ is an integral scheme, then there is an isomorphism $CaCl(X)\to Pic(X)$ taking $[D]$ to $[\mathcal{O}_X(D)]$. Does anyone know any simple examples where the above map ...
- 11.6k
5
votes
2
answers
1k
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Special divisors on hyperelliptic curves
I was reading a proof that used the following result
Let $C$ be a hyperelliptic of genus $\ge 3$ and $\tau \colon C \to C$ the hyperelliptic involution. If $D$ is an effective divisor of degree $g-1$...
- 3,968
7
votes
7
answers
2k
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Nef divisors with few global sections
Are there nef divisors D on a complex projective manifold X such that $h^0(X,D)$ is less than or equal to $\dim X$?
Edit: In fact I'm interested in nef line bundles D, not just divisors.
user2529
27
votes
5
answers
7k
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blowing up, -1 curves, effective and ample divisors
Lets say we're on a smooth surface, and we blow up at a point.
Is there a simple explicit computation that shows to me the fact that the exceptional divisor E has self intersection -1 ? I don't ...
- 271
3
votes
3
answers
732
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Nefness of $h-e$ in the blowup of $\mathbb{P}^n$
Let $S$ be the blow up of $\mathbb{P}^n$ in a point $P$. Let $h$ be the class of the pullback of an hyperplane of $\mathbb{P}^n$ and $e$ the class of the exceptional divisor. Why is the divisor $l=h-e$...
- 31
3
votes
4
answers
1k
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Examples of divisors on an analytical manifold
I am trying to understand divisors reading through Griffith and Harris but it is difficult to come up with any particular interesting example. I have browsed through Hartshone's book but everything is ...
- 2,215
10
votes
2
answers
2k
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Anticanonical divisor of the blow up of P^2 in 9 points
Let $S$ the blow up of $P^2$ in nine points. Why is the anticanonical divisor $-K_S$ not semiample?
- 427
4
votes
1
answer
1k
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A working generalization of Weil divisors
Hartshorne defines Weil divisors under the hypotheses "Noetherian integral separated scheme regular in codimension 1", which, for example, ensures that the divisor of a rational function is a finite ...
- 11.3k
6
votes
1
answer
320
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Is there a canonical notion of principal divisor on a discrete dynamical system?
I hope this question is well-posed.
Let (X, f) be a discrete dynamical system such that every x in X has finite period, i.e. there is some n such that f^n(x) = x. Let Div(X) be the free abelian ...
- 118k
20
votes
2
answers
10k
views
does a line bundle always have a degree
For curves there is a very simple notion of degree of a line bundle or equivalently of a Weil or Cartier divisor. Even in any projective space $\mathbb P(V)$ divisors are cut out by hypersurfaces ...
- 3,968
7
votes
3
answers
585
views
Weil divisors on non Noetherian schemes
Let X be an integral scheme that is separated (say over an affine scheme). Define a Weil divisor as a finite integral combination of height 1 points of X, where the height of a point of X is the ...
- 3,968