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Results tagged with algebraic-surfaces
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user 2083
An algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of dimension four as a smooth manifold.
7
votes
Accepted
Algebraic equivalence VS Numerical Equivalence - An Example.
Numerical and algebraic equivalence coincide up to torsion for codimension $1$ cycles in non-singular projective varieties over $\mathbb C$. Also, the group $Alg_{\tau}^1(X)/Alg^1(X)$ can be identifi …
- 30.6k
4
votes
1
answer
975
views
Torsion line bundles with non-vanishing cohomology on smooth ACM surfaces
I am looking for an example of a smooth surface $X$ with a fixed very ample $\mathcal O_X(1)$ such that $H^1(\mathcal O(k))=0$ for all $k$
(such thing is called an ACM surface, I think) and a globally …
- 30.6k
5
votes
Q-factorial and rational singularities on surfaces
About the converse: one does need all the assumptions Karl mentioned in his answer. There are $2$-dim. complete local rings which are UFD but does not have rational singularity. One such example (due …
- 30.6k
13
votes
Vector bundles on $\mathbb{P}^1\times\mathbb{P}^1$
It may be a bit unfair to compare $X=\mathbb P^1 \times \mathbb P^1$ to $\mathbb P^1$. EDIT: I removed a too optimistic statement about restricting vector bundles on $\mathbb P^3$ to a smooth hypersur …
- 30.6k