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Results tagged with algebraic-surfaces
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user 46104
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.
3
votes
Accepted
On surfaces with $p_g=0$, $q=1$, and $K^2=-3$
Xiao Gang is taking a configuration of six lines in the plane with 4 triple points $x,z_1,z_2,z_3$ and three double points $y_1,y_2,y_3$. He considers a general quartic through the seven points which …
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3
votes
Torsion of the Picard group for surfaces isogenous to a product
I think the answer to the question as it is stated is negative: if I do understand correctly the question I can prove that, if $G$ is abelian and both curves $C_i/G$ are rational, this is never true …
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1
vote
Accepted
Relation between curves in a complete linear system contained in another
In the following I suppose that by "curve" in a linear system you mean "effective divisor" vithout any claim about being irreducible and/or reduced.
1) Curves in |L'| are exactly the pull-back of cur …
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4
votes
Isotrivial fibrations over $\mathbb P^1$
If the genus of the fibre is not 0, by Theorem 2.1 in Serrano's paper "Fibrations on algebraic surfaces" any isotrivial fibration is birational to $(A \times B)/G \rightarrow B/G$ where $G$ is a finit …
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2
votes
Disjoint curves in an algebraic surface
The answer is clearly affirmative for $b=1$, which unfortunately corresponds only to 101 surfaces, including ${\mathbb P}^2$. I doubt that there is a single other case in which it works.
Indeed I do …
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2
votes
Accepted
Is each rationally chain connected surface rational?
By the comments of abx you only need an answer to question $3$. You get easily an answer if you assume the results in the classical book of Beauville on algebraic surfaces.
Namely, if $S$ is uniruled …
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5
votes
0
answers
335
views
Deformations of a blow up
My question is related to this question, but I'm looking for something a bit more explicit.
Let $S$ be a smooth surface over $\mathbb C$, fix a point $s\in S$ and take the blow up $\beta \colon S' \r …
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5
votes
Accepted
Does $h^1(D)=0$ imply numerical connectedness on K3 surfaces?
Interesting question. I think the answer is yes, let me try to prove it.
As you noticed, the ideal sheaf sequence shows that $h^1(D)=0$ is equivalent to the fact that $H^0({\mathcal O}_D)$ is 1-dimen …
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2
votes
Classification of quartic surfaces
A fine classification of the quartic surfaces that are not normal is in
Tohsuke URABE, "Classification of Non-normal Quartic Surfaces", TOKYO J. MATH. VOL. 9, No. 2, 1986, 265-295
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