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Results tagged with ag.algebraic-geometry
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user 75616
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
3
votes
Vanishing of higher direct image of a morphism with generic fiber $\Bbb{P}^1$
Let me supplement Laurent Moret-Bailly's answer with a positive result.
First though, even if $f$ is proper and $R^if_* O_X$ vanishes generically, you cannot prove that $R^i f_* O_X=0$ just by invoki …
2
votes
Accepted
Elliptic K3 surface with a section of infinite order
(I am just posting my comment as an answer at the OP's request.)
To make Sergey's answer even more concrete, try an example such as the Fermat quartic in $\mathbf P^3$. Here an elliptic fibration is …
2
votes
Accepted
Picard group of families of smooth projective varieties
The answer to the first question is yes. As explained at
Geometric generic fibre
the geometric generic fibre $f^{-1}(o) \times_{k(Y)} \overline{k(Y)}$ is isomorphic as a scheme to a very general fibre …
20
votes
1
answer
2k
views
Geometric generic fibre
This is a pretty elementary question about schemes, but it came up in the course of research, so let's try it here rather than MSE.
Question 1: Are the fibres of a family of complex varieties isom …
10
votes
Accepted
Asymptotic $\int_M \mathrm{exp}[\mathbf{e}\left(n -\frac{t}{2\pi i}\right)] \left( 1 + \frac...
OK, let's see if I can put my money where my commenting mouth is. Let me say at the outset that I have no idea where such an integral comes from, but I claim that it doesn't matter to answer the quest …
7
votes
projectivity with assumption of big and semi-amplness
Edit: I forgot about the Kaehler condition, so my answer is not relevant to the OP's question. As the comment of Henri shows, the correct answer is "yes". I will leave my original answer here (in modi …
3
votes
An example of "all non-torsion rational points on an elliptic curve are integral points''?
This is not possible for a nontrivial subgroup of the free part of $E(\mathbf Q)$, because of Siegel's theorem.
6
votes
Algebraic Geometry for Topologists
(I guess my opinion is no more worthy of being an answer than the opinions in the comments, but it's verbose, so let me put it in the answer box anyway.)
As a beginning PhD student I knew a reasonabl …
4
votes
Accepted
K3 surfaces and density of rational curves
The OP requested that I add my comment as an answer:
Your argument in the last paragraph seems to assume that any dense subset of $S$ contains a nonempty open set. That is not true.
Specifically in …
1
vote
Accepted
Sections of a linear system splitting as a product of degree one polynomials
A union of $d$ hyperplanes has multiplicty at most $d$ at every point, and a hyperplane in $\mathbf P^n$ can be made to pass through at most $n$ general points. So the conditions
$m_i \leq d$ for ea …
11
votes
Accepted
Is the automorphism group of a Calabi-Yau variety an arithmetic group
The answer for $K3$ surfaces is no. A counterexample, where the group is not even commensurable with an arithmetic group, was given by Totaro in Example 6.3 of this paper.