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Results tagged with ag.algebraic-geometry
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user 115211
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
5
votes
0
answers
172
views
Extension of Kollár's Vanishing Theorem to non-projective varieties
Kollár's Vanishing Theorem claims the following:
Theorem: Let $f\colon X \to Y$ be a surjective morphism of connected smooth projective varieties over a field $k$ of char. $0$. Then the complex $ …
- 2,841
5
votes
Accepted
Is a localization of a reduced finitely generated algebra analytically unramified?
Let me expand my comment as an answer. There is a notion of excellent rings, for a precise definition look here https://stacks.math.columbia.edu/tag/07QS (and see Chapter 13 of Matsumura's book "Commu …
- 2,841
6
votes
A proper flat family with geometrically reduced fibers
The claim is correct, and, actually, even more is true. The map $R \to \mathrm{H}^0(X, \mathcal O_X)$ is (finite) etale, but the difficult part is really to prove flatness. Before going to the actual …
- 2,841
5
votes
When does glueing affine schemes produce affine/separated schemes?
There is a general criterion that explains when a gluing of two separated schemes is separated.
Proposition: Let $X_1, X_2$ be a separated $S$-schemes, $U_i$ open subschemes in $X_i$ (for $i=1, 2 …
- 2,841
3
votes
Accepted
Compatibility between the functors of $\mathcal{O}_X$-modules and $\mathcal{D}_X$-modules
[All functors in this answer are assumed to be derived]
These commutativities basically boil down to unraveling all definitions. The actual computations are quite annoying (but totally possible), so I …
- 2,841
10
votes
1
answer
464
views
Example of a $p$-divisible group that is not representable by a formal scheme
Let $R$ be a ring such that $p^nR=0$ for some integer $n$, and $G$ be a $p$-divisible group over $R$.
We think of a $p$-divisible groups as an fppf sheaf $G\colon \mathrm{Alg}^{op}_{R}\to \mathbf{Gps} …
- 2,841
5
votes
1
answer
450
views
Tate conjecture and finiteness of Brauer group
What is the exact relation between the Tate conjecture for divisors on $X$ and finiteness of the Brauer group of $X$? And what is the reference for these relations?
More precisely, let $X$ be a smooth …
- 2,841
11
votes
Accepted
Picard group of connected linear algebraic group
$\DeclareMathOperator\Pic{Pic}$The statement is false over most imperfect fields, even for smooth affine group schemes.
In particular, it is false over any separably closed imperfect field $k$. I will …
- 2,841
8
votes
Accepted
Finite subgroup scheme and Neron model of an abelian variety
In general the scheme $\mathcal A[l]_{\mathcal O}$ is not finite because of the following lemma.
Let $f:X\to Y$ be a separated quasi-finite flat morphism of noetherian schemes. Then it is finite …
- 2,841
3
votes
Accepted
Is Frobenius on $R^\circ/p$ surjective for general perfectoid rings $R$?
[Probably this question is no longer interesting to the author. But since I faced the same problem while trying to learn basics of perfectoid spaces I decided to write down an argument here]
We start …
- 2,841
11
votes
Algebraic vs analytic normality
Francesco Polizzi's answer is perfectly fine, but let me try to explain the technique which helps to relate a lot of "local" properties of locally finite type schemes over $\mathbf C$ to their counter …
- 2,841
4
votes
Accepted
Map from local systems to holomorphic line bundles on a curve
I think the following theorem answers your question.
Theorem: Let $X$ be a smooth, proper connected curve over $\mathbf C$ with a line bundle $\mathscr L$. Then $\mathscr L$ admits a flat connecti …
- 2,841
17
votes
Accepted
Example of a smooth projective family of varieties in characteristic $p$ where the Hodge num...
Update: The details of this construction are now available in my blog post with Sean Cotner on Thuses.
I was recently interested in exactly the same question. But I failed to find any reference where …
- 2,841
2
votes
Accepted
Hodge decomposition of the symmetric product of a curve
Look at Example $1.1$ in this paper for a nice formula.
You can also compute them by hands (and, hopefully, prove the desired formula) by identifying $\mathrm{H}^{p,q}(\operatorname{Sym}^n X)$ with …
- 2,841
6
votes
Accepted
Cartier Divisor generated by Global Sections
First of all, you definitely need to assume that your curve is proper to make sense of $\chi_k(\mathcal O_X)$. If it is not, then $H^i(X,\mathcal O_X)$ is not finite dimensional over $k$, so $\chi_X(\ …
- 2,841