# Questions tagged [ag.algebraic-geometry]

for questions on algebraic geometry, including algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

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### Forcing quasi-compactness

I wonder what kind of conditions on a morphism of schemes imply, in a non-trivial fashion, quasi-compactness of the morphism. Some examples
Finiteness of surjective etale morphisms
Is a universally ...

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### Serre functors for non-proper categories

One usually defines a Serre functor to be a functor on a $k$-linear category $\mathcal{C}$ which has finite dimensional $Hom$s over $k$. In that case, the standard definition is that a Serre functor $...

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### Generalization of a result of Frey

In Proposition (2) in the paper [1], in below, it is proved that:
Assume that $K$ is a number field and that $C$ has infinitely many points of degree $\leq d$ over $K$. (Here $d\geq 2$ is an integer)...

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### Cohomological flatness in spectral algebraic geometry

If $f:X\rightarrow \mathrm{Spec}\,R$ is a proper, flat, cohomologically flat in degree 0 morphism of schemes with Noetherian target, then $O_X(X)$ is a flat $R$-module. Now if we switch to the world ...

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### Hyperelliptic Jacobians with (or without) CM

Let $C$ be a hyperelliptic curve $y^2 = f(x)
$ defined over $\mathbb{Q}$, where $f(x) \in \mathbb{Q} [x]$ is a polynomial of degree $n=5$ or $6$, and $J = Jac(C)$ its Jacobian.
I know Zarhin's result [...

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### Contractible affine surfaces of log Kodaira dimension 2

The first examples of contractible smooth affine algebraic surfaces (over the complex numbers) of log Kodaira dimension 2 were constructed in a famous paper of Ramanujam
https://www.jstor.org/stable/...

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### Holomorphic conformal structures on Hirzebruch surfaces

Let $X$ be a complex surface. A generalized holomorphic conformal structure on $X$ is given
by a tensor $\omega\in H^0(X, Sym^2\Omega^1_X\otimes L)$, where $L$ is a line bundle. Taking determinants ...

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### Equivariant Sheaf: Explanation on Stalks

I have a question about the explanation of the data defining a so called equivariant sheaf $F$ on a scheme X from wiki: https://en.wikipedia.org/wiki/Equivariant_sheaf. Let denote by $\sigma: G \...

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### Is this unipotent group, over characteristic 2, connected?

Let $E_{ij}(x)\in \mathrm{Mat}_{7\times7}(\overline{\mathbb{F}}_2)$ be the matrix with zeros everywhere, except for the value $x$ at $ij$. Set $$a(x)=1+E_{12}(x)+E_{34}(x)+E_{56}(x),\quad b(y)=1+E_{23}...

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### Equivalence De Rham and Dolbeault groupoids

I believe there is an error or incompleteness in Goldman's and Xia's proof of the equivalence of the De Rham and Dolbeault groupoids, contained in
Rank One Higgs Bundles and Representations of ...

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### Flat cohomology and finite direct sum

Let $X$ be a scheme (we can assume $X$ is smooth over a field $k$). Let $\mathcal F_1$ and $\mathcal F_2$ be two sheaves of abelian groups on $X$.
Is it always true that $H^i_{\text{flat}}(X, \...

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### A proper flat family with geometrically reduced fibers

Let $R$ be a Noetherian commutative unital ring. Let $f:X\rightarrow Y=\mathrm{Spec}\,R$ be a proper flat morphism of schemes with geometrically reduced fibers. We want to prove that the induced map $...

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### Infinitesimal deformation of strict transform

Let $X$ be an affine, complex surface with isolated singularities and $\pi:\widetilde{X} \to X$ be a resolution of singularities (not necessarily minimal) i.e., $\widetilde{X}$ is non-singular and $\...

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### EGA I (Springer), Proposition 0.4.5.4 [on hold]

I do not understand one argument in the proof of Proposition 0.4.5.4. in the new version by Springer of EGA I.
When proving that the functor $F$ is representable by $(X, \xi)$, where we obtained $X$ ...

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### Fundamental group of the Grothendieck ring scheme

Let $K_0(V_k)$ be the Grothendieck ring of $k$-varieties with $k$ a field. Let S$_k$ be its affine scheme. Is this a connected scheme for any field ? (I understand that this could be a very naive ...