# Questions tagged [ag.algebraic-geometry]

Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

20,024
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Let $k$ be a global field and $C$ a smooth projective curve over $k$ which is not isotrivial. Then there is a well-known trichotomy:
If $g(C) = 0$ and $C(k) \neq \emptyset$, then $C \cong \mathbb{P}^...

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0
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Let $M$ be the moduli space of pricipal $G-$bundles, where $G$ is a complex semisimple Lie group on a smooth projective curve over the field of complex number of genus $\ge 2$.
Question: Are $h^1(M, \...

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It is known that if $X$ is a smooth Fano Manifold of Picard number one, then $H^0(X, \Omega_X^1) = 0$ and $H^1(X, \Omega_X^1) \simeq \mathbb{C}$.
Question: Are the same results hold true if $X$ is a ...

9
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Let $X$ be a smooth proper algebraic curve over $\mathbb{C}$. Say a complex local system $\mathbb{V}$ on $X$ is motivic if there exists a dense Zariski-open subset $U\subset X$, and a smooth proper ...

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Let $\mathcal{E}$ be a coherent sheaf on an irreducible scheme $S$ ($S$ can be pretty nice, say noetherian of finite type), and let $\mathbf{P}(\mathcal{E}):=\mathrm{Proj}(\mathrm{Sym}(\mathcal{E}))$ ...

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this is not a homework question and I consider this problem quite difficult and confusing. I tried hard to solve it for 2 days, sure I found solutions, but they are not the same that the one provided ...

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I am trying to understand if there is a good notion of a $K$-theory attached to the etale topology on a nice scheme $X$ (say smooth projective goem connected curve over a finite field is enough for me)...

3
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By the result of A. Klyachko, there is categorical equivalence of toric vector bundles i.e. t-equivariant vector bundles on toric varieties and filtered vector spaces. For instance, on a toral surface,...

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Let $D$ be a Weil divisor on a normal toric variety $X$ with fan $\Sigma$ that is invariant under the action by the torus $T$. Then Proposition 4.3.2 of the textbook Toric Varieties by Cox, Little and ...

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There is an interesting trick used in Chen-LeBrun-Weber's paper on the extremal Kähler metrics of $\mathbb{CP}^2\#2\overline{\mathbb{CP}^2}$, and I would like to know whether it can be (has been?) ...

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In Lê-Ramanujam's paper The invariance of Milnor’s number implies the invariance of the topological
type, they prove what the title says for families with isolated singularities and constant Milnor ...

7
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2
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For a complex manifold $M$, I'm trying to understand (from a differential geometry point of view) what its category of coherent sheaves is. If I understand correctly, then the sheaf of holomorphic ...

11
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6
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Say we begin with an explicit elliptic curve over $\mathbb{C}$, say: $y^2=x(x-1)(x-2)$. According to abstract reasoning this elliptic curve has an (several, in fact) unramified cover with group $C_n$. ...

9
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Every non-singular complete surface is projective. On the other hand, there are non-projective complete surfaces (see e.g. Excercise II.7.13 of Hartshorne) - and there are such examples where the ...

2
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This is a question for those familiar with the section conjecture, so I'll do away with the definition of a ramification map in this case. Here is the definition of a ramification map from an etale ...

3
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$\DeclareMathOperator\Var{Var}\DeclareMathOperator\Motives{Motives}$Let us assume for the moment that we have a "nice" category of motives, that is for fields $k$ we have a contravariant ...

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Given smooth algebraic stacks $\mathcal{X}$, $\mathcal{Y}$ what are the linear deformations $\operatorname{Def}^1(f: \mathcal{X} \to \mathcal{Y})$ of a morphism $f:\mathcal{X} \to \mathcal{Y}$?
In ...

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1
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The purity of Brauer group states that for a smooth (quasi-)projective variety $X$ over a field $k$, removing a closed subscheme $Z \subset X$ of codimension at least $2$ ensures that the restriction ...

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2
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Let $N$ be a positive integer, and let $f$ be a newform for $S_2(\Gamma_0(N))$. Then by Shimura's construction, the variety $J_0(N)$ has a quotient $A_f$ which is an abelian variety attaced to $f$.
$$\...

6
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2
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Is there an English translation available for Deligne's Cohomologie Etale (Arcata) that is now part of the SGA 4 1/2 ?? Atleast for the first two sections - Grothendieck Topologies and Etale Topology.

64
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I am reading a nice booklet (in Italian) containing the exchange of letters that André and Simone Weil had in 1940, when André was in Rouen prison for having refused to accomplish his military duties.
...

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Let $X$ be a smooth projective variety of dimension $n$ over the field of complex numbers and $E$ be globally generated vector bundle of rank $n$. Choose a general $n$-linearly independent sections ...

3
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1
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In the article titled Tilting Exercises (See http://arXiv.org/abs/math/0301098v3) the authors define a notion of tilting perverse sheaves on an algebraic vareity $X$ with respect to stratification $\...

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Let $p \in \mathbb{Z}$ be a prime and let $\mathbb{F}_p$ be the field with $p$ elements.
Let $\chi: \mathbb{F}_p \to \mathbb{C}$ be an additive character, and let $f \in \mathbb{F}_p [x_1, \dots, x_n]$...

2
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0
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Let $k$ be a number field and $X$ a smooth geometrically connected variety over $k$. We denote by $H(k,X)$ the set of sections $G_k \rightarrow \pi_1(X)$, where $G_k$ is the absolute Galois group of $...

7
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1
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I am trying to understand the intuition for Luna's Étale Slice Theorem in the affine setting over $\mathbb{C}$.
Here is the setup. Let $X$ be an affine algebraic variety and $G$ a reductive group ...

2
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1
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502
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In what follows by projective variety I will mean the zero locus of homegeneous polynomials in some projective space. Anyway, feel free to deal with a sufficiently good scheme if you want.
Let $X$ be ...

3
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2
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Nine different types of singularities are possible on a cubic surface, according to Wikipedia. How exactly is the "type" of singularity defined? I know that the number corresponding to the ...

1
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1
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Let $X=X_{d_1}\cap\cdots \cap X_{d_m}\subset \mathbb{P}^n$ be a smooth complete intersection of smooth hypersurfaces $X_{d_i}$ of degree $d_i$ with $1<d_1\leq ...\leq d_m$ and $m>1$. For ...

3
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1
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In $G=\operatorname{PGL}(4,5)$ there are two elementary abelian $2$-subgroups of order $16$ denoted by $E_{1}$ and $E_{2}$ with $N_{G}(E_{1})=E_{1}.\operatorname{Sp}(4,2)$ and $N_{G}(E_{2})=E_{2}.(2^{...

1
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Let $X$ be a smooth projective geometrically connected curve over a field $k$ and $x\in X$ a geometric point.
Let $G$ be connected reductive group over a $k$ and let $\underline{G} = G\times X$ étale-...

16
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3
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There are many Grothendieck topologies used in algebraic geometry, with complex interrelations. Generally in one of these topologies, a cover of schemes is a family of maps which is jointly surjective ...

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Let $M= T^*(G/P)$ where $G$ is an algebraic group and $P$, a parabolic subgroup. Let $\mathbb{g}$ be its Lie algebra. Then there is moment map $\mu: M \to \mathbb{g}^*$, where $\mathbb{g}^*$ is the ...

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We assume $k=\mathbb{C}$. Consider $X=\mathbb{P}^1\times\mathbb{P}^1$ and a smooth divisor $D=V(s)$ for some $s\in H^0(X,\mathcal{O}(d,d))$ for $d>1$.
Let $\pi:S\rightarrow X$ be the $d$-th ...

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Let $(M, \omega)$ be a symplectic manifold. A Lagrangian fibration is a surjective map $\pi :(M, \omega) \to B$ whose regular fibres are Lagrangian submanifolds of $(M, \omega)$, which implies that $\...

0
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1
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189
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Every non-singular complex projective cubic surface has $27$ lines. Many such surfaces contain points where three lines intersect (called Eckardt points). There are even surfaces with many Eckardt ...

6
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1
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Over a field $ k $ of characteristic zero, actions of $ \widehat{\mathbb{G}_{a}} $ on a variety $ \operatorname{Spec}(A) $ are obtained from $ A $-derivations $ \delta $ of $ A $ over $ k $ via the co-...

3
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Let $X$ be a Fano manifold (different from $\mathbb{P}^n$) of dimension $n$ and Picard rank one. If its cotangent bundle $T^*X$ admits $n$ independent Poisson commuting regular functions then $T^*X$ ...

4
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If you could recommend a few papers for someone entering derived algebraic geometry outside of the classical category theory, algebraic geometry, and algebraic topology sequences? What would they be?

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The $\ell^p$ unit sphere $\{x\in\Bbb R^n\mid |x_1|^p+\cdots+|x_n|^p=1\}$ with $p\in\Bbb N$ is a semi-algebraic set, and its polar dual is
$$(*)\quad \{x\in\Bbb R^n\mid |x_1|^q+\cdots +|x_n|^q=1\},$$
...

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Let $X=\mathbb{P}^n$ over $\mathbb{C}$ for $n>1$, we know that there exists a line on $\mathbb{P}^n$ say $L$. Could we have a coherent sheaf $\mathcal{F}$ other than $\mathcal{O}_L$ (and that for ...

0
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89
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Let $X$ be a smooth projective curve over the complex numbers.
Question: When the moduli space of principal $G-$bundles over $X$ is smooth Fano of Picard rank one ?

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I have a feeling this question will be left hanging on StackExchange, so I've decided to just post it here, let me know if it's not suitable.
First of all, I'm trying to show that given an immersion ...

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Let $f : Y \to X$ be a finite morphism of degree $d$ of normal projective varieties over $k$ of dimension $n$. In Lehn and Huybrechts' book "The Geometry of Moduli Space of Sheaves", there ...

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1
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Consider a smooth hypersurface $X\subset\mathbb{P}^{n+1}$ of degree $d$ over a nice field (such as $\mathbb{C}$), we know that the cone $C(\operatorname{id}:\mathcal{O}_X\rightarrow\mathcal{O}_X)=0$. ...

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Today, somebody posted on the nLab a link to Kirti Joshi's preprint on the arXiv from last month: https://arxiv.org/abs/2210.11635
In that preprint, Kirti Joshi claims that
he agrees with Scholze and ...

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4
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Assume $X$ is smooth "simply connected" complex projective variety and $Y\subset X$ a smooth hyperplane section. ( $Y= X\cap H$, $H\subset \mathbb{P}^n$).
Let's $NE(X)$ be the cone of effective 1-...

2
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A Weil cohomology theory is a functor that assigns to a smooth projective variety $X$ of dimension $d$ over a field $k$ a graded ring of cohomology groups with values in a field $K$ of characteristic $...

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If $f: X \to S$ is a proper map from a complex manifold to a disc, $Y=f^{-1}(0)$ is a divisor with strictly normal crossings and the action of monodromy on $X_t=f^{-1}(t)$ for some (hence any) $t \neq ...

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Belyi's theorem states that every curve defined over $\mathbb{\bar Q}$ is almost modular (obtained from $\mathbb{H}^2/\Gamma,\ \Gamma$ a finite index subgroup of $PSL(2,\mathbb{Z})$), after ...