# Questions tagged [ag.algebraic-geometry]

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

19,642
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### Simplicial approximation theorem for toric varieties

Given abstract simplicial complexes $K$ and $L$, one constructs topological spaces $|K|$ and $|L|$. Simplicial approximation theorem says for any continuous map $f: |K|\to |L|$ that there exists ...

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56
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### Locally symmetric spaces dependence on number field

A special case of locally symmetric spaces is the moduli space of abelian varieties of a given dimension $g$ (over a given base field $k$), lets call it $\mathcal{A}_g$ and is a $k$-scheme. For any ...

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97
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### Completion of ring as direct limit

If $X$ is a variety and $x \in X$, there are several ways to look locally around the point $x$:
Localisation: taking the direct limit over open immersions around $x$.
Henselisation: taking the direct ...

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90
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### The genus of hyperplane sections

Let $S$ be a connected smooth projective surface over $\mathbb{C}$.
Let $\Sigma$ be the complete linear system of a very ample divisor $D$ on $S$, and let $d=\dim(\Sigma)$ be the dimension of $\...

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### Computer algebra tools for finding real dimension of an algebraic variety

I have a system of polynomial equations with the unknowns being real numbers. The set of solutions is infinite. What software can I use to compute the real dimension of the solution set?
The CAD-based ...

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103
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### Simple modules of quantum planes

Let $k$ be an algebraically closed field.
Let $R := k\langle x,y \rangle/(yx-qxy) (q \in k^*)$.
We often call $R$ a quantum plane.
If $q$ is a primitive $n$-th root, then for any $(\zeta, \xi) \in k^* ...

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109
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### Fibers of the coarse moduli space map

Let $\mathcal{X}$ be a Deligne-Mumford stack over a field $k$ which admits a coarse scheme $c : \mathcal{X}\rightarrow X$. This will be the case if $\mathcal{X}$ is separated and locally of finite ...

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### Why are compact arithmetic surfaces defined through quaternion algebras (usually) only over $\mathbb{Q}$?

As in Chapter 6.2 of "Introduction to arithmetic groups" (by D.W. Morris), compact arithmetic surfaces could be defined through quaternion algebras $\mathbb{H}^{a,b}_F=\big(\frac{a,b}{F}\big)...

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### Surjective sheaf homomorphisms induced by morphisms of schemes

Let $S$ be a scheme and $X\to Y$ be a morphism over $S$. Then we have an induced homomorphism of sheaves $h_X=\mathrm{Hom}_S(-, X)\to h_Y=\mathrm{Hom}_S(-, Y)$ over the small étale site $S_{étale}$.
...

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### How to simplify this homotopy totalization coming from an arc-cover into a pullback?

My question concerns the proof of Proposition 4.2 in Bhatt-Mathew’s paper on the arc-topology, but my confusion is completely general and anyone familiar with limits in $\infty$-categories would know ...

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### Let $V$ be a variety. A point $P \in V$ is nonsingular iff $\dim_k(M_P/M_{P}^{2})=\dim(V)$ [closed]

First of all, we consider $k$ to be an algebraically closed field, and by $M_P$ I denote the maximal ideal of the coordinate ring $k[V]$ at $P$.
As for the statement, I have managed to understand how ...

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### Abelian subvarieties corresponding to vector subspaces

Let $S$ be a connected smooth projective surface.
Let $C$ a smooth curve on $S$
In page 9 of the paper "https://arxiv.org/abs/1704.04187v1" a read the following:
Let
\begin{equation*}
r: ...

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146
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### Purity for proper varieties

Let $X$ be a proper, geometrically connected, geometrically integral variety over $\mathbf{F}_q$. There exists a finite field extension $k/\mathbf{F}_q$ of degree $d$ and an alteration $X'\to X_k$ ...

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141
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### Where can I find that Weil suggested a cohomology theory for characteristic $p>0$?

I have seen that in Grothendieck's paper "THE COHOMOLOGY THEORY OF ALGEBRAIC VARIETIES", he says "The need of a theory of cohomology for 'abstract' algebraic varieties was first ...

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146
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### Compact generation of Voevodsky motives

Let $DM(k,{\mathbb{Q}})$ be the derived category of Voevodsky motives over a field $k$ with $\mathbb{Q}$-coefficients.
As a triangulated category is $DM(k,{\mathbb{Q}})$ known or expected to be ...

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### Does the space of hyperplanes in the Grassmannian have a name?

A way of defining the Grassmannian $Gr(k,n)$ is to consider the space of $k\times n$ matrices mod $GL(k)$ transformations on the rows. I'm interested in the space of $k\times 2n$ matrices mod $GL(k)$ ...

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134
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### A group in a neighbourhood of a Zariski dense subgroup

By Borel's theorem, lattices in simple Lie groups are Zariski dense. I expect that a small (in metric sense) deformation of a lattice in a Lie group is also Zariski dense.
Suppose we have a Zariski ...

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74
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### How to compute Selmer set?

Let $X$ be an affine variety and $G$ an affine algebraic group (for example $\operatorname{PGL}_n$). How do I compute the Selmer set
$$ \operatorname{Sel}_\zeta(\mathbb{Q},G) = \{\tau \in H^1(\mathbb{...

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67
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### The dual of the Lefschetz operator under a perturbation

Let $(X, \omega)$ be a compact Kähler, or more generally, Hermitian manifold. Let $L_{\omega} : \Omega^k(X) \to \Omega^{k+2}(X)$ denote the Lefschetz operator given by $$L_{\omega}(\alpha) : = \omega \...

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81
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### Free closed group action on varieties

Suppose we are given a reductive group $G$, its closed subgroup $H$ (not necessarily reductive), an affine $G$-variety $X$ and its closed subvariety $Y$ such that
(1) The $G$ action on $X$ is free and ...

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56
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### Smoothness of homomorphisms between graded algebras

Let $A$ be a finitely generated $\mathbb{C}$-algebra. Let $S^{\bullet}=\bigoplus_{i\geq 0} S^i$ and $T^{\bullet}=\bigoplus_{i\geq 0} T^i$ be two graded $A$-algebras such that $S^0=T^0=A$ and $S^1, T^1$...

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### Non-trivial extension and tangent bundle isotropic Grassmannian

Let $V$ be a $2n$-dimensional vector space endowed with a nondegenerate skew-symmetric form $q:V \to V^\vee$. We define the isotropic Grassmannian to be
$$
X:=G_q(k,V)=\left\{[W] \in \mathbb P \left( \...

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2
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184
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### Rational solutions to $P(x,y)=0$ for $P$ reducible over ${\mathbb C}$

There are facts in Mathematics that are so "obvious" and "well-known" that no-one includes a proper proof. An example is:
Theorem: If polynomial $P(x,y)$ with rational coefficients ...

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82
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### Constructions of motivic complex that is only supported on positive degrees

It is expected by Beilinson-Soule vanishing conjecture that negative motivic cohomology groups are zero so the motivic complexes are supported on nonnegative degrees. My question is about ...

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109
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### Bounded torsion of quotients of affine formal models

$\DeclareMathOperator\Sp{Sp}$Let $X=\Sp(A)$ be a connected smooth affinoid rigid space over a discretely valued non-archimedean field $K$. Let $\mathcal{R}$ be a valuation ring of $K$, and fix a ...

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349
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### Pairing of cotangent and tangent bundles

I am reading the survey paper: "The de-Rham Witt complex and Crystalline cohomology" by Luc Illusie.
In math line (2.1.12), Illusie considers the pairing $\langle-,-\rangle:\Omega_{X/S}^1\...

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184
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### Approaching the Riemann-Roch Theorem for algebraic curves

I am using "Algebraic Curves: An Introduction to Algebraic Geometry" by William Fulton as a guidline for approaching the Riemann-Roch Theorem for algebraic curves. I have two questions:
...

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103
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### Cycle class/cohomology class of subvarieties in flat families

Let $X$ be a projective variety over $\mathbb C$ and $T$ an irreducible projective $\mathbb C$-scheme.
Let $a,b$ be closed points of $T$.
Suppose we have a flat family $Z\to X\times T\to T$ such that ...

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1
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67
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### Is the polar dual of a semi-algebraic convex body also semi-algebraic?

Call a convex body $C\subset\Bbb R^n$ semi-algebraic if it can be written as
$$(*)\quad C=\bigcap_{i\in I}\, \{x\in \Bbb R^n\mid p_i(x)\le 0\}$$
with polynomials $p_i\in\Bbb R[X_1,...,X_n]$ and a ...

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126
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### Complex quadric as a symmetric space

It is known that a smooth complex quadric is a symmetric space. For example, it is
$$\operatorname{Spin}(n+2)/G$$
where $G$ is the maximal parabolic subgroup.
I want a reference for more details and ...

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148
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### Difference between $K(1)$-local K theory and l-adic completion of etale $K$ theory

Let $X$ be an scheme. Fix a prime $l$ which is invertible in $X$. Consider the $K(1)$-localization at prime $l$ of algebraic K theory $L_1K(X)$ and $l$-adic completion of etale K theory $K^{et}(X)$.
...

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134
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### Is the restriction of the Cartan 3-form on conjugacy classes exact?

Let $G$ be a complex semisimple group and $\mathcal{O} \subset G$ a conjugacy class, i.e. $\mathcal{O} = \{gag^{-1} : g \in G\}$ for some $a \in G$. Let $\Omega$ be the Cartan 3-form on $G$ defined by
...

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103
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### Tannakian recovery of a group from other tensor abelian categories

Classical Tannakian reconstruction recovers an affine group scheme $G$ over $k$ from the category of its linear representations over a field $k$ (as the automorphism group of the forgetful functor to ...

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### When does a holomorphic symplectic manifold compactify to a Poisson manifold?

Let $X$ be a complex manifold endowed with a holomorphic closed 2-form $\omega$ whose associated map $\omega : TX \to T^*X$ is invertible. Can we always embed $X$ as an open subset of a compact ...

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114
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### Reference Request: sheves of abelian groups over a smooth projective variety

Can someone point some good reference (books or lecture notes) for these topics:
Let X a smooth projective variety over an algebraically closed field
Sheaves of abelian groups over X
Quasi-coherent ...

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108
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### Projectivization of normal bundle

We work over the field of complex numbers.
Let $Y \subset \mathbb P^N$ be a smooth projective variety.
Consider the set $I_Y^0 \subset \mathbb P^N \times (\mathbb P^N)^\vee$ of pairs $(y,H)$ such that ...

3
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139
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### Boundedness indices in Voevodsky's smash nilpotence conjecture in family

Let $X$ be a smooth projective variety over an algebraically closed field $k$. Voevodsky introduced the following notion : an algebraic cycle $Z$ in $X$ is smash nilpotent if there exist $N>0$ such ...

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389
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### What is the dual of the stable infinity category of perfect complex on smooth proper variety?

Fix a commutative ring $R$.
Lurie proved that smooth proper $R$-linear stable infinity categories are dualizable in $\text{Cat}^\text{perf}_{R,\infty}$.
For a smooth proper variety $X$ over $R$, what ...

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### What's a right parameter space of abelian varieties over a non algebraically closed fields?

Let $k$ be a field of characteristic not 2 or 3. Then the set of elliptic curves over $k$ can be parametrized by the affine variety $S=D(4a^3+27b^2)\subset\mathbb{A}^2_k$ via the family $E\to S$ where ...

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### Projection map of the Hirzebruch surface

Consider the Hirzebruch surface $\mathbb{F}_0:=\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1})\xrightarrow{\pi}\mathbb{P}^1$. We know that
$$\pi_*\mathcal{O}_{\mathbb{F}_0}(n)\...

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235
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### Decomposition of vector bundles on the inertia stack of a DM stack

Let $X$ be a tame DM stack over $\mathbb{C}.$ Let $IX$ denote the inertia stack of $X.$ Let $K(IX)$ denote the Grothendieck group of vector bundles on $IX.$ By the discussion on page 20 of Toen's ...

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228
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### Connected components of inertia stacks

Let $k$ be a field. Let $X$ be a connected tame DM stack over $k.$ Let $IX$ be the inertia stack of $X.$ Then $IX$ is a disjoint union of connected components.
Is this always a finite union? If not, ...

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1
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231
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### Existence of rigid objects in the derived category of a smooth projective variety

Let $X$ be a smooth projective variety (say over $\mathbb{C}$). An object $F \in D^b(X)$ is said to be rigid if $\mathrm{Ext}^1(F,F)=0$. I was wondering if we can always find a rigid object on a ...

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### Is the theta characteristic attached to an etale double cover of a plane quintic arising from a cubic threefold even in characteristic two?

We work over an algebraically closed field $k$ of characteristic $2$. Let $X$ be a cubic threefold realized as a conic bundle via $f: X \to \mathbb{P}^2$ (after blowing up some line). Let $C \to \...

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167
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### A quantity computed from weights of representations -- Have you seen it?

The following quantity has come up in some work my collaborators and I are doing on equivariant D-modules, and in that particular context it seems to be very significant (i.e. it's the only "...

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158
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### Degenerations of rationally connected varieties

Let $X$ be a smooth projective rationally connected variety over $\mathbb{C}((t))$ and $R=
\mathbb{C}[[t]].$
Does there exist a proper regular scheme $\mathcal{X} \to \mathrm{Spec}(R)$ whose special ...

4
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288
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### Coincide between Chern-connection and Levi-Civita connection

I am a beginner in complex geometry and I am going to show Levi-Civita connection $\nabla$ and the Chern connection $D$ are the same on the holomorphic tangent bundle $T^{1,0}M$ on Kahler manifold. By ...

2
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86
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### Control on the locus of bad reduction for divisors

Let $X$ be a smooth variety over a number field $K$ and let $\mathcal X$ be a normal, projective model of $X$ over the ring of integers $O_K$.
Now assume that $D\subset X$ is an irreducible divisor ...

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105
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### Counting Points on a Plane Curve

I want to find the number of points of $\displaystyle \mathcal{C}_{h} \cap A^{2}( F_{q})$, (char(F) is not equal to 2,3), where $\displaystyle h$ is a rational function defined as $\displaystyle h( x) ...

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899
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### Algebraic geometry over the complex numbers, and beyond

My question basically is very simple: when did mathematicians start to do algebraic geometry "outside the complex numbers" ?
In the old days, algebraic geometry was solely done over the ...