Questions tagged [ag.algebraic-geometry]

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

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Is a complex algebraic set with a Zariski dense subset of algebraic points already defined over the algebraic numbers?

This is a cross-post! For the original post on SE (9 upvotes, no answer) see: https://math.stackexchange.com/questions/4475853/is-a-complex-algebraic-set-with-a-zariski-dense-subset-of-algebraic-...
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Intermediate extensions of pure perverse sheaves (BBD 5.4.3)

I am working my way through "Faisceaux pervers" by Beilinson, Bernstein and Deligne and can't wrap my head around Corollary 5.4.3. To me it seems that one of the hypotheses is extraneous, ...
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1 vote
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63 views

Real analytic map with connected fibers

Let $X,Y$ be compact real analytic varieties. Suppose $Y$ is connected and there is a surjective analytic map $f:X\to Y$ such that each fiber of $f$ is connected. How to prove that $X$ is connected as ...
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3 votes
1 answer
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Normal schemes and Serre's criterion

Serre's criterion says that for a scheme to be normal is equivalent to it being $R_1$ (i.e. regular in codimension $1$) and $S_2$ (i.e. regular functions on $X-Y$ extend to $Y$ if $Y$ has codimension ...
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0 votes
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77 views

Irreducibility of plane algebraic curves

Given a plane algebraic curve $$ y^n + a_1(x)y^{n-1} + \dots +a_{n-1}(x) + a_n(x)y = 0, $$ with a branch point $P_0=(0, y_0)$ of order $n$. Can we prove that this curve is irreducible? What if the ...
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4 votes
1 answer
204 views

Does the main theorem of elimination theory with $\mathbb{Z}$-coefficients imply that projective varieties are complete?

I have a question concerning the completeness of projective varieties. Let $k$ be an algebraically closed field. By the "main theorem of elimination theory" I mean the following result: Let $...
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Does an isogeny between tori induce an isomorphism of the Lie algebras of their lft Néron models?

Let $f:T_1 \to T_2$ be an isogeny of tori over a number field $K$. Does $f$ induce an isomorphism of the Lie algebras of the lft Néron models of $T_1$ and $T_2$ ? Are there some interesting properties ...
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8 votes
3 answers
689 views

Is every smooth projective variety contained in a chain of smooth projective varieties of increasing dimension?

Let $X ⊆ \mathbb{P}^n$ be a smooth projective variety (over $\mathbb{C}$). I think we can find a chain of irreducible varieties $X = X_0 ⊆ X_1 ⊆ X_2 ⊆ \cdots ⊆ X_k = \mathbb{P}^n$ whose dimension ...
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1 vote
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61 views

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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1 vote
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75 views

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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4 votes
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127 views

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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111 views

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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64 views

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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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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4 votes
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131 views

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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1 vote
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166 views

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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1 answer
128 views

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 $h_X=\operatorname{Hom}_S({-}, X)\to h_Y=\operatorname{Hom}_S({-}, Y)$ over the small 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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-2 votes
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113 views

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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101 views

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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  • 439
1 vote
1 answer
160 views

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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0 answers
144 views

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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3 votes
0 answers
149 views

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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130 views

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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3 votes
1 answer
138 views

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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2 votes
0 answers
76 views

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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2 votes
0 answers
71 views

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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1 answer
82 views

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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2 votes
0 answers
59 views

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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  • 419
1 vote
0 answers
63 views

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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  • 369
2 votes
2 answers
191 views

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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2 votes
0 answers
84 views

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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3 votes
1 answer
109 views

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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4 votes
1 answer
352 views

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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0 votes
1 answer
187 views

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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106 views

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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  • 111
2 votes
1 answer
67 views

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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  • 10k
1 vote
0 answers
129 views

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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1 vote
1 answer
149 views

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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  • 37
2 votes
1 answer
135 views

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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2 votes
0 answers
104 views

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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  • 83
3 votes
1 answer
181 views

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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0 votes
0 answers
115 views

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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  • 21
-1 votes
0 answers
109 views

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 ...
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  • 369
3 votes
0 answers
140 views

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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2 votes
1 answer
390 views

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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  • 37
1 vote
1 answer
236 views

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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  • 2,297
0 votes
0 answers
130 views

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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  • 9
1 vote
0 answers
238 views

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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  • 1,079
2 votes
1 answer
230 views

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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