# Questions tagged [ag.algebraic-geometry]

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

16,812
questions

**2**

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91 views

### Outer Galois representations and Tate modules of Jacobian varieties

Let $X$ be a proper smooth curve over a field $k$. Then we have an exact sequence of profinite groups
\begin{equation*}
1 \to \pi_1(X_{\overline k}) \to \pi_1(X) \to G_k \to 1,
\end{equation*}
...

**17**

votes

**1**answer

517 views

### Is a direct sum of flabby sheaves flabby?

Consider a family of flabby (= flasque) sheaves $(\mathcal F_i)_{i\in I}$ of abelian groups on the topological space $X$.
My question : is their direct sum sheaf $\mathcal F=\oplus _{i\in I} \mathcal ...

**2**

votes

**0**answers

113 views

### Resolution of pairs in characteristic p

Let $R$ be a complete DVR of characteristic $p$, say $R=\mathbb{F}_p[[t]]$, and $X$ be a reduced scheme of finite type over $R$. Let also $X_s$ denote the special fiber of $X$. If I understand ...

**2**

votes

**0**answers

91 views

### Is the fiber product of two connected complex varieties over a connected base connected?

If $X,Y,Z$ are connected varieties over $\mathbb{C}$, with morphisms $X\rightarrow Z, Y\rightarrow Z$, is it true that the fiber product $X\times _Z Y$ is connected?
The statement is true for $S=Spec(\...

**8**

votes

**0**answers

144 views

### Which field extensions do not affect Chow groups?

Let $X$ be a (say, smooth projective) variety over a field $k$. For which $K$ it is known that the ("ordinary", that is, not higher) Chow groups of $X$ map onto that of $X_K$ bijectively?
...

**2**

votes

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117 views

### Homotopy Kan extensions, formally coherent functors and derived Schlessinger criterion

Let $k$ be a finite field. Denote by $discArt_k$ the category of Artinian rings with residue field $k$ and $Art_k$ the category of Artinian simplicial rings. Consider a functor $\mathcal{F}:disArt_k\...

**1**

vote

**1**answer

106 views

### Finding an irreducible region of a space given a group of transformations

Given some $d$ dimensional torus, (i.e. just a $d$-dimensional hypercube with periodic boundary conditions) I'll call $\Omega$, and a group of transformations $G$ of $\Omega$, I want to find the ...

**32**

votes

**2**answers

890 views

### Residues in several complex variables

I am trying to educate myself about the basics of the theory of residues in several complex variables. As is usually written in the introduction in the textbooks on the topic, the situation is much ...

**6**

votes

**1**answer

174 views

### Bounded non-symmetric domains covering a compact manifold

This question is somewhat related to this other question of mine.
I was wondering which are the known examples of bounded domains $\Omega$ in $\mathbb C^n$ admitting a compact free quotient.
By a ...

**2**

votes

**0**answers

87 views

### Whether a particular fiber product of varieties is integral

Let $X,Y,Z$ be irreducible projective varieties over $\mathbb{C}$ (you can assume all of them are normal), and let $f:X\rightarrow Z, g: Y\rightarrow Z$ be two birational projective morphisms, such ...

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votes

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61 views

### Hyperplane which does not contain any associated point of qc sheaf $\mathcal{F}$

I have a question about an argument on $m$-regularity
from 'Fundamental Algebraic Geometry' by Fantechi on page 114, Chapter
5.2: Castelnovo-Mumford regularity. The statement is:
Let $k$ be a field ...

**3**

votes

**0**answers

109 views

### Uniqueness of $\delta$-structure on a $p$-torsion ring

I was working through Bhargav's notes on $\delta$-rings and prismatic cohomology, specifically lecture 2, page 2, point 5 where he claims that the ring $\mathbb Z[x]/(px,x^p)$ has a unique $\delta$-...

**1**

vote

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58 views

### Automorphisms of the completion of a strict henselian local ring $R$ which come from automorphisms of $R$

Let $A\rightarrow R$ be a local homomorphism of Noetherian strict henselian local rings with completions $\hat{A},\hat{R}$.
Let $u\in R^\times, x\in R$ be such that there is a unique $\hat{A}$-linear ...

**4**

votes

**0**answers

121 views

### Finite maps to normal varieties have fibers with bounded number of points

Let $f\colon X\rightarrow Y$ be a dominant, finite, and proper map of normal varieties of degree $d$ over an algebraically closed field $k$. Let $y\in Y$ be any closed point.
Question. Is it true that ...

**3**

votes

**0**answers

63 views

### Decomposability and analytification of coherent sheaves

Let $X$ be an affine (algebraic) complex variety and $f:Y \to X$ be a finite morphism. Given any coherent sheaf $\mathcal{F}$ on $X$, we denote by $\mathcal{F}^{an}$ the analytification of $\mathcal{F}...

**2**

votes

**0**answers

49 views

### Reference request: boundedness for semistable principal bundles on a family of curves

We work over an algebraically closed field $k$.
Let $G$ be a reductive group and $X$ be a smooth projective curve over $k$. It is proven in [1, Theorem 1.2] that the moduli of semi-stable principal $G$...

**3**

votes

**0**answers

188 views

### Kummer theory if $\ell = p$

Background. Let $k$ be a field and let $\ell$ be an integer which is divisible in $k$. Then one has a short exact sequence of abelian étale sheaves
$$ 0 \to \mu_\ell \to \mathbb{G}_m \xrightarrow{(\,\...

**3**

votes

**0**answers

144 views

### A characterization for a commutative ring with a special intersection property for prime ideals

Let $R$ be a commutative ring with $1$ with the property that for any infinite family $\{P_i\}_{i\in I}$ of distinct prime ideals of $R$ we have $\cap_{i\not= j} P_i\subseteq P_j$ for all but fnitely ...

**1**

vote

**0**answers

63 views

### Projection from closure of locally closed subscheme is Etale

Let $S$ an arbitrary scheme and denote by $\Delta: S \to S \times S$ the diagonal immersion and $p_i: S \times S \to S$ the both projections to first resp second factor. (in following we will wlog ...

**3**

votes

**1**answer

113 views

### Let $R$ be a local ring where 2 is invertible. Must there exist a faithfully flat $R$-algebra where the squaring map is surjective?

Let $R$ be a local ring where 2 is invertible. Must there exist a faithfully flat $R$-algebra where the squaring map $x\mapsto x^2$ is surjective?
This is certainly true for fields. For DVR's, you can ...

**9**

votes

**1**answer

282 views

### Extending a holomorphic vector bundle: a reference request

Let $Y$ be a complex manifold, $X\subset Y$ a compact submanifold, and $E\to X$ a holomorphic vector bundle. Can $E$ be extended
to a bundle over an open neighborhood of $X$ in $Y$? (Four years ...

**3**

votes

**1**answer

241 views

### “Universal coefficent theorem” for pro-étale cohomology

In algebraic topology, for any space with finite homology type, the universal coefficient theorem states that for any abelian group $G$, we have
$$H^n(X,G)\cong \left( H^n(X,\mathbb{Z})\otimes G\right)...

**1**

vote

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133 views

### Affine scheme as algebraic space

We working in the following with Knutson's definition of an algebraic space
(ie via equivalence relation; there is also another equivalent def via
sheaves but let us work here with the following one):
...

**1**

vote

**0**answers

79 views

### Spaces intersecting a plane non-trivially in $G(3,6)$

I want to understand the Schubert variety $\Sigma\subseteq G(3,6)$ representing 3-dim subspaces intersecting a given 2-dim subspace non-trivially. Is it smooth? How to describe $det(T_{\Sigma})$?

**3**

votes

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79 views

### Period domain closure and mixed Hodge structures

The moduli space of Hodge structures is the period domain
$$D\ = \ \coprod_{V,\psi} \text{Hom}_{\mathbf{R}\text{ alg.gp.}}(\mathbf{C}^*,G_\mathbf{R})/G_\mathbf{Z}$$
where $G\subseteq \text{GL}(V)$ are ...

**5**

votes

**1**answer

149 views

### Closure of the product of subfunctors

Background:
Let $X: \textbf{CRing} \to \textbf{Set}$ be a presheaf on the category of affine schemes and $Z \subseteq X$ a subfunctor. One defines $Z$ to be closed if for every ring $A$ and every ...

**1**

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132 views

### Picard scheme of family of quartic surfaces

Recall that a quartic surface in $\mathbb{P}^3_\mathbb{C}$ has $N = 35$ coefficients. Let $U$ be the open subset of $\mathbb{P}^{N-1}$ parametrising smooth quartic surfaces and let $Q \to U$ be the ...

**3**

votes

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101 views

### Locally a Deligne-Mumford stack has a finite etale covering

This is a part of the proof of corollary 6.1.1 of Laumon, Moret-Bailly's "Champs Algeriques".
Let $S$ be a scheme and $\mathscr{X}$ a non empty quasi-separated Deligne-Mumford stack over $S$...

**5**

votes

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149 views

### GAGA for vector bundles over Riemann surfaces

Serre’s GAGA theorem gives an equivalence of categories between algebraic and analytic coherent sheaves over a complex projective variety. The proof relies on the finiteness of the cohomologies of ...

**5**

votes

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163 views

### Algebraic Space: Two equivalent constructions

According to Wikipedia
there are two common ways to define algebraic spaces:
they can be defined as either quotients of schemes by étale
equivalence relations,
or as sheaves on a big étale site that ...

**3**

votes

**0**answers

213 views

### Galois representations and pro-étale Site

On a scheme, we can define the pro-étale site. This is an improvement over the étale site in that we can define the $\ell$-adic cohomology as the sheaf cohomology of the constant sheaf $\underline{\...

**3**

votes

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81 views

### Is the determinantal ideal of the span of a linearly independent set of rank-one matrices radical?

Let $k$ be an algebraically closed field, and let $X_1,\dots, X_n \in M_m(k)$ be rank-one matrices that are linearly independent over $k$. For a fixed integer $1 \leq r \leq m$, consider the ideal $I \...

**1**

vote

**0**answers

115 views

### Fixed point stack for a torus action

In this paper, M. Romagny defines for an action of a group scheme $G$ on a stack $X$ the fixed point stacks $X^G$ associated
to the group action on a stack and in Theorem 3.3 he proves that if
the ...

**3**

votes

**1**answer

211 views

### Prime ideals of formal power series ring that are above the same prime ideal

Let $R$ denote a commutative ring with identity and let $R[[X]]$ denote the
ring of formal power series over $R$ in an indeterminate $X$. If $I$ is an ideal of $R$,
then $I[[X]]$, the set of power ...

**1**

vote

**0**answers

61 views

### Blowing up of a stable curve

I'm struglling with 1.12 of Deligne-Mumford's "The irreducibility...".
Let $R$ be a DVR with algebraically closed residue field, $\pi$ a prime of $R$, $X$ a stable curve (in the sense of D-...

**1**

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**0**answers

60 views

### Weak Lefschetz theorem for Lef line bundles

I'm studying
M. A. A. de Cataldo, L. Migliorini - The Hard Lefschetz Theorem and the topology of semismall maps, Ann. sci. École Norm. Sup., Serie 4 35 (2002) 759-772.
The premises are the following....

**1**

vote

**0**answers

79 views

### Sifted sets can't accumulate on a curve

Let $f,g,h$ be elements in $\mathbb{Z}[x,y]$, each geometrically integral and at least two of them are distinct. Without loss of generality, suppose that $f$ is not proportional to $g$ over $\mathbb{C}...

**3**

votes

**1**answer

285 views

### What sort of object represents skyscaper sheaves on the etale site of $\mathbb{Z}_p$?

By SGA 4 IX Proposition 2.7, any constructible sheaf $\mathcal{F}$ on a qcqs scheme $X$ can be represented as an equalizer of two etale maps between representable (by schemes) sheaves. This would ...

**9**

votes

**2**answers

386 views

### fiberwise-quasi-compact implies quasi-compact?

Let $f\colon X\to \mathbb{A}^n_{\mathbb{C}}$ be a morphism of $\mathbb{C}$-schemes. Suppose $f$ is (a) separated, (b) flat, (c) locally of finite type, (d) all fibers are quasi-compact, is $X$ ...

**5**

votes

**0**answers

119 views

### Differential birational equivalence

Suppose the base field algebraically closed and of zero characteristic.
There are two fascinating questions in the intersection of ring theory and algebraic geometry (for which an excellent discussion ...

**2**

votes

**0**answers

137 views

### Extending etale morphisms

Let $Y$ be an affine, integral, Gorenstein surface. Let $y \in Y$ be a closed point such that there exists a finite, etale morphism $f: X \to Y\backslash \{y\}$ from an integral variety $X$ to the ...

**2**

votes

**1**answer

212 views

### Cohomological Brauer group vs classical

Let $X$ be a smooth scheme over $\mathbb{C}$.
A $O_X$-algebra $A$ is called Azumaya algebra on $X$
if locally it's ismorphic to matrix algebra: ie for
every $p \in X$ there exist open $U \subset X$ ...

**1**

vote

**0**answers

107 views

### Subgroup of $PGL(n(n-1)/2, \mathbb K)$ preserving the grassmannian $Gr(2, n)$

How can we determine the subgroup of $PGL(\wedge^2 \bar{\mathbb Q}^n)$ which preserves the grassmamnnian $Gr(2, n)$ embedded as a projective variety in $\wedge^2(\bar{\mathbb Q}^n)$ via the Plucker ...

**5**

votes

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278 views

### Definition of the cotangent complexes of Artin stacks

I am studying the notion of the cotangent complexes of Artin stacks reading LMB's book and Olsson's paper. According to them, the cotangent complexes are defined as projective systems in their derived ...

**1**

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119 views

### Understanding an example: splitting of a sequence restricted to the divisor

Set up:
Let $X$ be a smooth projective variety and $D\hookrightarrow X$ a smooth divisor. Let $E$ and $F$ be two holomorphic vector bundles with an isomorphism $\phi:F|_D\cong E|_D$. Define a new ...

**1**

vote

**1**answer

47 views

### Infinite uniform dimension $\Rightarrow$ infinitely many idempotents in a localization of a quotient

Let $R$ be a commutative ring with $1$ such that its uniform dimension is infinity, equivalently, $Sup\{k \mid R$ contains a direct sum of $k$ nonzero ideals$\}=\infty. $ How can we construct an ...

**13**

votes

**1**answer

297 views

### A royal road to Coulomb branches of 3D $\mathcal{N}=4$ gauge theories

So, I've been very interested recently with the developements of the (now mathematically precise) theory of Coulomb branches - in particular because of its recent applications on representation theory ...

**1**

vote

**1**answer

156 views

### Finding the dimension of the intersection of two real algebraic varieties

Suppose we have two polynomials $p, q \in \mathbb{R}^3$ and we are interested in their simultaneous zeros. Parameter counting tells us that the zero set most probably is going to be a one dimensional ...

**0**

votes

**0**answers

121 views

### Counting points in elliptic curves

Given an elliptic curve over $\mathbb Z_n$
Is it $\#P$ hard to compute $\# E(\mathbb Z_n)$?
Is it $PP$-hard to compute $\# E(\mathbb Z_n)\leq\frac n2$?
Is it $\oplus P$ hard to compute $\# E(\...

**3**

votes

**0**answers

272 views

### While solving the 1988 IMO problem 6, I have questions about new solutions without using Vieta Jumping [closed]

I think most of you may know the well-known problem:
"Let $x$ and $y$ be positive integers such that $xy + 1$ divides $x^{2} + y^{2}$. Show that $\frac {x^{2} + y^{2}}{xy + 1}$ is the perfect ...