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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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*} ...
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1answer
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
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0answers
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
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0answers
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
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0answers
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
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0answers
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
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1answer
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
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2answers
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
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1answer
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
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0answers
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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0answers
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
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0answers
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 $\delta$-...
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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 ...
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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
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0answers
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
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0answers
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
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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
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0answers
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 ...
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0answers
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
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1answer
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
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1answer
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
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1answer
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)...
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0answers
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): ...
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0answers
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 $det(T_{\Sigma})$?
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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
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1answer
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 ...
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0answers
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
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0answers
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
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0answers
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
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0answers
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
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0answers
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
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0answers
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 \...
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0answers
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
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1answer
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 ...
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0answers
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-...
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0answers
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....
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0answers
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
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1answer
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
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2answers
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
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0answers
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
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0answers
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
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1answer
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 $U \subset X$ ...
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0answers
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
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0answers
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 ...
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0answers
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
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1answer
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
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1answer
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
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1answer
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 ...
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0answers
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
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0answers
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 ...