# Questions tagged [ag.algebraic-geometry]

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

**4**

votes

**1**answer

107 views

### Henselianizations over countable index sets

Let $A$ be a ring, $I\subset A$ a finitely generated ideal.
The henselianization $A^h$ of $A$ along $I$ is the universal $A$-algebra that is henselian along $I$ and can be presented as a direct limit ...

**0**

votes

**0**answers

81 views

### Ramified covering morphisms

I found the following theorem in the book Revêtements
Étales et Groupe Fondamental$-$Alexander Grothendieck.
Théorème 10.11. — Soit $f : X\longrightarrow Y$ un morphisme quasi-fini séparé. On ...

**3**

votes

**0**answers

108 views

### Is there a Seifert–van Kampen theorem for etale fondemental group?

Is there a Seifert–van Kampen theorem for etale fondemental group? (for example for varieties over a non-algebraically closed field) Any example is welcome.

**2**

votes

**1**answer

169 views

### Maps from a scheme over the dual numbers to constant schemes

Let $X$ be a smooth scheme over $k[t]/(t^2),$ where $k$ is a field of characteristic 0 (the case when $X$ is a projective curve is already interesting). Let $X_{0} \to X$ denote $X$ with the reduced ...

**10**

votes

**2**answers

547 views

### Is a scheme Noetherian if its topological space and its stalks are?

Is a scheme being Noetherian equivalent to the underlying topological space being Noetherian and all its stalks being Noetherian?

**1**

vote

**1**answer

61 views

### Existence of meromorphic 2-forms over normal surface singularities

Let $(X,o)$ be an isolated normal surface singularity. Denote by $U:=X\backslash \{o\}$. I am looking for conditions on $(X,o)$ under which there exists a holomorphic section $\omega \in H^0(U, \Omega^...

**3**

votes

**0**answers

41 views

### Intersecting ideal with smaller base field

Let $K\subsetneq L$ be a field extension and let $I\subset L[x_1,\ldots,x_n]$ be an ideal. Is there a trick/machinery/theory for determining the ideal $I\cap K[x_1,\ldots,x_n]$?
In the case of an ...

**3**

votes

**0**answers

96 views

### Sections of sheaves on limit spaces

Let $\{U_{\nu}\}_{\nu\in I}$ be an inverse system of topological spaces over a filtered index set $I$ with continuous transition maps.
Let $A_0$ be a sheaf of abelian groups on $U_{\nu_0}$, for some $...

**7**

votes

**0**answers

139 views

### Is $\mathbb{C}^n\setminus V(f)$ homotopy equivalent with a “large ball complement”?

Let $f\in\mathbb{C}[x_1,\dots,x_n]$, and let $V(f)$ denote the vanishing locus. Is it true that for large enough $N$, there is a homotopy equivalence
$$\mathbb{C}^n\setminus V(f)\simeq B(0,N)\setminus ...

**0**

votes

**1**answer

88 views

### Poles of equivariant meromorphic functions on Riemann surfaces

Let $p:\Sigma\to \mathbb{P}^1$ be the cyclic cover of $\mathbb{P}^1$ with Galois group $\Gamma$. Let $\Gamma\cdot p$ be a free $\Gamma$-orbit on $\Sigma$. Given any character $\chi$ of $\Gamma$, does ...

**2**

votes

**1**answer

124 views

### What is the topological/smooth analogue of Nagata compactification

A celebrated theorem of Nagata and subsequent refinements to schemes and algebraic spaces say that over a not-completely-monstrous base scheme, any separated morphism can be openly immersed in a ...

**0**

votes

**2**answers

401 views

### Defining algebraic manifold without referring to schemes

Let $M$ be a complex manifold admitting an atlas with each chart biholomorphic to $\mathbb{C}^n$ and transition maps being rational functions.
Is it true that there exists a smooth integral ...

**12**

votes

**0**answers

226 views

### Cohomology of real analytic coherent sheaves

Let $M$ be a real analytic variety
(if someone is concerned about distinction between
"real analytic spaces" and "real analytic varieties"
in real analytic geometry, let's assume that $M$
is both "...

**0**

votes

**0**answers

34 views

### Testing relation between two integers in a linear equation [on hold]

This is what I read:
One can test if there exist integers x and y such that
$c\cdot x + j = d \cdot y + k$,
if $(k-j)\mod(\gcd(c,d)) = 0$
How can one see that?
--
PS: I understand ...

**3**

votes

**0**answers

83 views

### How to construct the espace etale (space of sections) for an arbitrary category?

I want to consider the sheaf of an arbitrary category (not only of sets, groups, modules and so on) on a topological space, using the language of etale space.
In all references I am reading (...

**0**

votes

**1**answer

78 views

### Relation between Hilbert function and complete intersection ideals

Consider $T=k[x_1,\ldots,x_n]$ ( $k$ alg. closed and of char $k=0$), and consider the ideal $$I=(x_1,x^{a_2}_2,\ldots,x^{a_n}_n)$$
with $2\leq a_2 \leq\ldots\leq a_n$. I want to prove that $$\sum_{i=...

**0**

votes

**0**answers

57 views

### Smallest collection of linear operators satisfying isometry property

Let $\mathscr{A}=\{(A_{1,1},A_{1,2},A_{1,3}),...,(A_{S,1},A_{S,2},A_{S,3})\}$ be a collection of linear operators $A_{n,k}:\mathbb{R}^2 \rightarrow \mathbb{R}^2$. For $u$ and $v\in (\mathbb{R}^2)^3$, ...

**6**

votes

**1**answer

131 views

### Irreducibility of Gelfand-Serganova strata

To keep the notations simple I'll restrict my attention to the complete flag variety although the question should be equally valid for partial flag varieties. Consider $G=SL_n(\mathbb C)$ with Borel $...

**2**

votes

**0**answers

78 views

### Generalizing characterizing crystalline representations of dimension 2 to certain special classes of crystalline representations of higher dimension

Let $A$ be an abelian variety defined over a number field $K$, and let $v$ be a prime of $K$ such that $A$ has good reduction modulo $v$. Let $\rho$ be the representation of $G_K = \text{Gal}(\...

**4**

votes

**0**answers

43 views

### Behavior of geometric fibers under strict transforms

Consider a digram of the form
$$
\begin{array}{ccc}
Z' & \rightarrow & Z \\
\downarrow & & \downarrow \\
X' & \rightarrow & X \\
\downarrow & & \downarrow \\
S' &...

**0**

votes

**0**answers

37 views

### Restriction of fractional ideal sheaf to irreducible component is torsion-free

I translate the question into commutative algebra:
Let $R$ be a one-dimensional, reduced ring (which is also finite free over some PID since the considered curve corresponding lies finitely over the ...

**3**

votes

**1**answer

142 views

### Is a factorial scheme with Noetherian stalks locally Noetherian?

Motivation: The Oka's coherence theorem tells us that the structure sheaf of a complex manifold is coherent. Taking into account the fact that coherence is a local property stable under finite direct ...

**6**

votes

**1**answer

258 views

### The cotangent bundle of a non-compact Riemann surface

Suppose that $M$ is a non-compact Riemann surface obtained by removing several points from a compact one. It is known that the holomorphic cotangent bundle of $M$ is trivial. Therefore there exists a ...

**15**

votes

**1**answer

289 views

### Curves over number fields with everywhere good reduction

My question is the following:$\newcommand{\Q}{\Bbb Q}
\newcommand{\Z}{\Bbb Z}$
What is known about number fields $K$ fulfilling the condition
$C_{g,K}$ "there is a smooth projective curve of ...

**1**

vote

**1**answer

130 views

### A question regarding de Franchis theorem

One form of de Franchis theorem for algebraic curves is the following: let $X$ be an algebraic curve (defined over $\mathbb{C}$ say) with genus $g > 1$. Then there are only finitely many (...

**5**

votes

**0**answers

77 views

### The structure of the algebraic cobordism cohomology of the motivic Eilenberg-Maclane spectrum

I have a question related to the stable motivic homotopy categories of Morel-Voevodsky. Consider the motivic Eilenberg-MacLane spectrum $H\mathbb{Z}$ and the algebraic cobordism theory $MGL$.
Let $a$ ...

**1**

vote

**0**answers

65 views

### Meromorphic mappings between complex projective spaces

Let $n>2$ and $\phi: \mathbb{P}_{\mathbb{C}}^n \setminus S \rightarrow \mathbb{P}_{\mathbb{C}}^n$ be a holomorphic map and $S$ a closed analytic subset of $\mathbb{P}_{\mathbb{C}}^n$ with ...

**0**

votes

**1**answer

116 views

### singularity of a hypersuface in $\mathbb{P}^3$

Let $X$ be an irreducible hypersurface defined by a polynomial $f$ of degree $5$ in $\mathbb{P}^3$. Let the homogeneous co-ordinates is given by $[x, y, z, w]$ and let $H$ be a hyperplane given by $w= ...

**5**

votes

**1**answer

221 views

### Resolution of a torsion sheaf

Let $J$ be the hyperplane divisor in $\mathbb{C}P^2$, and let $i:C \hookrightarrow \mathbb{C}P^2$ be the closed immersion of a smooth generic curve of degree 2. We know that $C\simeq \mathbb{C}P^1$, ...

**5**

votes

**0**answers

321 views

### Is it true that all smooth group schemes can be deformed?

Consider for instance the map $\mathbb Z/p^2 \to \mathbb Z/p$ and suppose we are given a group smooth scheme $G$ over $\mathbb Z/p$. Is it always possible to lift it to a smooth group scheme $G'$ over ...

**4**

votes

**1**answer

223 views

### Stack being represented by a scheme/manifold

On page $10$ of the survey article Algebraic stacks, by T. Gomez (arXiv:math/9911199), we have following result
If a stack has an object with an automorphism other than the identity, then the ...

**2**

votes

**0**answers

143 views

### Deformation invariance of homotopy type

Let $\mathscr{X}\to \Delta$ be a flat family of projective varieties over the unit disk so that each fiber $X_t$ has canonical singularities and its canonical sheaf $\omega_{X_t}$ is $\mathcal{Q}$-...

**-1**

votes

**0**answers

86 views

### Disjoint union of points of the affine line---an fpqc cover of the affine line

For any field $k$, we have a faithfully flat covering
$$
\varphi : \coprod _{x \in \mathbf{A}^1_ k} \mathop{\mathrm{Spec}}(\mathcal{O}_{\mathbf{A}^1_ k, x}) \to \mathbf{A}^1_ k.
$$
Is it quasi-compact ...

**0**

votes

**0**answers

81 views

### A reference for studying special ring

A topological space $X$ is called profinite if it is compact, Hausdorff, and has a basis of open–closed sets. Also a commutative ring $R$ with 1 is called a topological ring it there is a topology on ...

**2**

votes

**0**answers

71 views

### Some notions of immersions of locally ringed spaces

Let $f:X\to Y$ be a morphism of locally ringed spaces. In this MSE answer, the first definition below is suggested.
Say $f:X\to Y$ is an $R$-immersion of locally ringed spaces if it's a topological ...

**2**

votes

**0**answers

99 views

### Fundamental Group of small Zariski open set

Let $Y$ be an integral affine variety over $\mathbb{C}$ and $K$ be its function field. How to find a sufficiently small Zariski open set of $Y$ such that it is isomorphic to $K(\pi,1)$? Here $\pi$ is ...

**5**

votes

**0**answers

83 views

### Reference Request: Cohomology and limits of coherent topoi. (Non-abelian case)

SGA 4 VI Discusses finiteness conditions one can impose on topoi to make limits behave correctly. I am not that familiar with SGA but it is my impression that this expose only discusses abelian ...

**6**

votes

**1**answer

236 views

### Degree of the variety of singular points

Let $V\subset \mathbb{A}^n$ be an irreducible affine variety. The set of singular points of $V$ is a subvariety $W$ of $V$; denote its components by $W_i$. How may we bound $\sum_i \deg(W_i)$ in terms ...

**4**

votes

**0**answers

204 views

### Subsets $E$ of $\mathbb{F}_{p^k}$ with vanishing polynomial sums

The following question arose in some discussions recently as a misunderstanding of another problem.
Question: Which subsets $E\subset \mathbb{F}_{p^k}$ satisfy the property that $ \sum\limits_{x\in E}...

**2**

votes

**0**answers

152 views

### Can etale-analytic comparison hold when etale-Cech comparison doesn't?

Assume we have a scheme over $\mathbb{C}$ and a constructible sheaf on $X$. We have a natural morphism from etale cohomology to derived functor cohomology in complex-analytic topology
$$
H_{et}(X, F)\...

**1**

vote

**0**answers

71 views

### Etale-analytic comparison without elementary fibrations

A theorem due to Artin states that for a smooth scheme $X$ of finite type over $\mathbb{C}$ and a locally constant constructible sheaf $F$ we have an isomorphism
$$
H^*_{et}(X, F)\approx H^*(X(\...

**5**

votes

**1**answer

111 views

### Flat limit (of twisted cubic) contained in surfaces

Let $H$ denote the irreducible component of $\text{Hilb}^{3t+1}\mathbb{P}^3$ whose general member corresponds to a non-singular twisted cubic. Let $C$ be a subscheme lying in the boundary of $H$ and ...

**7**

votes

**2**answers

218 views

### $2$-fiber product is a scheme then map of stacks is representable

Ariyan Javanpeykar said here in comments that,
$X\times_{\mathcal{X}}X$ being a scheme is equivalent to representability of $X\rightarrow \mathcal{X}$.
Context is as in this question.
Suppose $p:...

**2**

votes

**0**answers

137 views

### Diagonal is representable then composition is representable

Let $\mathcal{X}$ be a stack over $S$ i.e., a stack over category of schemes over $S$ (which we denote by $Sch/S$) which comes with a functor $\mathcal{X}\rightarrow Sch/S$. Consider the diagonal map ...

**3**

votes

**1**answer

233 views

### Diagonal is representable then any morphism is representable

Ariyan Javanpeykar said here in comments that,
If the diagonal is representable, then isn't any morphism $S\rightarrow \mathcal{X}$ with $S$ a scheme representable?
I could not find the statement (...

**3**

votes

**1**answer

134 views

### Direct limit of strict henselizations

Assume we have a map $A \rightarrow A'$ of strictly henselian local rings, such that the induced map between spectra $S'\rightarrow S$ is essentially smooth. Is is true that $S'$ is a direct limit of ...

**1**

vote

**0**answers

54 views

### generalization of Bruhat decomposition and $G$-orbits in $(G/B)^n$

Let $G$ be a connected reductive group over an algebraically closed field $k$ and $B$ be one Borel group of $G$. The Bruhat decomposition describes $G$-orbits in $(G/B)^2$ by Weyl group, which is ...

**3**

votes

**0**answers

132 views

### Finiteness of $H^2(X,\mu_n)$

Let $X$ be a proper curve over $k$ (algebraically closed) of characteristic $p>0$.
When is $H_{fl}^2(X,\mu_n)$ is a finite group?
It's true when $X$ is smooth but are there any more general ...

**-2**

votes

**0**answers

68 views

### Rational maps induced by natural transformations

I am reading the paper Function Spaces and Continuos Algebraic Pairings for Varieties of Friedlander and Walker. More precisely, I am working on the following result.
In the proof of the proposition ...

**4**

votes

**1**answer

127 views

### Constructing algebraic groups of type E6 with split Tits algebras

Let us assume our base field $k$ has characteristic zero.
From a series of papers by Borel and Siebenthal it is known that there is an embedding of groups
$A_2 \times A_2 \times A_2$ into $E_6$.
...