Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [ag.algebraic-geometry]

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

4
votes
1answer
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
0answers
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
0answers
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
1answer
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
2answers
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
2answers
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
2answers
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
0answers
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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 $E_6$. ...