**0**

votes

**0**answers

63 views

### Pro-constructible subset of scheme intersects very dense subsets?

Let $X$ be a scheme, let $D$ be a very dense subset of $X$ and let $Y$ be a pro-constructible subset of $X$. Is it true that $Y \cap D \neq \emptyset$?
If $Y$ is just constructible, this is true.
...

**0**

votes

**0**answers

28 views

### normality of truncated arc space

Let $X=Spec(A)$, with $A$ a normal $k$-algebra of finite type, $k$ is a field.
For any integer $n$, let $X(k[t]/(t^{n}))$ the $n$-th truncated arc space, is it also normal?
Same question for ...

**7**

votes

**0**answers

80 views

### Weierstrass division theorem for henselian rings

Let $A$ be an henselian local noetherian ring. There is an old result of Lafon ("Anneaux henséliens et théorème de préparation" (1967)), which says that if $A$ is analytically normal and of ...

**0**

votes

**1**answer

163 views

### Artin approximation of a diagram

Let consider $f:(X,x)\to (Z,z)$ and $g:(Y,y)\to (Z,z)$ morphisms of pointed $k$-schemes of finite type ($k$ is a field). Suppose that there exists a map on the level of formal neighborhoods ...

**3**

votes

**1**answer

364 views

### on the local structure of schemes

Let $X$ be an integral finite type scheme over $\mathbb{C}$. Let $x\in X$, such that there exists a neighborhood $U$ of $x$, such that the sheaf of differentials $\Omega^{1}_{U}$ decomposes into:
...

**0**

votes

**0**answers

36 views

### closed subscheme associated to a closed rigid subspace

Let me start by saying that I know almost nothing about rigid geometry and I have no idea whether my question is easy or not.
Start with a ring $R$ which is local, complete for the topology defined ...

**8**

votes

**1**answer

434 views

### Can one check formal smoothness using only one-variable Artin rings?

Let $f:X\rightarrow Y$ be a morphism of schemes over a field $k$. Can one check that $f$ is formally smooth using only Artin rings of the form $k^{\prime}\left[t\right]/t^{n}$, where $k^{\prime}$ is ...

**3**

votes

**1**answer

227 views

### What if the base change of an algebraic space is representable

Let $k\subset L$ be an extension of fields of characteristic zero.
Suppose that $X/k$ is an algebraic space such that $X\otimes_k L$ is representable by a finite type $L$-scheme.
I am sure there are ...

**7**

votes

**1**answer

347 views

### Is the functor of points of a scheme cofinally small?

Background: In functorial algebraic geometry one would like to consider the category of all functors $\mathsf{CRing} \to \mathsf{Set}$ and define/characterize the category of schemes as a full ...

**2**

votes

**0**answers

149 views

### Stable Lefschetz fibrations

Let $S$ be a non-singular complex projective surface, then construct the Lefschetz fibration $\pi:\widetilde S\longrightarrow\mathbb P^1$ associated to $S$ (we have a birational morphism ...

**1**

vote

**0**answers

112 views

### Covering a finite set of points of height 1 by an affine open

Let $R$ be a Noetherian ring and let $X$ be a finite type, separated $R$-scheme that is normal and integral. Let $x_1, \dotsc, x_n \in X$ be points of height $1$. Does there exist an open affine $U ...

**1**

vote

**1**answer

168 views

### Relative identity component for group algebraic spaces

Let $S$ be a locally noetherian scheme and let $G$ be a separated and smooth $S$-group algebraic space of finite presentation. Does there exist an open sub-(group algebraic space) $G^0 \subset G$ ...

**3**

votes

**2**answers

280 views

### Push-forward of a quasi-coherent graded algebra under a proper map

Let $f\colon X \rightarrow Y$ be a proper morphism with $Y$ Noetherian (and even affine, if you wish), and let $\mathscr{A} = \bigoplus_{n \ge 0} \mathscr{A}_n$ be a quasi-coherent graded ...

**1**

vote

**0**answers

73 views

### (Affine) Schemes and the point of view of morphisms with values in a field

Let $A$ be a commutative ring with unity. If $\varphi_1 : A \rightarrow K_1$ and $\varphi_2 : A \rightarrow K_2$ are ring morphisms from $A$ to fields, $\varphi_1$ and $\varphi_2$ are said to ...

**24**

votes

**2**answers

2k views

### Ring-theoretic characterization of open affines?

Background
Recall that, given two commutative rings $A$ and $B$, the set of morphisms of rings $A\to B$ is in bijection with the set of morphisms of schemes $\mathrm{Spec}(B)\to\mathrm{Spec}(A)$. ...

**0**

votes

**0**answers

232 views

### Affine communication lemma and finite limits in the category of rings

Let $X$ be a scheme and $\mathrm{Spec}(B) = V \subseteq X$ be an open affine subset. When using the affine communication lemma (c.f. Theorem 6.3.2, Vakil's notes, Foundations of Algebraic Geometry), ...

**1**

vote

**1**answer

206 views

### Smoothness and smoothness over formal neighborhood

Let $f:X\rightarrow Y$ a locally finitely presented map. Let $x\in X$ and $y=f(x)$.
We assume that the map on the level of fomal neighborhoods $X_{x}\rightarrow Y_{y}$ is formally smooth, can we find ...

**14**

votes

**0**answers

301 views

### Does every relative curve have a Picard scheme?

More precisely:
Let $X \to S$ be a smooth proper morphism of schemes such that the geometric fibers
are integral curves of genus $g$. Must the fppf relative Picard functor
$\operatorname{\bf ...

**3**

votes

**1**answer

183 views

### Flatness of Normalization of regular schemes

I have a followup to the following question: Flatness of normalization.
Suppose that $X$ is a regular scheme (of finite type over a $\mathbb{C}$ if one wants) and $X'$ is the normalization of $X$ in ...

**18**

votes

**3**answers

2k views

### A book on locally ringed spaces?

Are there enough interesting results that hold for general locally ringed spaces for a book to have been written? If there are, do you know of a book? If you do, pelase post it, one per answer and a ...

**2**

votes

**0**answers

87 views

### motivic integration and jacobian ideal

When we consider the change of variables in motivic integration, we have a birational map $f:Y\rightarrow X$ with Y smooth and we have to consider two invariants the order of the Jacobian ideal of $X$ ...

**0**

votes

**0**answers

68 views

### on lifting elements in a tangent space

Let X a normal integral scheme over a base field scheme, assumedd to be singular and an integer $n$
Let $\mathcal{O}=k[[t]]$, we consider the arc space $X(\mathcal{O})$ which is a $k$- pro-scheme and ...

**1**

vote

**1**answer

122 views

### Some questions about ruled surfaces defined over $\overline{\mathbb Q}$

definitions:
A non-singular complex projective surface $S$ is a ruled surface if it is birationally equivalent to $C\times_{\text{Spec} \mathbb C}\mathbb P^1_{\mathbb C}$ where $C$ is a non-singular ...

**1**

vote

**2**answers

234 views

### When the contraction is a morphism defined over $\overline{\mathbb Q}$

Suppose that $S$ is a complex projective surface defined over $\overline{\mathbb Q}$, namely there exists a surface $S_{\overline{\mathbb Q}}$ over $\overline{\mathbb Q}$ such that:
...

**2**

votes

**1**answer

159 views

### About $\mathbb P^1_\mathbb C$ contained in a surface

Suppose that $X$ is a non-singular projective surface over $\mathbb {\overline Q}$ ( $X$ is a $\mathbb {\overline Q}$-scheme...) and suppose that there is an embedding:
$$j:\mathbb P^1_{\mathbb ...

**2**

votes

**1**answer

197 views

### Minimal model of a non-singular complex projective surface defined over $\overline{\mathbb Q}$

Suppose that $S$ is a non-singular complex projective surface that is defined over $\overline{\mathbb Q}$, namely $S\cong\text{Proj}\frac{\mathbb C[T_1,T_2,\ldots,T_n]}{(f_1,\ldots,f_n)}$ where $f_i$ ...

**2**

votes

**0**answers

79 views

### Can we classify reductive group schemes over curves

Let $C$ be a smooth quasi-projective connected curve over the complex numbers.
Can one classify all reductive group schemes over $C$?
Certainly, you have the trivial ones (coming from pulling-back ...

**2**

votes

**1**answer

235 views

### Can the Grothendieck ring of varities over a field $k$ be defined for non separated schemes?

The Grothendieck ring of varieties over a field $k$ is the abelian group generated by isomorphim classes $[X]$ of separated, reduced $k$-schemes $X$ of finite type with the relation
$[X]=[Y] + ...

**4**

votes

**0**answers

173 views

### Tannaka categories and reductive groups

The group associated to a Tannaka category $T$ over a field is pro-reductive if and only if $T$ is semi-simple.
Pro-reductive groups make sense over any scheme.
Is there an extension of the theory ...

**2**

votes

**0**answers

131 views

### infinite dimensional germs of schemes and tangent spaces

(The question of the type "how to define?")
Let $(R,\mathfrak{m})$ be a local ring over a field $k$ of zero characteristic. Consider the matrices over this ring, $Mat(m,R)$. I think of $Mat(m,R)$ as ...

**0**

votes

**1**answer

100 views

### curve through a point avoiding an hypersurface

Let $H$ be a closed hypersurface in $\mathbb{A}^{n}$, $n$ big enough over $\mathbb{C}$. Let $U$ be the complementary open subset.
Let $x\in H$, Is it possible to find an curve ...

**5**

votes

**1**answer

246 views

### Existence of affine hulls

(This question is inspired by Matthieu Romagny's answer to my previous question about base change properties of affine hulls.)
Given a scheme $S$, it is well-known (cf. EGA I.9.1.21) that the ...

**3**

votes

**1**answer

265 views

### Affine hulls and base change

Let $S$ be a scheme. We consider the functor, called affine hull, from the category of quasicompact and quasiseparated $S$-schemes to the category of affine $S$-schemes, defined as a left adjoint to ...

**0**

votes

**2**answers

234 views

### Weil restriction

I've already asked a similar question in SE, without success, so I've decided to post here a more general version of my question.
Let $f: Y \to X$ be a finite etale morphism of smooth proper ...

**6**

votes

**3**answers

956 views

### Do Disjoint Unions and Fiber Products Commute?

Do disjoint unions and fiber products commute?
In other words, is the following statement true?
Statement: Let $C$ be a category with (infinite) coproducts and fiber products. Let {$U_{i}$} be a ...

**3**

votes

**0**answers

86 views

### Examples of subspaces singled out by modular forms

I am wondering what subspaces of modular varieties defined as the zero locus of modular forms have been studied in the literature.
To be more clear let me explain the example I have in mind.
Let ...

**2**

votes

**1**answer

424 views

### Base change through a field automorphism

Note:For a correct comprehension of the question see the "important edit" at the end.
Consider a projective variety over $\mathbb C$, $X=\textrm{Proj}\frac{\mathbb ...

**4**

votes

**1**answer

156 views

### Which valuations of a field yield codimension $1$ subschemes of their 'models'

For a field $F$ (for example, a one generated by a finite number of its elements) there is a directed set of its 'models' (in this case those are 'arithmetic' schemes whose fraction field is $F$). It ...

**1**

vote

**1**answer

142 views

### Conjugate surfaces: informations about the orbits

Consider a complex algebraic variety $X$ (namely a $\mathbb C$-scheme, of finite type, geometrically integral and separated); if $\sigma\in\textrm{aut}(\mathbb C)$, then is well defined the complex ...

**3**

votes

**0**answers

123 views

### Are there any useful Grothendieck topologies for which the H1 of $GL_n$ is not the set of rank $n$ vector bundles

Let n be a positive integer and X a scheme. Then for all the Grothendieck topologies I know (Zariski, etale, fppf) the set $H^1(X,GL_n)$ is the set of (isomorphism classes of) rank $n$ vector bundles.
...

**3**

votes

**2**answers

195 views

### Minimal fields of isomorphism for varieties

Let $V$ be an algebraic variety over a field $K$. Is there a constant $d = d(V) \in \mathbb{N}$ such that for any variety $W$ defined over $K$ and isomorphic to $V$ over the algebraic closure of $K$, ...

**-2**

votes

**1**answer

200 views

### Schemes over $K_s$ and over $\bar{K}$

Let $K$ be a field. Let $X$ be a scheme over $K$. We denote by $K_s$ and by $\bar{K}$ the separable closure and the algebraic closure of $K$ respectively.
By base change we have the schemes $X_{K_s}$ ...

**5**

votes

**1**answer

563 views

### For a morphism f from a regular scheme, should there exist an open subscheme U of the target such that fibre of f at each point of U is regular

For a finite type morphism $f:X\to S$, $X$ is a regular scheme, should there always exist an open (dense) subscheme $U\subset S$ such that the fibre of $f$ at each Zariski point of $U$ is regular? All ...

**6**

votes

**1**answer

779 views

### A naive algebraic geometry question

Suppose $X$ is a scheme over a ring $A$, $B$ is an $A$-algebra, and $X\times_AB$ is affine. I am looking for conditions on $A$ and $B$ (and perhaps the structure morphism of $X$ over $A$) that will ...

**0**

votes

**0**answers

65 views

### Group schemes decomposition

Given an abelian group scheme of finite type $(G,+)$ over $\mathbb{F}$ connected, and given two connected closed subgroup schemes of finite type $G$ over $\mathbb{F}$ connected $H$, $N$ of $G$. ...

**0**

votes

**1**answer

126 views

### Closed immersion of closed fiber?

Suppose $f: X \rightarrow Y$ is a morphism of schemes, and $y \in Y$ is closed point.
We know the first projection morphism $p_1: X \ $ x$_{Y} \ k(y) \rightarrow X$ is a homeomorphism onto ...

**0**

votes

**1**answer

198 views

### Confusion with the field of definition of a variety [closed]

Fix a field extension $k\subseteq K$ (assume that the fields have characteristic $0$) and consider the two following definitions:
Now let's restrict our attention to a closed subscheme ...

**3**

votes

**1**answer

334 views

### Spreading out flat morphisms of schemes

In EGA IV, Chapter 8, projective systems of schemes (and morphisms between them) are considered. Let $(S_{\lambda})_{\lambda \in L}$ be a projective system of schemes and let $S$ be the projective ...

**2**

votes

**0**answers

86 views

### Quasi-finite morphisms of stacks

Let $f:X\to Y$ be a morphism of ``nice" stacks over $\mathbf C$ such that the induced morphism on coarse moduli spaces is quasi-finite. Is $f$ quasi-finite?
By a "nice" stack I mean a smooth finite ...

**9**

votes

**2**answers

375 views

### Is the category of schemes wellpowered? regularly wellpowered?

Wellpowered means that for every scheme $X$, the subobject lattice of monormophisms $Y \to X$ is essentially small; regularly wellpowered means that for every scheme $X$, the regular subobject lattice ...