**1**

vote

**0**answers

96 views

### In how many ways can one extend the zero section of the affine line with a double origin

Let $X$ be the affine line with a double origin over Spec $\mathbb Z$. Let $X_\eta$ be its generic fibre, the affine line with a double origin over Spec $\mathbb Q$.
Let $0$ be one of the origins of ...

**6**

votes

**0**answers

202 views

### Is the stack of varieties with a big line bundle algebraic

In Starr's paper https://www.math.stonybrook.edu/~jstarr/papers/moduli4.pdf the folk result that the fibred category of pairs $(X\to S, L)$, where $S$ is an affine scheme, $X\to S$ is flat proper ...

**6**

votes

**1**answer

244 views

### Isotrivial families with non-zero Kodaira spencer map

Let $S$ be a smooth quasi-projective curve over the complex numbers. Let $P$ be a closed point in $S$. Let $f:\mathcal X \to S$ be a polarized family of smooth projective connected varieties. To this ...

**0**

votes

**0**answers

87 views

### Irreducible component of a scheme over a dvr

Let $\mathcal M$ be a (reduced) quasi-projective scheme over a dvr (of mixed caracteristics), $R$. Suppose that the generic fiber $\mathcal M_{\eta_R}$ is (nonempty) smooth and irreducible of ...

**-3**

votes

**1**answer

237 views

### The scheme $y^n = x^{2n}$ for $n$ a rational number [closed]

Let $n\geq 1$ be an integer. If $A$ is a ring, then the spectrum of $A[x,y]/(y^n - x^{2n})$ is a well-defined (affine) scheme, say $X_n$. This scheme describes the "variety" given by the equation ...

**9**

votes

**1**answer

226 views

### Torsors trivializing over a fixed finite etale cover

Let $S$ be an integral regular scheme and let $T\to S$ be a finite etale morphism. Let $G$ be a smooth affine finite type group scheme over $S$.
Is the set of $S$-isomorphism classes of $G$-torsors ...

**0**

votes

**0**answers

101 views

### Zariski closure construction (over a field) implicitly uses reduced induced structure?

Let $X$ be a $k$-scheme of finite type, $k$ a field, and $\Sigma$ a subset of $X(k)$. Since the $k$-points of $X(k)$ may be identified with the physical points $x\in X$ for which the structural map ...

**4**

votes

**2**answers

314 views

### Infinitesimal deformations of a fibration

Let $f:X\rightarrow Y$ be a morphism of normal projective varieties over an algebraically closed field with connected fibers.
Assume that both $Y$ and the general fiber of $f$ admit a non-trivial ...

**14**

votes

**1**answer

385 views

### Geometric generic fibre

This is a pretty elementary question about schemes, but it came up in the course of research, so let's try it here rather than MSE.
Question 1: Are the fibres of a family of complex varieties ...

**3**

votes

**1**answer

189 views

### The stack of group algebraic spaces

The fibred category $\mathcal A$ of algebraic spaces over a scheme $S$ is a stack (over the category of affine schemes with the etale topology). This is proved in Laumon and Moret-Bailly's book (see ...

**3**

votes

**1**answer

174 views

### Stacks with a small coarse moduli space

Let $k$ be a field of characteristic zero.
Let $X$ be a finite type algebraic stack over $k$ with a coarse (or good) moduli space $M$.
Suppose that $M$ is isomorphic to a point, i.e., $M = Spec k$.
...

**4**

votes

**0**answers

165 views

### Properties of schemes determined by field valued points [closed]

Are there any interesting cases (interesting here is interpreted rather loosely here) where you can show $X$ has property $P$ whenever all $X(K)$ have property $P$ where $K$ runs through all fields?
...

**3**

votes

**0**answers

105 views

### Is there a difference between the inertia stack and the universal automorphism group

Let $\mathcal M$ be a stack representing some moduli problem. Let $\mathcal X\to \mathcal M$ be the corresponding universal family.
What is the difference between the inertia stack $I\to \mathcal M$ ...

**1**

vote

**1**answer

146 views

### On functors which are generically representable

Let $F$ be a set-valued (contravariant) functor on the category of schemes. Let $F_{\mathbb Q}$ be the associated functor on the category of schemes over $\mathbb Q$.
Suppose that $F_{\mathbb Q}$ is ...

**3**

votes

**1**answer

138 views

### Sections of morphisms up to fppf covering

Let $f:X\to S$ be a finite type affine morphism of schemes where $S$ is an integral noetherian affine regular scheme whose function field is of characteristic zero.
Assume that all geometric fibers ...

**1**

vote

**0**answers

41 views

### Locally free sheaves of algebras vs. algebra bundles

It is well known that there is a bijective correspondence between locally free sheaves of modules and vector bundles (cf. ...

**0**

votes

**0**answers

47 views

### constructibility for pushforward

Let consider a quasicompact open $j:U\rightarrow\mathbb{A}^{\mathbb{N}}$ over a field $k$, Is there an example where $Rj_{*}\mathbb{Z}/n\mathbb{Z}$ is not constructible, where $n$ is prime to the ...

**0**

votes

**0**answers

125 views

### If the quotient of an algebraic space $X$ by a finite group is a scheme, is $X$ a scheme?

If the quotient of an algebraic space $X$ by a finite group $G$ is a scheme, is $X$ already a scheme? Here $G$ is just a finite group, but I'd like to know the answer when $X$ is defined over ...

**2**

votes

**0**answers

198 views

### Finiteness of the connected components of a stack

Let $X$ be an algebaic stack over a scheme $S$, for any $S$-scheme $Y$ we can consider the groupoid $X(Y)$ of $Y$-points. Denote by $\pi_0(X(Y))$ the set of isomorphism classes of the groupoid.
Are ...

**1**

vote

**1**answer

203 views

### On Q-Cartier Divisors

I have my question on Q-Cartier Weil divisor.
People say $D$ is Q-Cartier divisor if $nD$ is Cartier for some $n \geq 1$. Especially for $n > 1$, I have never seen the `rigorous' definition of ...

**5**

votes

**1**answer

197 views

### Evaluation maps for moduli of stable maps

Let $\overline{M}_{0,n}(\mathbb{P}^N,d)$ be the moduli space of stable maps of degree $d$ from curves of genus zero with $n$-marked points to $\mathbb{P}^N$.
Consider the product of the evaluation ...

**2**

votes

**0**answers

64 views

### completion of non-finitely generated ideal

Let consider $A=k[x_{1},x_{2}...]$, the polynomial ring with countably many indeterminates.
Then we can consider the completion ...

**3**

votes

**0**answers

133 views

### going down theorem

Typical maps that satisfy "going down theorem" are flat morphisms and integral extensions of normal rings that are integral.
Let $Spec(B)\rightarrow Spec(A)$ be a finite type morphism of k-noetherian ...

**2**

votes

**1**answer

74 views

### Are the fibers of this morphism geometrically regular?

Let $A\rightarrow B$ be a local morphism of complete noetherian rings making $B$ a formally smooth $A$-algebra. Does the induced morphism $\textrm{Spec}(B)\to\textrm{Spec}(A)$ have geometrically ...

**0**

votes

**0**answers

79 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

43 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 ...

**9**

votes

**0**answers

115 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

423 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 ...

**0**

votes

**0**answers

45 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 ...

**3**

votes

**1**answer

253 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

386 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 ...

**6**

votes

**1**answer

661 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:
...

**2**

votes

**0**answers

174 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

81 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 ...

**3**

votes

**2**answers

311 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

**1**answer

187 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) $G^0 \subset G$ ...

**1**

vote

**0**answers

118 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 ...

**0**

votes

**0**answers

264 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

217 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 ...

**18**

votes

**1**answer

457 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

**0**answers

100 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

74 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

126 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

246 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

165 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

203 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 $f_i$ ...

**3**

votes

**1**answer

195 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 ...

**2**

votes

**0**answers

83 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

245 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

188 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 ...