**4**

votes

**1**answer

128 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

227 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

180 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

858 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 ...

**2**

votes

**0**answers

56 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

398 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

153 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

133 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

109 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

170 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

190 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

556 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

759 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

58 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

115 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 ...

**1**

vote

**0**answers

102 views

### complex automorphisms acting on projective varieties

Consider a complex projective variety $X=\operatorname{Proj}\frac{\mathbb C[T_1,\ldots,T_n]}{(f_1,\ldots,f_n)}$ with $f_1,\ldots,f_n$ homogeneous polynomials. If $\sigma\in\operatorname{Aut}(\mathbb ...

**0**

votes

**1**answer

180 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

318 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

66 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

348 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 ...

**3**

votes

**0**answers

222 views

### Given a morphism of schemes, when does bijective + isomorphic tangent spaces = isomorphism?

Let $f: X \to Y$ be a morphism of schemes over a field $k$ such that $f$ induces (1) a bijection between their closed points, and (2) an isomorphism of their Zariski tangent spaces.
Under these ...

**3**

votes

**0**answers

135 views

### Does the following first order approximation of the Kapranov-Vasserot infinitesimal loops still do any job?

Let $X$ be a scheme over, say, a field $k$. Let us denote $\mathrm{Spec}(k[\varepsilon])$ by $T$ and its (unique) $k$-point by $0\in T$. Call the first order infinitesimal cone $C_{T,0}(X)$ over $X$ ...

**8**

votes

**1**answer

393 views

### Classical algebraic varieties VS $k$-schemes VS schemes

We know that there is an equivalence of categories between the two following categories:
$1)$ Classical varieties over $k$, where $k$ is an algebraically closed field. (Informally I mean locally ...

**1**

vote

**3**answers

1k views

### Properties stable under base change in algebraic geometry

I remember to have seen a big list in the EGA of properties $(P)$ such that:
if $f : X \to Y$ has $(P)$ then, $f_{(S')} : X_{(S')}\to Y_{(S')}$ has $(P)$, where $f_{(S')}$ is the morphism $f$ after a ...

**1**

vote

**0**answers

73 views

### Holomorphic convergence conditions on $\mathbb C((z))$-valued points of a group $G$

Let $G$ be a complex, connected, simply connected, semisimple group. I'm trying to compare the following two spaces: The free loop space $LG$ of $G$, and the $\mathbb C((z))$-valued points of $G$, ...

**3**

votes

**1**answer

461 views

### How to compute cohomology groups of a closed subscheme Z of projective space, defined by a homogeneous polynomial of degree d?

Let $Z = \mathrm{Proj}\,k[x_{0},x_{1},\ldots,x_{r}]/f$ be a closed subscheme of degree $d$, i.e., $f$ is a homogeneous polynomial of degree $d$, and ...

**0**

votes

**0**answers

93 views

### Normal cones and the geometry of closed subschemes

Let $S$ be a closed subscheme of a smooth variety $M$ and suppose its ideal sheaf factors as $\mathscr{I}_S=\mathscr{I}_{S_1}\cdot \mathscr{I}_{S_2}$ for closed subschemes $S_1$ and $S_2$. Then what ...

**9**

votes

**1**answer

369 views

### Is there a direct proof that affine schemes are fppf quasi-compact?

Let $A$ be a (commutative) ring. A family $(B_i)_{i\in I}$ of $A$-algebras is said to be an fppf cover if it satisfies three properties: (1) each $B_i$ is flat as an $A$-module, (2) each $B_i$ is ...

**2**

votes

**1**answer

282 views

### Examples of Quot schemes

I'm studying Quot schemes, that I denote with $Quot_{N,X,P}$, with $N \in \mathbb{Z}$, $X \subset \mathbb{P}^d$ and $P \in \mathbb{Q}[t]$. So, I'm looking for explicit examples of Quot schemes. Could ...

**3**

votes

**2**answers

210 views

### Induced maps of an automorphism of a curve on the tangent ot its jacobian and on its differential forms

Let $C$ be a smooth projective curve of genus $g$ over a field $k$ and $J$ be its jacobian (defined over $k$). Let $\sigma: C \rightarrow C$ be a $k$-automorphsm of $C$. This automorphism $\sigma$ ...

**2**

votes

**0**answers

108 views

### Examples of non-projective morphisms with projective fibres

Let $X\to S$ be a morphism of noetherian schemes such that, for all $s$ in $S$, the morphism $X_s\to $ Spec $k(s)$ is projective.
Then it doesn't follow that $X\to S$ is projective in general. In ...

**0**

votes

**2**answers

237 views

### Pullback of a constant sheaf

Let $\varphi:X\to Y$ be a surjective morphism of schemes which are connected and of finite type.
Let $A$ be an abelian group, $\mathscr{F}$ be the constant sheaf on $X$ with fibers $A$ and ...

**3**

votes

**1**answer

341 views

### Orbits of group scheme action

I am interested in orbits of the action of a group scheme on a scheme and I'm particularly interested in the following special case: Let $k$ be an algebraically closed field, let $G$ be an affine ...

**11**

votes

**1**answer

733 views

### Functorial characterization of open subschemes?

Given a morphism of schemes f: U → X, can one determine when f is an isomorphism of U onto an open subscheme of X in terms of some induced functors between the categories of quasicoherent modules ...

**1**

vote

**1**answer

79 views

### Cohen-Macaulayness of the scheme of centralizer

Let $G$ be a simply connected group over an algebraically closed field $k$, and
$I:=\{(g,\gamma)\in G\times G\vert~ g\gamma=\gamma g\}$
the scheme of centralizer.
Is $I$ a Cohen-Macaulay scheme ...

**3**

votes

**2**answers

261 views

### Varieties that do not extend to flat families

Is it easy to give an example of a function field $K$ and a smooth proper variety $X$ over $K$ that does not extend to a flat scheme over $B$, where $B$ is a smooth proper variety with function field ...

**5**

votes

**3**answers

182 views

### Subset of Spec(A) realized as inverse image of some Spec(B)

Let $A$ be a (commutative) ring and $U$ be an arbitrary subset of $\mathrm{Spec}(A)$. Do there exist a ring $B$ and a ring homomorphism $\varphi\colon A\to B$ such that ...

**4**

votes

**0**answers

203 views

### Differential Geometry of (Non-Abelian) Gerbes in the language of Brylinski

Context
In an effort to have a definition for connection on a non-abelian gerbe, in the style of Brylinski, I am reading Breen and Messing's Differential Geometry of Gerbes [BM]. It seems that there ...

**3**

votes

**0**answers

171 views

### Does a section of a morphism of schemes give a subscheme?

Let $f:X\rightarrow Y$ be a morphism of schemes (or algebraic spaces), and $s:Y\rightarrow X$ is a section to $f$, i.e. $fs=1_{Y}$.
Question: Is $s$ an (closed) immersion?

**1**

vote

**1**answer

151 views

### Are Isom-schemes geometrically connected

This question is about properties of Isom-schemes that are well-known over algebraically closed fields.
Let $K$ be a field of characteristic zero, let $C$ be a smooth projective geometrically ...

**3**

votes

**0**answers

56 views

### on Neron defect of smoothness for groups schemes

Let $G$ a semisimple simply connected group over $\mathbb{C}$.
Let $\gamma\in G(\mathbb{C}[[t]])$ such that $\gamma$ is regular semisimple on $G(\mathbb{C}((t)))$.
We consider $I_{\gamma}$ the group ...

**0**

votes

**0**answers

47 views

### open subset in constructible set of divisors

Let a smooth projective curve $X$ over $\mathbb{C}$.
Let a pair $(x, D)$ a pair xith a closed point $x$ and $D$ an effective divisor on $X$, such that $d_{x}:=m_{x}(D)\neq 0$.
Let $N=\deg (D)$ and ...

**1**

vote

**1**answer

118 views

### Infinite residue field extensions and algebraic closure of residue fields

Let $X$ be a $K$-scheme of finite type over a field $K$, let $L$ be an extension field of $K$, let $X_L := L \times_K X$, and let $p:X_L \rightarrow X$ be the projection. For each $x \in X_L$ we get ...

**3**

votes

**1**answer

190 views

### When is the gluing of two finite type affine Z-schemes affine?

Let $A$ and $B$ be finitely generated $\mathbf{Z}$-algebra. Suppose that there exists two coprime integers $m$ and $n$ and an isomorphism of $\mathbf{Z}$-algebra
...

**2**

votes

**1**answer

173 views

### Closed points of field extension of k-scheme under projection

I really couldn't figure out the answer to the following question: Let $X$ be a scheme of finite type over a field $k$ and let $K$ be an extension field of $k$. Let $X_K := K \times_k X$ be the base ...

**0**

votes

**0**answers

61 views

### change of topologies and functoriality

Let X be a scheme and $\epsilon_X:X_{FL}\to X_{et}$ be the morphism of topoi from the big flat topos to the small etale topos. Let $f:X\to Y$ be a morphism of schemes. I denote $f_{top}$ the induced ...

**2**

votes

**0**answers

64 views

### semicontinuity of the conductor defined by Temkin

We say a principal pair $(X,\mathcal{I})$ where $X=Spec(A)$ is affine scheme and $\mathcal{I}=\tilde{I}$ where $I\subset A$ is a principal ideal generated by $\pi$ wich is a non zero divisor.
For a ...

**0**

votes

**0**answers

194 views

### Projective spaces with nonconstant regular functions

I can construct a scheme by patching that represents a projective space over an arbitrary ring. I can also prove that, if the ring is a Jacobson domain, the only regular functions on it are constants.
...

**2**

votes

**1**answer

468 views

### Can I conclude that a morphism of vector bundles is zero if it is so fiberwise?

Let $f: X \rightarrow Y$ be a flat morphism of locally noetherian schemes and $\varphi: \mathcal U \rightarrow \mathcal V$ a map of vector bundles (locally free sheaves of finite rank) on $X$.
I want ...

**0**

votes

**0**answers

116 views

### Reducibility of fibers over closed points implies reducibility of the generic geometric fiber?

Suppose that $f\colon X\to Y$ is a proper (or even projective) morphism of (reduced) algebraic varieties over an algebraically closed field $k$. If fibers of $f$ over all closed points of $Y$ are ...