2
votes
1answer
213 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
0answers
159 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
0answers
104 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
1answer
95 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
1answer
227 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
1answer
250 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
2answers
198 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 ...
2
votes
0answers
73 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
1answer
399 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 ...
1
vote
1answer
136 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 ...
4
votes
1answer
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 ...
3
votes
0answers
111 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
2answers
177 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
1answer
194 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 $X_{K_s}$ ...
0
votes
1answer
116 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
0answers
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
1answer
182 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 ...
2
votes
0answers
69 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 ...
6
votes
1answer
761 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 ...
9
votes
2answers
350 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
0answers
227 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
0answers
136 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
1answer
400 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 ...
0
votes
0answers
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 ...
2
votes
0answers
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
2answers
238 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
1answer
355 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 ...
2
votes
1answer
284 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
2answers
264 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 ...
4
votes
0answers
204 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
0answers
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
1answer
155 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
0answers
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
0answers
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
1answer
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 ...
2
votes
1answer
175 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 ...
3
votes
1answer
193 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
0answers
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 ...
2
votes
1answer
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
0answers
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 ...
2
votes
0answers
112 views

scheme of sections over complete local ring

Let $f:X\rightarrow S= Spec(k[[\pi]])$ a finite type faithfully flat morphism. Let $U\subset X$ be an open subset such that $U$ is smooth and surjective on $S$. We consider the $k$-scheme ...
1
vote
0answers
101 views

smooth morphism from a finite type source

Let $f: X\rightarrow Y$ a smooth morphism over a field $k$. We assume that $X$ is locally of finite type, does it imply that $Y$ is also locally of finite type?
1
vote
0answers
78 views

ind scheme and Jacobson property

Let $G$ a semisimple group over $k$ and $k$ algebraically closed. Let $G(k((t)))$ the corresponding ind-scheme, does it satisfies the Jacobson property, say closed points are dense in it?
0
votes
0answers
95 views

on the fibers over closed points

Let $X$ and $S$ $k$-schemes of finite type . ($k$ a field) and $U$ an open subset of $X$ Let $f:X\rightarrow S$ a $k$-morphism of finite type. We assume that for any closed point $s\in S(\bar{k})$, ...
2
votes
0answers
140 views

fpqc, formal smoothness

Based on Possible formal smoothness mistake in EGA, let $X$ and $Y$ $k$-schemes ($k$ a field), let $f:X\rightarrow Y$ a fpqc morphism such that $f$ is formally smooth and $X$ formally smooth, do we ...
2
votes
1answer
191 views

flat and finite type morphisms

Let $f:X\rightarrow Y$ a faithfully flat morphism between $k$-schemes. We assume that the fibers are locally of finite type, do we have that $f$ is locally of finite type?
4
votes
0answers
113 views

formal smooth morphism with a formal smooth source

Let $f:X\rightarrow Y$ a morpism between $k$-schemes ( $k$ a field). We suppose that X is formally smooth and f is formally smooth and surjective. Do we have that $Y$ is formally smooth? Or if it's ...
2
votes
0answers
115 views

descent for formally smooth maps

Let $f:X\rightarrow Y$ a morphism between schemes and $Y'\rightarrow Y$ a fpqc morphism such that the base change $f'$ of $f$ to $Y'$ is formally smooth, does it imply that $f$ is formally smooth?
1
vote
0answers
69 views

closed subscheme of ind scheme

Let $X$ a ind-scheme of ind-finite type and ind-affine. (e.g, take a k- smooth, affine scheme of finte type $T$, $C$ a smooth projective curve over $k$ and $x$ a closed point, then $X=T(C-x)$ verifies ...
0
votes
2answers
187 views

open immersion between affine spaces

Let $j:\mathbb{A}^{n}\rightarrow\mathbb{A}^{n}$ an open immersion over a field $k$. Is it an isomorphism?