All Questions
16 questions
18
votes
3
answers
2k
views
Is there an example of a variety over the complex numbers with no embedding into a smooth variety?
Is there an example of a variety over the complex numbers with no embedding into a smooth variety?
- 10.5k
9
votes
1
answer
943
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:
$\...
- 3,472
9
votes
1
answer
531
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 ...
- 10.5k
6
votes
1
answer
931
views
Is every variety an image of a smooth variety?
Let $X$ be a finite type scheme over a field $k$.
Is it true that there exists a surjective morphism $f : Y \rightarrow X$, where $Y$ is smooth over $k$?
In other words, is every such scheme a ...
- 485
3
votes
0
answers
294
views
Formal smoothness implies local freeness of the sheaf of relative differentials
What is the least restrictive finiteness assumptions guaranteeing that for a formally smooth morphism of schemes $f:X\rightarrow Y$, the sheaf of relative differentials $\Omega _{X/Y}$ would be ...
- 953
2
votes
0
answers
148
views
etale locally infinitesimal lifting property
For a morphism $X\rightarrow Y$ of qcqs schemes, one has the usual notion of formal smoothness which says that for a pair $(R,I)$ with $I^2=0$, if there is a point $y\in Y(R)$ such that $y_{\vert R/I}$...
- 3,472
2
votes
0
answers
192
views
Morphism between jet spaces smooth
In this article "Introduction to Jet Schemes and Arc Spaces" S. Ishii introduces the spaces of $m$-jets:
Let $X$ be a variety over algebraically closed field $k$. The space $X_m$ of $m$-jets ...
- 6,028
2
votes
0
answers
138
views
Formally smooth maps of schemes and factorization systems
I am thinking about how formally smooth maps of schemes relate to factorization systems.
Let $C$ be the category of schemes. Let $E$ be the class of morphisms of schemes consisting of closed ...
user30211
2
votes
0
answers
394
views
Blow up along a section of a smooth morphism
Let $C$ be a ground locally notherian and quasiprojective scheme. Let $\pi:S\to B$ be a $C$-morphism of finite type $C$-schemes. We call $\pi$ a $C$-smooth morphism if the morphisms $S\to C$ and $B\to ...
- 43
2
votes
0
answers
141
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 $X(k[[\pi]]...
- 3,472
2
votes
0
answers
250
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 ...
- 3,472
2
votes
0
answers
236
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?
- 3,472
1
vote
1
answer
342
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 ...
- 3,472
1
vote
0
answers
91
views
Factorizations of closed embeddings of smooth schemes
All schemes will be of finite type over a field $k$. Say I have a closed embedding $\iota: X \hookrightarrow Y$ of smooth $S$-schemes for some scheme $S$ (in particular it is a regular embedding). ...
- 595
1
vote
0
answers
128
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?
- 3,472
0
votes
0
answers
329
views
Smooth morphisms under base change, Qing Liu's proposition 4.3.38
I have a concern about the first assertion in the proof of proposition 4.3.38 of Qing liu's "Algebraic Geometry and Arithmetic Curves". Referring to smooth morphisms, he says "The ...
- 149