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location Rennes, France
age 63
visits member for 4 years, 9 months
seen 8 hours ago

Apr
13
answered Fibrations of projective varieties
Apr
11
comment Does this $\mathbb{Z}_p$-algebra morphism induce a closed immersion on the generic fiber?
Take $R=\mathbb{Z}_p$, $B=$ a nontrivial finite extension of $\mathbb{Q}_p$, $f$ mapping $x$ to zero. The image of $f[1/p]$ is $\mathbb{Q}_p$ but $f\bmod p$ is surjective since $B/(p)$ is zero.
Apr
6
comment Is a normal proper relative curve over a DVR projective?
If I remember correctly, Artin's contraction results give rise to algebraic spaces, not schemes (except, for instance, over a finite field). So, what kind of ``curve'' are we looking for?
Apr
2
comment Writing down gerbes explicitly over the projective line
Try the gerbe of square roots of $\mathscr{O}(1)$.
Mar
30
comment When does an algebraic space that is a torsor over a scheme have to be a scheme?
The paper cited by Ariyan (which I have not read) is presented as a simplified version of the example in Raynaud's thesis (LNM 119, XIII.3.2). See also loc. cit. VI.2.5 for Q3: if $S$ is normal and noetherian and $G$ is smooth with connected fibers on $S$, then every $G$-torsor is locally quasiprojective over $S$.
Mar
27
comment Is surjectivity for morphisms of schemes local on the domain?
It is wrong indeed.
Mar
26
comment How singular can the Stein factorization of a proper map between smooth varieties be?
Moreover, I don't see why $X\to\hat{X}\to Y$ is the Stein factorization of $X\to Y$.
Mar
26
answered How singular can the Stein factorization of a proper map between smooth varieties be?
Mar
25
comment Connectedness of fibers for flat, proper morphism
I agree with abx's comment if by "connected" you mean "geometrically connected".
Mar
24
answered Adeles and twisted adeles
Mar
22
comment Number of elements in a fiber
@ulrich: Normality guarantees that $\mathrm{Spec}\,(A')$ is irreducible, which is necessary to conclude that it is dominated by each $\mathrm{Spec}\,(B_i)$. So, I guess "$A$ unibranch" would suffice.
Mar
21
comment Number of elements in a fiber
Other method: by Zariski's Main Theorem we may assume $B$ finite over $A$. Let $A'$ be the completion of $A$ and $B'=A'\otimes_A B$. Then $B'$ is a product of $r$ complete noetherian local rings $B_i$ with $\dim B_i=\dim B=\dim A$. Clearly $r$ is the number of points in the closed fiber, and for dimension reasons each $\mathrm{Spec}\,(B_i)$ must dominate $\mathrm{Spec}\,(A)$, hence $r\leq d$.
Mar
13
comment Covering of schemes and flatness
I suspect there is a ground field. Is it assumed algebraically closed?
Mar
10
comment What is the meaning of “field of constants”?
In this paper at least, my guess is the same as yours.
Mar
9
comment Is every surjective, birational transformation of projective varieties automatically proper?
If $\phi$ is proper, then $U$ is proper (because $Y$ is), so $U=X$.
Mar
3
comment Definition of internal field objects
Never thought about this before, but it does sound right. So, for instance, for any scheme $S$, $\mathbb{A}^1_S$ is a field object in the category of $S$-schemes.
Feb
26
comment Pro-constructible subset of scheme intersects very dense subsets?
@user68750: It is the union of all proper closed subsets.
Feb
26
comment Pro-constructible subset of scheme intersects very dense subsets?
If $X$ is integral and Jacobson, the generic point is pro-constructible and the set of closed points is very dense.
Feb
25
answered Implicit Function Theorem on Singular Varieties
Feb
24
answered reference for “curves over S are locally the base change of a curve over S' which is finite type over R”