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location Rennes, France
age 63
visits member for 4 years, 8 months
seen 7 hours ago

6h
comment Is surjectivity for morphisms of schemes local on the domain?
It is wrong indeed.
1d
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 $X\to Y$.
1d
answered How singular can the Stein factorization of a proper map between smooth varieties be?
2d
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”
Feb
24
comment reference for “curves over S are locally the base change of a curve over S' which is finite type over R”
More trivially, you can take for $S$ a disjoint union of points $s_n=\mathrm{Spec}(k_n)$ ($n\in\mathbb{N}$ and $k_n$ a field) and $X=\coprod_n X_n$ where $X_n$ is smooth projective of genus $n$ over $k_n$.
Feb
10
comment A quantity associated with an algebraic variete
I don't understand what you mean by "interpretation". The definition seems geometric enough to me.
Feb
10
comment Is it decidable whether a finite type scheme is proper?
I think any approach (including Jason's, which is quite natural) would involve deciding first whether $X$ is separated.
Jan
29
comment cohomological obstructions and rational points
@DanielLoughran: the Diophantine undecidability of $\mathbb{Q}$ is not known, although generally believed.
Jan
25
comment Extending a model to a given compactification of its generic fiber
@OlivierBenoist: right, but you just have to check that this $R$-scheme is separated, e.g. by the valuative criterion.