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location  Rennes, France  
age  63  
visits  member for  4 years, 8 months 
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6h

comment 
Is surjectivity for morphisms of schemes local on the domain?
It is wrong indeed. 
1d

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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

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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 
Proconstructible subset of scheme intersects very dense subsets?
@user68750: It is the union of all proper closed subsets. 
Feb 26 
comment 
Proconstructible subset of scheme intersects very dense subsets?
If $X$ is integral and Jacobson, the generic point is proconstructible 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 
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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 
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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. 