bio | website | |
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location | Rennes, France | |
age | 63 | |
visits | member for | 4 years, 9 months |
seen | 8 hours ago | |
stats | profile views | 4,892 |
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 |
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Writing down gerbes explicitly over the projective line
Try the gerbe of square roots of $\mathscr{O}(1)$. |
Mar 30 |
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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 |
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Is surjectivity for morphisms of schemes local on the domain?
It is wrong indeed. |
Mar 26 |
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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 $X\to Y$. |
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 |
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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 |
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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 |
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Covering of schemes and flatness
I suspect there is a ground field. Is it assumed algebraically closed? |
Mar 10 |
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What is the meaning of “field of constants”?
In this paper at least, my guess is the same as yours. |
Mar 9 |
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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 |
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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 |
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Pro-constructible subset of scheme intersects very dense subsets?
@user68750: It is the union of all proper closed subsets. |
Feb 26 |
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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” |