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 Yearling
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Apr
24
revised Irreducible algebraic sets via irreducible polynomials
added 181 characters in body
Apr
24
answered Irreducible algebraic sets via irreducible polynomials
Apr
23
comment Number of rational points in a non-smooth variety
Yes a similar property holds whenever $X$ is geometrically integral. Search google for the "Lang-Weil estimate".
Apr
18
comment A cyclic subgroup as a decomposition group
Yes, this is a simple application of the Chebotarev density theorem.
Apr
2
awarded  Yearling
Mar
31
answered Equation with norms of cyclic extensions of coprime degrees
Mar
16
comment Is it true that singular fibers of elliptic fibrations that have the same Kodaira type are isomorphic schemes?
Good point, I had over-looked this case.
Mar
16
comment Is it true that singular fibers of elliptic fibrations that have the same Kodaira type are isomorphic schemes?
Anyway, for singular fibres, assuming that $S$ and $S'$ are smooth projective curves over an algebraically closed field $k$ of characteristic $0$, the answer should be yes. Most cases consist of a collection of smooth genus $0$ curves meeting in some configuration of points. As there is a unique smooth genus $0$ curve over $k$ up to isomorphism (namely $\mathbb{P}^1$), the result is clear. The remaining cases are that of a plane cubic nodal or cuspidal curve. Again, such a curve is unique over $k$ up to isomorphism, hence the result follows.
Mar
16
comment Is it true that singular fibers of elliptic fibrations that have the same Kodaira type are isomorphic schemes?
What is your definition of "special fibre"? For me, for some scheme over a spectrum of a DVR, the special fibre is the fibre over the closed point; this can certainly be smooth. What are your $S$ and $S'$? Smooth projective curves over an algebraically closed field?
Mar
16
comment Is it true that singular fibers of elliptic fibrations that have the same Kodaira type are isomorphic schemes?
Not in general: Just take two non-isomorphic smooth fibres. These have the same Kodaira type $I_0$.
Mar
14
revised Hilbert symbol averages
added 15 characters in body
Mar
14
revised Hilbert symbol averages
added 474 characters in body
Mar
14
answered Hilbert symbol averages
Mar
7
answered Classification of cubic surfaces in $\mathbb{P}^3$
Feb
29
answered Which groups are Galois over some p-adic field?
Feb
23
answered Convergence of zeta functions for schemes of finite type over the integers
Feb
16
awarded  Disciplined
Feb
11
comment Standard term for parametrisation where heights of parameters and values are correlated
Can you give some examples to illustrate the kind of phenomenon you are interested in? At the moment the question is very vague. I think it seems doubtful there could be a reasonable answer - problems about the height of the smallest rational point or counting rational points of bounded height can be incredibly difficult, even for rational varieties. A prime example being cubic surfaces (yes, even the rational cubic surfaces are incredibly difficult for such problems).
Feb
8
comment splitting property of etale covering
For a counter-example to Q1, why not take $X = \mathrm{Spec}(\mathbb{Z}_p)$ and $Y \to X$ a totally ramified non-abelian Galois covering? Due to being totally ramified, the fibre over the closed point is just the the trivial extension of residue fields, hence clearly split.
Feb
3
comment Examples of NIP fields of characteristic $p$
This question might attract more interest if you included the definition of a NIP field.