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. |