1
$\begingroup$

I am confused with general notion of integral models of algebraic varieties. Let us focus on, say, algebraic curves.

If $X$ is a not necessarily projective algebraic curve over a number field $K$, is the set $S$ of bad primes for $X$ intrinsic?

Here is an example for giving a feeling for my usage of the word 'intrinsic': let $E_1$ and $E_2$ be elltipic curves over $\mathbb{Q}$ defined by the equations $y^2=x^3+x$ and $v^2=u^3+16u$ respectively. Then $E_1\simeq E_2$ over $\mathbb{Q}$. But, if I did not make any mistake, it seems like that 2 is a bad prime for $E_2$ which is not for $E_1$. With this example, I think that the set of bad primes for a curve is intrinsic meaning that even though $X$ and $X'$ are isomorphic curves over $K$, the set of bad primes may not be the same. Am I right or am I confused with some part of the definition of the primes of bad reduction?

One further question is, let us say that $X/K$ be as before and $S$ the set of primes of bad reduction (well, I am implicitly assuming that $X$ is smooth). With this setting, I may be able to say that there exists a smooth $S$-integral model $\mathcal{X}$ of $X$. Then my second quesiton is

Let $\mathcal{X}'$ be another smooth $S$-integral model of $X$. What can we say difference of $\mathcal{X}$ and $\mathcal{X}'$? For example, can we say that $\mathcal{X}$ is isomorphic to $\mathcal{X'}$ up to extending $S$?

Improve this question
$\endgroup$
5
  • $\begingroup$ In your elliptic curve example, you are really talking about your curves as if they live over $\mathbb Z$, not $\mathbb Q$. I think for curves, it follows from Neron-Ogg-Shafarevich that there is a "best" possible S to take that is minimal and it's exactly those primes for which the galois rep on tate module of jacobian of curve is unramified. $\endgroup$
    – Asvin
    Commented Jun 22, 2021 at 4:53
  • $\begingroup$ For your second question, you can indeed say that they are isomorphic up to extending S points because they are isomorphic over the generic point and you can "spread out" generic isomorphisms. $\endgroup$
    – Asvin
    Commented Jun 22, 2021 at 4:54
  • $\begingroup$ @Asvin Thank you for your comment. Could you elaborate more on your second comment? I mean the "spreading out generic isomorphism" part. Any reference would be helpful for me. $\endgroup$
    – User0829
    Commented Jun 22, 2021 at 5:00
  • 1
    $\begingroup$ I think Poonen's "Rational points on varieties" has a section on spreading out. The basic idea is simply that if you have a finite type morphism defined over $\mathbb Q$, there can only be finitely many primes in all the denominators and as long as you avoid those, you can extend integrally. And then you need to show that various things can be checked generically (like being an isomorphism). $\endgroup$
    – Asvin
    Commented Jun 22, 2021 at 5:09
  • $\begingroup$ @Asvin Thank you very much! $\endgroup$
    – User0829
    Commented Jun 22, 2021 at 5:10

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .