ag.algebraic geometry - Existence of (smooth) models

I'm happy to present my example of a smooth projective surface $X$ over
$K=mathbb{Q}_p$ ($p$ prime) such that $X(K)neqemptyset$, whose
$l$-adic cohomology groups are unramified (for all primes $l$) and which still
has bad reduction : there is no smooth $mathbb{Z}_p$-scheme whose generic
fibre is $X$. (The method works for any finite extension of $mathbb{Q}_p$ and was worked out a few years ago.)

The surface $X$ is going to be a conic bundle over $mathbb{P}_1$ with four
degenerate fibres, so it is a rational surface in the sense of being $bar
K$-birational to $mathbb{P}_2$. It will be clear that the example is not
isolated.

If $p$ is odd, let $dinmathbb{Z}_p^times$ be a unit which is
not a square, and take $d=5$ if $p=2$, so that $K(sqrt{d})|K$ is the unramified quadratic extension.

Let $e_1, e_2$ be two distinct units of $K$. We take $X$ to be the surface in
$mathbb{P}({cal O}(2)oplus{cal O}(2)oplus{cal O})$ (coordinates $y:z:t$)
over $mathbb{P}_1$ (coordinates $x:x'$) defined by the equation
$$
y^2-dz^2=xx'(x-e_1x')(x-e_2x')t^2.
$$
I claim that this $X$ has all the properties stated above, if $v_p(e_1-e_2)>0$.

First, $X(K)neqemptyset$ because each degenerare fibre is a pair of
intersecting lines conjugated by $mathrm{Gal}(bar K|K)$.

Secondly, the $l$-adic cohomology is unramified because the action of
$mathrm{Gal}(bar K|K)$ on the Picard group $mathrm{Pic}(bar{X})$ of $bar
X=Xtimes_Kbar K$ factors via the quotient $mathrm{Gal}(K(sqrt{d})|K)$.

Finally, $X$ has bad reduction because its Chow group $A_0(X)_0$ of $0$-cycles of degree
$0$ is $mathbb{Z}/2mathbb{Z}$ (cf. prop. 1 of arXiv:math/0302156), and a theorem of Bloch (th. 0.4, On the Chow groups of certain
rational surfaces, Annales scientifiques de l'École Normale Supérieure, Sér.
4, 14 no. 1 (1981), p. 41-59, available at Numdam) asserts that if a conic bundle has good
reduction, then its Chow group of $0$-cycles of degree $0$ is $0$.

Addendum (in response to a question in an email I received). One can show moreover that no smooth projective surface $Y$ over $mathbf{Q}_p$ which is $mathbf{Q}_p$-birational to $X$ can have good reduction. This follows from the facts recalled above and the theorem of Colliot-Thélène and Coray (which can be found in Fulton's Intersection theory) : $A_0(Y)_0$ is isomorphic to $A_0(X)_0$.