2 pctn

# Good reduction and blow-ups.

Let $X$ be a projective variety over $\mathbb{Z}$, and suppose that $X$ has everywhere good reduction. Let $Y$ be the blow-up of $X$ at an integral point.

Then is it the case that $Y$ also has everywhere good reduction?

The example situation that I have in mind is the following (My my main motivation is del Pezzo surfaces).

Take $X= \mathbb{P}^2$ over $\mathbb{Z}$. This clearly has good reduction everywhere. Next let $Y$ be the blow-up of $\mathbb{P}^2$ at the integral point $(0:0:1)$. This can be realised as the subvariety of $\mathbb{P}^2 \times \mathbb{P}^1$ (with variables $x_0,x_1,x_2$ and $y_1,y_2$) given by the equation $x_1 y_2 = x_2 y_1$. Then $Y$ has everywhere good reduction (at least if my caluclations are correct).

I am curious to know if this happens after successively blowing up more integral points to obtain other del Pezzo surfaces. Note however that I am not claiming that all del Pezzo surfaces have everywhere good reduction!

Thanks in advance!

1

# Good reduction and blow-ups.

Let $X$ be a projective variety over $\mathbb{Z}$, and suppose that $X$ has everywhere good reduction. Let $Y$ be the blow-up of $X$ at an integral point.

Then is it the case that $Y$ also has everywhere good reduction?

The example situation that I have in mind is the following (My main motivation is del Pezzo surfaces).

Take $X= \mathbb{P}^2$ over $\mathbb{Z}$. This clearly has good reduction everywhere. Next let $Y$ be the blow-up of $\mathbb{P}^2$ at the integral point $(0:0:1)$. This can be realised as the subvariety of $\mathbb{P}^2 \times \mathbb{P}^1$ (with variables $x_0,x_1,x_2$ and $y_1,y_2$) given by the equation $x_1 y_2 = x_2 y_1$. Then $Y$ has everywhere good reduction (at least if my caluclations are correct).

I am curious to know if this happens after successively blowing up more integral points to obtain other del Pezzo surfaces. Note however that I am not claiming that all del Pezzo surfaces have everywhere good reduction!

Thanks in advance!