Regular maps between quasi-projective varieties defined over a global field

Question

Let $K$ be a global field, and $f: V \to W$ be a regular map, where $V,W$ are quasi-projective varieties, and everything is defined over $K$. Is it true that $f(V(O_v)) \subset W(O_v)$ for all but finitely many $v$? If yes, what is the idea of proof?

The way I understand V(O_v) is as follows. Let $V\subset\mathbb P^N$ be the the variety \begin{equation}\label{qvar} \left(\cap_{i=1}^r\{f_i=0\} \right)\setminus\left(\cap_{j=1}^s\{g_j=0\} \right) \end{equation} where $f_1,\ldots,f_r$ and $g_1,\ldots,g_s$ are homogeneous polynomials in $K[X_0,\ldots,X_N]$. Assume each polynomials is in $O_v[X_0,\ldots,X_N]$ with some coefficient lying in $O_v^*$. Then \begin{equation} V(O_v)=\left\{[x_0:\ldots:x_N] \in\mathbb P^N(K_v): \begin{array}{l} x_i\in O_v \text{ for all } i\\ x_{i_0}\in O_v^* \text{ for some } i_0\\ f_i(x_0,\ldots,x_N)=0 \text{ for all } i\\ g_j(x_0,\ldots,x_N)\in O_v^* \text{ for all } j \end{array} \right\}, \end{equation}

Although $f$ can be locally given by polynomial maps, I don't know how the condtions such as $g_j(x_0,\ldots,x_N)\in O_v^*$ can be preseved?

Answer 1

Let $V = \{x \in \mathbb{P}^n |\forall i \in I, f_i(x) = 0, \exists j \in J,g_j(x) \ne 0\}$ and $W = \{y \in \mathbb{P}^m |\forall i \in I', f'_i(x) = 0, \exists j \in J', g'_j(x) \ne 0 \} $ be quasi-projective varieties over a global field $K$ and $\phi: V \to W$ be a regular map defined over $K$. We show that, for all but finitely many places $v$ of $K$, $\phi(V(O_v)) \subset W(O_v)$. Here, $O_v$ is the ring of integers of the completion $K_v$ of $K$ at $v$ (assumed without loss of generality to be non-archimedian), $V(O_v) = \{x = (x_0:\ldots:x_n) \in V(K_v) | \forall i, x_i \in O_v, \exists i, x_i \in O_v^*, \exists j \in J, g_j(x) \in O_v^*\}$, and likewise for $W$.

We begin by covering $V$ by open subsets $U_{\ell} =\{x \in V | r_{\ell}(x) \ne 0\}$ such that, on $U_{\ell}$, $\phi(x) = (p_0^{(\ell)}(x):\ldots:p_m^{(\ell)}(x))$, where the $p_i^{(\ell)}$ are homogeneous polynomials with coefficients in $K$. The fact that the $U_{\ell}$ cover $V$ means that whenever $x \in \mathbb{P}^n$ is such that $f_i(x) = r_{\ell}(x) = 0 \forall i,\ell$, then we must have $g_j(x) = 0$ for all $j$. Thus, by the Nullstellensatz, $g_j^N = \sum u_{ij}f_i + \sum v_{\ell j}r_{\ell}$ for some polynomials $u_{ij},v_{\ell j}$ and integer $N >0$.

Now we use that, on $U_{\ell}$, $\phi(x) = (p_0^{(\ell)}(x):\ldots:p_m^{(\ell)}(x)) \in W$, so $g'_j(p_0^{\ell}(x):\ldots:p_m^{\ell}(x)) \ne 0$ for some $j \in J'$. So, for any $x \in P^n$ satisfying $f_i(x)=0, \forall i \in I, g'_j(p_0^{(\ell)}(x),\ldots,p_m^{(\ell)}(x)) = 0 \forall j \in J'$ we must have $g_j(x)r_{\ell}(x) = 0$ for all $j \in J$. Again, by the Nullstellensatz, for all $j \in J$ $(r_{\ell}g_j)^M = \sum z_{ij}^{(\ell)}f_i + \sum_{j \in J'} w_j^{(\ell)}g'_j(p_0^{(\ell)},\ldots, p_m^{(\ell)})$, for some polynomials $z_{ij}^{(\ell)},w_j^{(\ell)}$ and integer $M > 0$.

We now take the finite set of places $S$ of $K$ including all archimedian places and all places where some coefficient of one of the polynomials considered above is non-integral. Let $v \notin S$ be a place of $K$.

Let $a \in V(O_v)$, then $g_j(a)^N = \sum v_{\ell}(a)r_{\ell}(a)$. By assumption, for some $j$, $g_j(a)^N \in O_v^*$ and that implies that $r_{\ell}(a) \in O_v^*$ for some $\ell$. Using these $j,\ell$, we get $(r_{\ell}(a)g_j(a))^M \in O_v^*$ and therefore, for some $j\in J'$ $g'_j(p_0^{(\ell)}(a),\ldots,p_m^{(\ell)}(a)) \in O_v^*$. This implies that $g'_j(\phi(a)) \in O_v^*$ for some $j$ and that $p_i^{(\ell)}(a) \in O_v^*$ for some $i=0,\ldots,m$ and finally that $\phi(a) \in W(O_v)$, as desired.