isomorphism of varieties

Definition 1.

Let $V_{\mathrm{1}}$ and $V_{\mathrm{2}}$ be algebraic varieties. We say that $V_{\mathrm{1}}$ and $V_{\mathrm{2}}$ are isomorphic, and write $V_{\mathrm{1}}\mathrm{\cong }{V}_{\mathrm{2}}$, if there are regular maps

$$\varphi :V_1\to V_2,\psi :V_2\to V_1$$

such that the compositions $\psi \mathrm{\circ }\varphi$ and $\varphi \mathrm{\circ }\psi$ are the identity maps on $V_{\mathrm{1}}$ and $V_{\mathrm{2}}$ respectively.

Definition 2.

Let $V_{\mathrm{1}}$ and $V_{\mathrm{2}}$ be varieties defined over a field $K$. We say that $V_{\mathrm{1}}\mathrm{/}K$ and $V_{\mathrm{2}}\mathrm{/}K$ are isomorphic over $K$ if $V_{\mathrm{1}}$ and $V_{\mathrm{2}}$ are isomorphic as in Definition 1 and the regular maps $\varphi$ and $\psi$ can be defined over $K$.