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

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

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