# isomorphism of varieties

###### Definition 1.

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

 $\phi\colon V_{1}\to V_{2},\quad\psi\colon V_{2}\to V_{1}$

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

###### Definition 2.

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 $K$.

Title isomorphism of varieties IsomorphismOfVarieties 2013-03-22 15:06:22 2013-03-22 15:06:22 alozano (2414) alozano (2414) 4 alozano (2414) Definition msc 14A10 JInvariantClassifiesEllipticCurvesUpToIsomorphism