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Let $f:V\dashrightarrow W$ and $g:W\dashrightarrow X$ be two rational dominant maps of affine varieties, where dominant means $f(\mathrm{dom} \, f)$ dense in $W$ ang $g(\mathrm{dom} \, g)$ dense in $X$, with respect to Zariski topology. How can be proved that the composition $g\circ f$ is also dominant?

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up vote 4 down vote accepted

If $U\subseteq X$ is open and nonempty then $V:=g^{-1}(U)$ is nonempty ($g$ dominant) and open (rational maps are Zariski-continuous), then $f^{-1}(V)$ nonempty ($f$ dominant), hence in total $(g\circ f)^{-1}(U)$ nonempty, i.e. $g\circ f$ dominant.

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