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Let $k$ be a field, $X$ and $Y$ be $k$-schemes, $X\rightarrow Y$ be a $k$-open immersion with dense image. Assume that $Y$ is an integral scheme. Suppose that:

  • $X$ is geometrically connected, is $Y$ geometrically connected then?
  • $X$ is geometrically reduced, is $Y$ geometrically reduced then?
  • $X$ is geometrically integral, is $Y$ geometrically integral then?
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