the connection between f.g. field extensions and varieties;

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Is there a connection between finitely generated field extension and varieties .

Few days back, I asked this Question,

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One person answered this question, but sadly he is no longer active on this site.I am attaching the image of his answer.enter image description here

Kindly mention some reference so that I can understand this connection between finitely generated field extension and varieties.

I would be thankful if someone can explain or give idea about this connection.