I think dimension of a quasi-projective variety is greater than transcendence degree of its regular function ring but I recently learn there are quasi projective variety with non-finitely generated ring of functions.can you give an example of a quasi projective variety $X$ such that $\dim(X)<\operatorname{tr.deg}(k[X])$

  • $\begingroup$ Rephrasing my earlier comment to: How do you define the "regular function ring" of a quasiprojective variety? This is something usually defined for affine or projective varieties, and the two notions differ accordingly. $\endgroup$ – Jesko Hüttenhain Apr 28 '18 at 13:08
  • $\begingroup$ a regular function is a rational function defined everywhere.(a quasiprojective variety sits in a projective variety and you can diffine rational function on it) $\endgroup$ – ali Apr 29 '18 at 7:52

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