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Let $k$ be any field, and we give $k^3$ the Zariski topology.

Then the question is how to compute the Zariski closure of the set $S=\{(x,y,z)\in k^3|xz=y, x+1=z^2, x\neq0\}$ in $k^3$.

The motivation is that I want to compute the blowup of the variety $\{(x,y)\in k^2|x^3+x^2=y^2\}$ at the point $(0,0)$.

Remarks: I have seen this question and this question. I also want more examples of computing the Zariski closure.

Merci beaucoup !

  • I assume you mean $k$ is any infinite field? If $k$ is finite then every set, including $S$ itself, is Zariski closed. For an infinite field, note that $S$ is just two points less than the clearly closed set ${(x,y,z)\in k^3\colon xz=y,, x+1=z^2}$ (those two points being $(0,0,\pm1)$; does that suggest what the Zariski closure of $S$ might be? – Greg Martin Aug 04 '19 at 07:47

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