This question orginally comes from the exercise 3.18 of Hartshorne, Algebraic Geometry. If $Y$ is a projective variety in $\mathbb{P}^n$ then $Y$ is projectively normal (w.r.t the embedding) if its homogenoeus coordinate ring $S(Y)$ is integrally closed.

We can easily prove that if $Y$ is projectively normal then $Y$ is normal. However, the converse, as Hartshorne claimed, is not true. He gave as an example the twisted quartic in $\mathbb{P}^3$.

My question are :

  • How can I prove that the twisted quartic is not normal ? I attempt to show that its homogenoeus coordinate ring is not integrally closed, but I could not find the counter example. Is there other way to show that the twisted quartic is not normal ?
  • Is projectively normal important ? I searched google for the information about that property, and it gave me alot of information related to algebraic geometry. What is its role in algebraic geometry ?
  • Could you please show me some other example of normal variety in some projective space that is not projectively normal?

Thank for reading my question!

  • 2
    $\begingroup$ I have the feeling that you are talking about this ring. If it is so, then try to prove that $t/s$ is not integral over $K[s^4,s^3t,st^3,t^4]$. $\endgroup$ – user26857 Jan 27 '13 at 19:44
  • $\begingroup$ @YACP : Thank you very much ! $\endgroup$ – Arsenaler Jan 28 '13 at 8:09

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