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Hartshorne EX I 3.18 b

Define a curve by [t^4,t^3s,ts^3,s^4]. It is actually a P^1. Why the curve [t^4,t^3s,ts^3,s^4] is not projectively normal in P^3?

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    $\begingroup$ Please avoid even the appearance of a "homework problem". If you're posting an exercise, please explain what you've thought about so far, and what you're confused about. Don't just ask the exercise. $\endgroup$ Mar 14, 2010 at 6:36

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The problem is that this embedding is not by a complete linear system, since you are missing the coordinate $t^2s^2$. If we consider the 4-uple embedding of ${\bf P}^1$ inside of ${\bf P}^4$ given by the parametrization $[t^4 : t^3s : t^2s^2 : ts^3 : s^4]$, then it is projectively normal. And in general, if the embedding of a variety by a very ample line bundle $\mathcal{O}(1)$ is projectively normal, then the same is true for $\mathcal{O}(d)$ for positive $d$, provided that we use a complete linear system.

To see that the curve you mentioned is not projectively normal, call the coordinates $a,b,c,d$. Then $c^2/d = t^2s^2$ is in the field of fractions, and satisfies the equation $x^2 - ad = 0$, for example.

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