Let $k$ be an algebraically closed field. Let $S=k[x_0,\ldots, x_4]$ be the ring of polynomials. We set $S^i$ to the graded piece of degree $i$ polynomials. Let $H$ be a hyperplane of $S^5$ with no base point. Then how can I prove that $H$ generates $S^6$? namely $S^1\cdot H=S^6$. Moreover, is the conclusion true in general (for arbitrary variables and degrees)?
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1$\begingroup$ This should be Example 1.8.15. (Green’s theorem) in Lazarsfeld's Positivity I (my e-copy...it may be slightly different in the refs) and there are several far reaching generaizations. $\endgroup$– HaconCommented Sep 23, 2020 at 15:03
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$\begingroup$ @Hacon: Thank you for pointing out the reference. $\endgroup$– user96145Commented Sep 24, 2020 at 18:02
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