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I was reading the proof of the Proposition (3.4, IV) in Hartshorne's Algebraic Geometry book.

Proposition: Let $D$ be a divisor on a curve $X$. Then:

(a) the complete linear system $\vert D \vert$ has no base points if and only if for every point $P\in X$, $$ dim \vert D-P \vert = dim \vert D \vert - 1;$$ (b) $D$ is very ample if and only if for every points $P, Q\in X$(including the case $P=Q$), $$ dim \vert D-P-Q \vert = dim \vert D \vert - 2.$$

In the proof, I could not understand the line in the proof which says that if $D$ satisfies the condition (b), then we have $$ dim \vert D-P \vert = dim \vert D \vert - 1$$ for every $P\in X.$

(It may be silly question.)

I would be thankful if someone could elaborate this sentence.

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  • $\begingroup$ I don’t have my copy of Hartshorne on me, but if you’re comfortable with the proof of (a) then just use the fact that very ample line bundles are base point free, and apply (a). $\endgroup$ Jun 24 '18 at 14:02
  • $\begingroup$ For any divisor $D$ and any point $P$ (with no condition), you always have an inequality, $\dim |D|-1\leq\dim |D-P|\leq \dim |D|$. Use this with (b) to deduce what you want. $\endgroup$
    – Mohan
    Jun 24 '18 at 14:22

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