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I am referring to Marten's theorem on the dimension of $W_d^r $ as in ACGH p. 192 . It seems to me that an even shorter proof can be given using Hopf's Theorem that if $\nu : A \otimes B \to C $ is any map of vector spaces A,B,C that is injective on each factor separately (over k algebraically closed) that one has $\dim(\nu(A \otimes B) \geq \dim(A) + \dim(B) - 1 $ . This can also be found on p. 108 of ACGH. It seems that Marten's result follows directly upon putting $ A = H^0(C,L), B = H^0(C, K_C \otimes L^{-1} \mbox{ and } C = H^0(K_C)$. Does anyone have a reference in the literature for a proof along those lines or perhaps am I making a silly mistake ?

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    $\begingroup$ what Hopf lemma do you refer to? $\endgroup$
    – roy smith
    Jun 29 '15 at 6:06
  • $\begingroup$ I've revised the question to include the statement of Hopf's Theorem, not Hopf's lemma as I originally wrote. $\endgroup$
    – meh
    Jun 29 '15 at 13:33

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