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Suppose $X$ is a smooth projective variety over $\mathbb C$. Then under what conditions, the natural map $H^0(X,mK_X) \otimes H^0(X,nK_X) \to H^0(X,(m+n)K_X)$ for $m, n \in \mathbb{Z}_{>0}$ is surjective?

My case is particularly simple: $X$ is a smooth curve of genus greater than $1$, and I wish $\otimes_{i=1}^m H^0(X, K_X) \to H^0(X, mK_X)$ to be surjective when $m \geq 3$.

However, I was unable to show this or give a conterexample.

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I assume you mean $H^0(X, K_X)^{\otimes m}$ rather than $\oplus_{i=1}^m H^0(X, K_X)$. If $X$ is a smooth projective connected complex curve of genus $g \geq 2$, then the map $$H^0(X, K_X)^{\otimes m} \longrightarrow H^0(X, m K_X),$$ is surjective for any $m \geq 0$, as long as $X$ is not hyperelliptic. This is a theorem of M. Noether.

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  • $\begingroup$ Great! Could you also point out the reference for this result? $\endgroup$ – Li Yutong Oct 5 '16 at 12:36
  • $\begingroup$ The original proof is in : M. Noether, "Uber die invariante Darstellung algebraicher Funktionen", Math. Ann., 17, 263–284 (1880). For a more modern proof, see p117 of Arbarello-Cornalba-Griffiths-Harris, "Geometry of algebraic curves", Springer-Verlag (1985). $\endgroup$ – js21 Oct 5 '16 at 12:43
  • $\begingroup$ A different proof can also be found in Stohr&Viana, "A variant of Petri’s analysis of the canonical ideal of an algebraic curve". Here, "Petri's analysis" refers to a result of Petri describing the kernel of the $m$-th map. $\endgroup$ – js21 Oct 5 '16 at 12:52
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In the case where $X$ is a smooth curve of genus $g\geq 4$, this map fails to be surjective for all $m\geq 2$, for reasons of dimension. On the one hand, Riemann-Roch shows that $h^0(X,mK_X) = (2m-1)(g-1)$ for $m\geq 2$. On the other hand, $\bigoplus_{j=1}^m H^0(X, K_X)$ has dimension $mg$; this less than $(2m-1)(g-1)$ as long as $g\geq 4$, I believe. When $g=3$, the same argument should show that the map is not surjective provided $m\geq 3$.

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  • $\begingroup$ Sorry, I should write $\otimes$ instead of $\oplus$. $\endgroup$ – Li Yutong Oct 5 '16 at 12:37

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