Suppose $X$ is a projective smooth, geometrically connected, hyperelliptic curves over a field k, we may ask $X(k)\neq\varnothing$. I want to know how to compute the $H^{0}(X,\Omega\_{X}^{1})$.


And there is theorem of Noether, which claims that:
Let $X$ be a non-hyperelliptic curve, then the map:
$H^{0}(X,K\_{X})^{\otimes2}\rightarrow H^{0}(X,K\_{X}^{\otimes 2})$ is surjective.


Can someone explain intuitively why the theorem is true and why hyperelliptic curves are excluded.



kahler differential on hyperelliptic curves
I just wanted to mention another way to recover the basis of holomorphic differentials and the failure of surjectivity given in the comments : an hyperelliptic curve of genus $g\geq 2$ can be expressed as a 2-to-1 cover of $\mathbb{P}^1$ ramified along $2g+2$ points, $f\colon X\to \mathbb{P}^1$. In particular, using double covers formulas we see that
$$ f\_\*\mathcal{O}\_X = \mathcal{O}\_{\mathbb{P}^1}\oplus \mathcal{O}\_{\mathbb{P}^1}(-g-1) \qquad K\_X \cong f^\*K\_{\mathbb{P}^1}\otimes f^\*\mathcal{O}\_{\mathbb{P}^1}(g+1) \cong f^\*\mathcal{O}\_{\mathbb{P}^1}(g-1) $$
so that
$$ H^0(X,K\_X) = H^0(\mathbb{P}^1,f\_\*K\_X) = H^0(\mathbb{P}^1,\mathcal{O}\_{\mathbb{P}^1}(g-1)\otimes f\_\*\mathcal{O}\_X) = H^0(\mathbb{P}^1,\mathcal{O}\_{\mathbb{P}^1}(g-1)) $$
and by the same reasoning
$$ H^0(X,K^2\_X) = H^0(\mathbb{P}^1,f\_\*K^2\_X) = H^0(\mathbb{P}^1,\mathcal{O}\_{\mathbb{P}^1}(2g-2)\otimes f\_\*\mathcal{O}\_X) = H^0(\mathbb{P}^1,\mathcal{O}\_{\mathbb{P}^1}(2g-2)) \oplus H^0(\mathbb{P}^1,\mathcal{O}\_{\mathbb{P}^1}(g-3)) $$
Hence, the multiplication map $H^0(X,K\_X)^{\otimes 2}\to H^0(X,K^2\_X)$ can be interpreted as
$$ H^0(\mathbb{P}^1,\mathcal{O}\_{\mathbb{P}^1}(g-1))^{\otimes 2} \to H^0(\mathbb{P}^1,\mathcal{O}\_{\mathbb{P}^1}(2g-2)) \oplus H^0(\mathbb{P}^1,\mathcal{O}\_{\mathbb{P}^1}(g-3)) $$
which is clearly not surjective as soon as $g\geq 3$, whereas it is surjective for $g=2$. However, one can prove in the same way that the map $H^0(X,K\_X)^{\otimes 3} \to H^0(X,K\_X^{\otimes 3})$ is not surjective when $g=2$.




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An argument for surjectivity can be found in Arbarello-Cornalba-Griffiths-Harris in their discussion of extremal curves.


