algebraic geometry ag – Hypereliptic curve (Book of Liu)

Leave $ Y: = mathbb {P} ^ 1_k $ Y $ X $ A hypereliptic curve. We work with definition 4.7 of "Algebraic geometry and arithmetic curves" of Liu (page 288)

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Namely there finite separable Map $ pi: X a Y $ of degree $ 2 $.

I have a Question About the step labeled in the Lemma 7.4.8 test:

The rational point $ y_0 $ of $ mathbb {P} ^ 1_k $ defines a Cartier divider (resp. the inverted cover corresponds to $ O_Y (y_0) $) and let $ D: = pi ^ * y_0 in Div (X) $ it's low recoil $ pi $.

We denote by $ O_X (D) $ the invertible sheaf corresponding to $ D $ and keep in mind that the author uses the notation $$ L (D) = H ^ 0 (X, O_X (D)) $$.

My question is why under the donation environment we have an inclusion

$$ H ^ 0 ( mathbb {P} ^ 1_k, O _ { mathbb {P} ^ 1_k} (y_0)) subset H ^ 0 (X, O_X (D)) $$

as indicated in the extract?

My idea was to use the fact that the complement between $ pi ^ * $ Y $ pi _ * $ gives rise to the morphism of $ O_Y $ modules

$$ O_Y (y_0) to pi_ * pi ^ * O_Y (y_0) = pi_ * O_X (D) $$

My question is whether and why in this context this map is injective. If this were true, then the exact left $ H ^ 0 (-) $-Functor would preserve the injectivity and we're done.

Or does Liu have a simpler argument in mind here? since Liu has not mentioned that in the previous excerpt that makes me assume that there is an easier reason for this inclusion