# 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) 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