My questions refer to two steps in the proof of Prop. 7.4.1 in Liu's "Algebraic Geometry" (page 285):
What does he mean by CANONICAL divisor $K$ of $X$? so, futhermore why does $deg (D) = deg(-K)=2$ hold?
That's not clear to me why the equation $l(D-2x_1)=1$ holds,
where $x_1 \in X(k)$ is a $k$-rational point of $X$ and $l$ is defined by
$l(D):= dim_k H^0(X, \mathcal{O}_X(D))$ (compare with page 279: Definition 3.19).
My attempts:
I want to apply the formula of Theorem 3.17 (page 279, too) which provides the equation
$\chi(\mathcal{O}_X(D-2x_1))= deg(D-2x_1)+\chi(\mathcal{O}_X)$
where $\chi_k(\mathcal{F}) := \sum _{i \ge 0} (-1)^i dim_k H^i(X, \mathcal{F})$ is the Euler-Poincare characteristic of a sheaf $\mathcal{F}$.
So I can reduce my problem to calculating $deg(D-2x_1)$ and $\chi(\mathcal{O}_X)$:
ad $deg(D-2x_1)$: Definition 7.3.1 at page 275 says
$deg_k(D-2x_1) := \sum_x mult_x(D-2x_1)[k(x):k]= deg_k(D) -2[k(x_1):k]$
So if we have $deg(D) =2$ (why? see 1. question), then I need $[k(x_1):k]=1$. But why does this hold, or equivalently why the residue field $k(x_1)$ isn't a true extension of $k$?
ad $\chi(\mathcal{O}_X)$:
Does $X$ have the arithmetic genus $g=0$? (the arithmetic genus $g$ is defined by $g:= 1 -\chi_k(\mathcal{O}_X)$ (see page 279).
Futhermore (if I would be able to get $\chi(\mathcal{O}_X(D-2x_1))= deg(D-2x_1)+\chi(\mathcal{O}_X)= 0 +1 =1$ then I have to conclude from $1=\chi(\mathcal{O}_X(D-2x_1))= dim_k H^0(X, \mathcal{O}_X(D-2x_1))- dim_k H^1(X, \mathcal{O}_X(D-2x_1))$ that finally $l(D-2x_1)=dim_k H^0(X, \mathcal{O}_X(D-2x_1))=1$ holds. How?
Here the page 275 with the used definitions:
