I have a few doubts about some algebraic geometry problems. This is the situation: X is a (smooth) curve on the projective plane $P^2(F)$, F being an algebraic closure of $F_2$ $X = Z(x_{1}^2x_{2}+x_1x^{2}_{2}+x^{3}_{0}+x^{3}_{2}$. $Y = X\cap U_2 = Z(y^2 + y + x^3 + 1)$ is affine. The point at infinity of $X$ is $(0:1:0)$. $X_0$ is the variety seen as defined over $F_2$ (this is not X ∩ P^2(F_2)). First I don't understand very well the distinction between X and $X_0$.. since $X$ is already given as zero-set of a polynomial with coefficient in $F_2$. The first question i can't answer is to list all the positive divisor of degree 2 on $X_0$. Two examples are: $2\infty, (0,z)+(0,z^2)$ where $z$ is such that $z^2+z+1=0$.
The second question is to find the divisor of the functions $x, y, x+1, y+1, x+y, x+y+1$. From theory i know that such divisors are supposed to have degree 0, but the divisor i've found have not. Isn't $(0:z:1) + (0:z^2:1) - \infty$ the divisor of $x$?
Please, somebody help me. Claudia