I've been studying the basics of Algebraic Geometry for coding theory using the Pless-Huffman book. However since this is mostly self study, and without good resources I still feel a little shaky on the subject as a whole. So I'm attempting to work through an example from the book and see how it works out, with hopefully some help.
Statement: Let X be the Fermat Curve over $F_2$ of genus 1 defined by $x^3+y^3+z^3=0$. Let $P_\infty$=(1:1:0) and $P^1_\infty$={$(\omega:1:0),(\overline\omega:1:0)$} be the points at infinity on X. Let D= k($P_\infty + P^1_\infty$) for some positive integer k.
Question1: Show that dim(L(D))=3k
Question2: Find the intersection divisor of the Fermat Curve with the curve x=0.
Question3: Find the intersection divisor of the Fermat Curve with the curve y=0.
Question4: Find the intersection divisor of the Fermat Curve with the curve z=0.
Question5: Compute ($x^i y^j$/$z^i z^j$) Where i and j are non-negative integers.
Question6: What is the restriction on div($x^i y^j$/$z^i z^j$ so that it is in L(D)
Attempt:
Q1: By Riemann-Roch theorem if deg(D)>2g-2 => dim(L(D))=deg(d)-g+1 where g is the genus. Since the genus is 1. deg(D)>2(1)-2 or deg(D)>0 using D from above we get deg(k$P_\infty$)+deg(k$P^1_\infty$)=k(1)+k(2). Since k>0 => deg(d)>0. So we can use the Riemann-Rock theorem.
So dim(L(D))=deg(k$P_\infty$)+deg(k$P^1_\infty$)-g+1 =>k+2k-1+1=3k.
Q2:The curve x=0 does not meet with the divisor(s) of the Fermat Curve. So by Bezout's Theorem the intersection divisor should be zero.
Q3: The curve y=0 again does not meet with the divisor(s) of the Fermat Curve. So by the same theorem the intersection divisor should be zero.
Q4: The curve z=0 meets both the divisors of the Fermat Curve. So then the intersection divisor should be $deg_D * D$ = 1$P_\infty$ + 2$P^1_\infty$ (Not too sure on this one due to multiplicity and degree)
Q5:div($x^i y^j$/$z^i z^j$) works a lot like a log from the examples in the book. So I get div($x^i y^j$/$z^i z^j$)= div($x^i y^j$) - div($z^i z^j$) = 0 - (i)div(z)+(j)div(z) (Not sure about this part since the examples in the book replace div(z) with the answer from Q4) => -(i+j)($P_\infty$+2$P^1_\infty$)
Q6: Since I am not sure about Q5, I cannot be too sure about Q6. But we look for when div($x^i y^j$/$z^i z^j$)+D is effective. But if Q5 is right that means that k$P_\infty$ + k$P^1_\infty$ -(i+j)($P_\infty$+2$P^1_\infty$)$\geq$0
So if at all possible I'd like some direction on Q4-Q6. Or corrections if I am doing it incorrectly.
Question 1 is fine.
Question 2 forgets the fact that there exist 9 projective points over $F_2$.
Namely:
(1:1:0) (1:$\omega$:0) (1:$\overline\omega$:0)
(1:0:1)(1:0:$\omega$)(1:0:$\overline\omega$)
(0:1:1)(0:1:$\omega$)(0:1:$\overline\omega$)
So the correct answer is (0:1:1)+(0:1:$\omega$)+(0:1:$\overline\omega$)
Similarly for question 3, and 4.
Question 5 starts out correctly with simplification.
div($x^i y^j / z^i z^j$) = i div(x)+ j div(y) - (i+t) div(z) where div(x) is the answer to 2, div(y) is the answer to 3, and div(z) is the answer to 4.
Question 6 again has the right idea, but the mistakes in 2-4 were messing up the results. div(x)+ j div(y) - (i+t) div(z) +D when solved shows that to be effective i,j $\geq$ 0 and k$\geq$i+j$\geq$0