I'm self-studying Miranda's Algebraic Curves and Riemann Surfaces and am uncertain of how I'm supposed to solve problem V.2c on linearly equivalent divisors:
Let $X$ be the projective plane cubic defined by the equation $y^2z=x^3-xz^2$. Let $p_0=[0:1:0]$, $p_1=[0:0:1]$, $p_2=[1:0:1]$, $p_3=[-1:0:1]$. Show that $2p_0\sim2p_i$ for each $i$. Show that $p_1+p_2+p_3\sim3p_0$.
It's clear from Plücker's formula that $g(X)=1$ and hence $X$ is isomorphic to a complex torus $\mathbb{C}/L$, for some $L=\mathbb{Z}+\mathbb{Z}\tau$. By Corollary 2.9 two divisors $D_1,D_2$ on $\mathbb{C}/L$ are linearly equivalent $\iff deg(D_1)=deg(D_2)$ and $A(D_1)=A(D_2)$ for $A:$ the Abel-Jacobi map on $\mathbb{C}/L$.
The degrees of the divisors in the problem are obviously equal, but I'm not sure how to show $A(D_1)=A(D_2)$ If you know the group law on $X$ or have an explicit isomorphism between $X$ and $\mathbb{C}/L$ then you can compute $A$, but this is far from a trivial result and is not covered by the text. Is there a different approach to the problem that I'm missing?
-Steve
You don't need anything fancy to solve this: just use the fact that all the divisors $L \cap X$, where $L$ is any line in the plane, are linearly equivalent.
Let $L_i$ be the tangent line at the point $p_i$: the above remark shows that $$3p_o \sim 2p_i+p_0$$ for each $i$, as you want. (Note that $p_0$ is an inflection point!)
Finally, taking $L$ to be the line $y=0$ (which intersects $X$ in $p_1 \cup p_2 \cup p_3$), you get the last statement.