special $g_3^1$ on elliptic curve

Question

Let $C$ be a planar smooth cubic curve. We know that projection from a point $p$ that does not lie on $C$ produces a 3-to-1 map $\pi: C\to \mathbb P^1$, which has 6 ramification points in general.

Now, I suppose I choose the projection center $p$ to be special — it is the intersection of two flex lines of $C$. So, the projection $\pi:C\to \mathbb P^1$ has 2 total ramification points, $a$ and $b$, and 2 simple ramification points, $c$ and $d$.

Question: Is it true that $a+b=c+d$?

It seems to hold in a few examples that I computed — Working with explicit equations, I find the line $\overline{ab}$ through $a$ and $b$, which intersects $C$ at a third point $e$, and then computed that $c,d,e$ are collinear. However, I don't know if it is true in general, and if it is yes, is there a coordinate-free proof?

Answer 1

Yes, this is true. Let $H$ be the divisor class of a line on $C$. We have $H\equiv \pi ^*p$ for any $p$ in $\mathbb{P}^1$; in particular: $$H\equiv 3a \equiv 3b\,.$$ On the other hand, the Riemann-Hurwitz formula gives $$2H\equiv 2a + 2b + c +d\, ,$$ from which you get $$c+d-a-b\equiv 2H -3a-3b \equiv 0\,.$$