(At least) In 2.8 Group law on a plane cubic of the book, he repreatedly used the notation | inbetween curves, e.g., $F | L$ where F is a curbic curve and L is a line in the projective space, but what does it mean?
I was trying to understand it as the usual notion of divisibility (so I read F|L as F divides L) but it doesn't make sense...
My guess
In Sec 2.12, he goes on as follows;
So, I started guessing it means the intersection, i.e., F intersects L?
This is restriction - you're restricting the equation $F=-Y^2Z+X^3+aXZ^2+bZ^3$ to the curve $L$ determined by the equation $Z=0$, which works by plugging in $Z=0$ to get $F|L = -Y^2\cdot 0 + X^3+aX\cdot 0^2 +b\cdot 0^3=X^3$. Frequently this has $L$ as a subscript, i.e. $F|_L$, but conventions are not quite universal.