I'm reading Klaus Hulek's algebraic geometry and have encountered this. It is the page 124 of the book and C means irreducible singular cubic plane curve.
Here the sentece says if C has more than two singularities, the line through two singular points intersect C in at least 4 points. Why is it so? Could anyone explain?
Best Answer
The explanation is already written in that text. The intersection multiplicity of the curve with a line passing through a singular point is larger than one ($\geq 2$). If you take a line passing through both singularities (assuming more than one) then you have $\geq 4$ multiplicity. But for a cubic we must have $\leq 3$.
The counting of $4$ is including multiplicity. So the intersection of $y=x^3$ and $y=0$ is one point (but $3$ counting multiplicity).