Here, a surface is a nonsingular projective surface over an algebraically closed field, and a curve is any effective divisor on that surface.
Let X be surface. Let C and D be two curves on X, and let $P \in C \cap D$ be a point of intersection of C and D. C and D are then said to meet transversally at P if the local equations $f, g$ of C and D on X are generated the maximal ideal $\mathfrak{m}_{P}$ of $\mathcal{O}_{P,X}$.
I understand what this is saying, but I am currently struggling with coming up with any explicit examples… Could anyone provide any explicit examples of such a transversal intersection?
Thank you in advance 🙂
Best Answer
There are various ones (distinct lines on the projective plane is definitely a class of examples), but it is worth to notice that generally two intersecting curves will usually intersect transversally. If you want to understand the definition, it is more instructive to look for non-transversal examples.
One such is the intersection of $y=0, y=x^2$ on the affine plane.