two plane curves intersecting in a single point

Question

do there exist plane curves $f,g\in\mathbb{C}[x,y,z]_d,\ d\ge 2$ that intersect in exactly one point in $\mathbb{P}^2$? Taking both to be products of $d$ distinct lines obviously works, but I cant come up with anything else.

I feel like this should just be obvious whether it exists or not.

Thx for any help

Answer 1

Of course they do exist. Consider, for instance, the curves $f=x^d+yz^{d-1}$ and $g=x^d+yz^{d-1}+z^d$.