A nodal projective curve in $\mathbb{CP}^2$ inherits a Kähler metric from the Fubini-Study metric, and hence a Riemannian metric. In particular, with respect to this metric, a line has constant Gaussian curvature 1; according to Vitter (page 826), the Fermat conic $\{x^2+y^2+z^2=0\}$ has constant curvature 1/2, but no other Fermat curve has constant curvature. In fact, Vitter explicitly computes the Gaussian curvature of the zero set of a given homogeneous polynomial.

Does there exist a flat plane cubic?

More generally,

Fix a degree $d>2$: does there exist a smooth, degree-$d$ plane curve with constant Gaussian curvature? Does there exist an irreducible degree-$d$ nodal curve with constant Gaussian curvature?

If such a curve existed, and it had total Milnor number $\delta$, its curvature would be $(3-d)/2 + \delta/d$.