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A smooth complex degree-two projective plane curve is a projective linear image of the degree-two Veronese map and is diffeomorphic to a sphere. Its class in ${\mathbb{CP}}^2$ is twice the class of a projective line. Its self-intersection number is four, so its oriented normal bundle has Euler class evaluating to four.

For a projective plane curve $C=V(F)$ and a nonsingular projective point $p$, the tangent is the line $F_x(p)x+F_y(p)y+F_z(p)z=0$. The coefficients are not all zero, and the Euler theorem for homogeneous functions makes this line pass through $p$. On an affine chart this is precisely the first-order part of a local defining equation, so the definition agrees with the usual tangent line and is invariant under projective transformations.

Two projective plane curves with no common irreducible component meet transversely at $p$ if both are nonsingular there and their projective tangent lines differ. This holds exactly when their intersection multiplicity is one. Indeed, in the local ring of the ambient smooth surface, independent linear terms of the two equations generate the maximal ideal and leave a quotient of length one. Conversely, length one forces those linear terms to span the two-dimensional cotangent space, proving both nonsingularity and distinctness of the tangents.