Transverse intersection of plane curves ($I_p(C,D)=1$)

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.