= Transverse intersection of plane curves
{title2=$I_p(C,D)=1$}
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.
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