Multiplicity of a plane curve at a point ($m_P(C)=\min \{i+j:a_{ij}\ne 0\}$)

Translate the point to the origin of an affine chart and expand a local defining polynomial as $\sum a_{ij}{x}^i{y}^j$. Its lowest nonzero total degree is the multiplicity. A smooth point of a variety has multiplicity one; a double point has multiplicity two. On a point blowup of a smooth algebraic surface, the total transform is the strict transform of an algebraic subvariety plus this multiplicity times the algebraic exceptional divisor.