= Multiplicity of a plane curve at a point
{title2=$m_P(C)=\min\{i+j:a_{ij}\ne0\}$}
Translate the point to the origin of an affine chart and expand a local defining <polynomial> as $\sum a_{ij}x^iy^j$. 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>.
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