= Plane curve singularity multiplicity bound
{title2=$\sum_Pm_P(m_P-1)\le(d-1)(d-2)$}
Blow up each distinct singular point of an <integral projective curve> in the <projective plane> once. The <arithmetic genus drop under a point blowup> and the <genus-degree formula> give arithmetic genus $(d-1)(d-2)/2-\sum_Pm_P(m_P-1)/2$ for the strict transform. Its <arithmetic genus> is nonnegative, proving the bound in every characteristic. Infinitely near singularities need not have been resolved.
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