= Arithmetic genus drop under a point blowup
{title2=$p_a(C')=p_a(C)-m(m-1)/2$}
For an <integral projective curve> on a <smooth projective surface>, blowing up a point of multiplicity $m$ gives $C'=\pi^*C-mE$. The <intersection formula for blowing up a surface> and <canonical divisor formula for a surface blowup> yield $C'^2=C^2-m^2$ and $K_{X'}\cdot C'=K_X\cdot C+m$. Substitute both in the <arithmetic adjunction formula on a smooth surface>. This does not require the transformed <curve> to be smooth.
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