= Rationality of a plane quartic with three double points
Three distinct points of multiplicity two use the entire <arithmetic genus> three of an <integral projective curve> of degree four. One point blowup at each gives <arithmetic genus> zero. The <normalization of an algebraic curve> then has genus zero and is the <projective line> over an <algebraically closed field>, by <rationality of a smooth projective genus-zero curve>. This allows double points with one branch as well as ordinary nodes.
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