For real singular cubic threefolds with no real singular points, rationality is determined by singularity type and by the existence of real lines, planes, or cubic scrolls, with cohomological obstructions in four exceptional cases.
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Rationality of singular cubic threefolds over $\mathbb R$
For real singular cubic threefolds with no real singular points, rationality is determined by singularity type and by the existence of real lines, planes, or cubic scrolls, with cohomological obstructions in four exceptional cases.