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In algebraic geometry, Du Bois singularities are singularities of complex varieties studied by Du Bois.

Schwede gave the following characterisation of Du Bois singularities. Suppose that ${\displaystyle X}$ is a reduced closed subscheme of a smooth scheme ${\displaystyle Y}$.

Take a log resolution ${\displaystyle \pi :Z\to Y}$ of ${\displaystyle X}$ in ${\displaystyle Y}$ that is an isomorphism outside ${\displaystyle X}$, and let ${\displaystyle E}$ be the reduced preimage of ${\displaystyle X}$ in ${\displaystyle Z}$. Then ${\displaystyle X}$ has Du Bois singularities if and only if the induced map ${\displaystyle {\mathcal {O}}_{X}\to R\pi _{*}{\mathcal {O}}_{E}}$ is a quasi-isomorphism.

Notes

1. Du Bois (1981)
2. Schwede (2007, p. 1, Thm 4.6)

References

• Du Bois, Philippe (1981), "Complexe de de Rham filtré d'une variété singulière", Bulletin de la Société Mathématique de France, 109 (1): 41–81, ISSN 0037-9484, MR 0613848
• Schwede, Karl (2007), "A simple characterization of Du Bois singularities", Compositio Mathematica, 143 (4): 813–828, arXiv:0903.4125, doi:10.1112/S0010437X07003004, ISSN 0010-437X, MR 2339829, S2CID 5737297

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