title: Du Bois singularity
text: # Du Bois singularity
  
  In algebraic geometry, Du Bois singularities are singularities of complex varieties studied by Du Bois (1981).
  
  Schwede (2007) gave the following characterisation of Du Bois singularities. Suppose that $X$ is a reduced closed subscheme of a smooth scheme $Y$.
  
  Take a log resolution $\pi :Z\to Y$ of $X$ in $Y$ that is an isomorphism outside $X$, and let $E$ be the reduced preimage of $X$ in $Z$. Then $X$ has Du Bois singularities if and only if the induced map ${\mathcal {O}}_{X}\to R\pi _{*}{\mathcal {O}}_{E}$ is a quasi-isomorphism.
  
  