algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

, Du Bois singularities are singularities of complex varieties studied by . gave the following characterisation of Du Bois singularities. Suppose that

X

is a reduced

closed subscheme This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. ...

of a

smooth scheme In algebraic geometry, a smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no singular points. A special case is the notion of a smoo ...

Y

. Take a log resolution

\pi: Z \to Y

X

Y

that is an isomorphism outside

X

, and let

E

be the reduced preimage of

X

Z

. Then

X

has Du Bois singularities if and only if the induced map

\mathcal_X \to R\pi_\mathcal_E

is a

quasi-isomorphism In homological algebra, a branch of mathematics, a quasi-isomorphism or quism is a morphism ''A'' → ''B'' of chain complexes (respectively, cochain complexes) such that the induced morphisms :H_n(A_\bullet) \to H_n(B_\bullet)\ (\text H^n(A^\bull ...

References

* * Singularity theory Algebraic geometry {{algebraic-geometry-stub