Monoidal transformation

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blowing up, -process

A special kind of birational morphism of an algebraic variety or bimeromorphic morphism of an analytic space. For example, let be an algebraic variety (or an arbitrary scheme), and let be a closed subvariety given by a sheaf of ideals . The monoidal transformation of with centre is the -scheme — the projective spectrum of the graded sheaf of -algebras . If is the structure morphism of the -scheme , then the sheaf of ideals on (defining the exceptional subscheme on ) is invertible. This means that is a divisor on ; in addition, induces an isomorphism between and . A monoidal transformation of a scheme with centre is characterized by the following universal property [1]: The sheaf of ideals is invertible and for any morphism for which is invertible there is a unique morphism such that .

A monoidal transformation of an algebraic or analytic space with as centre a closed subspace can be defined and characterized in the same way.

An important class of monoidal transformations are the admissible monoidal transformations, which are distinguished by the condition that is non-singular and is a normally flat scheme along . The latter means that all sheaves are flat -modules. The importance of admissible monoidal transformations is explained by the fact that they do not worsen the singularities of the variety. In addition, it has been proved (see [1]) that a suitable sequence of admissible monoidal transformations improves singularities, which allows one to prove the theorem on the resolution of singularities for an algebraic variety over a field of characteristic zero.

Admissible monoidal transformations of non-singular varieties are particularly simple to construct. If is a monoidal transformation with a non-singular centre , then is again non-singular and the exceptional subspace is canonically isomorphic to the projectivization of the conormal sheaf to in . In the special case when consists of one point, the monoidal transformation consists of blowing up this point into the whole projective space of tangent directions. For the behaviour of various invariants of non-singular varieties (such as Chow rings, cohomology spaces, the -functor, and Chern classes) under admissible monoidal transformations see [2][5].


[1] H. Hironaka, "Resolution of an algebraic variety over a field of characteristic zero I, II" Ann. of Math. (2) , 79 (1964) pp. 109–326
[2] P. Berthelot (ed.) A. Grothendieck (ed.) L. Illusie (ed.) et al. (ed.) , Théorie des intersections et théorème de Riemann–Roch (SGA 6) , Lect. notes in math. , 225 , Springer (1971)
[3] A. Grothendieck, I. Bucur, C. Honzel, L. Illusie, J.-P. Jouanolou, J.-P. Serre, "Cohomologie -adique et fonctions . SGA 5" , Lect. notes in math. , 589 , Springer (1977)
[4] I. Porteous, "Blowing up Chern classes" Proc. Cambridge Philos. Soc. , 56 : 2 (1960) pp. 118–124
[5] Yu.I. Manin, "Lectures on the -functor in algebraic geometry" Russian Math. Surveys , 24 (1969) pp. 1–89 Uspekhi Mat. Nauk , 24 : 5 (1969) pp. 3–86


The word "s-process" appeared for the first time in [a1].


[a1] H. Hopf, "Schlichte Abbildungen und lokale Modifikationen 4-dimensionaler komplexer Mannigfaltigkeiten" Comm. Math. Helv. , 29 (1954) pp. 132–156
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Monoidal transformation. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article