Monoidal transformation

blowing up, $ \sigma $- process

A special kind of birational morphism of an algebraic variety or bimeromorphic morphism of an analytic space. For example, let $ X $ be an algebraic variety (or an arbitrary scheme), and let $ D \subset X $ be a closed subvariety given by a sheaf of ideals $ J $. The monoidal transformation of $ X $ with centre $ D $ is the $ X $- scheme $ X ^ {1} = \mathop{\rm Proj} ( \oplus _ {n \geq 0 } J ^ {n)} $— the projective spectrum of the graded sheaf of $ {\mathcal O} _ {X} $- algebras $ \oplus _ {n \geq 0 } J ^ {n} $. If $ f : X ^ {1} \rightarrow X $ is the structure morphism of the $ X $- scheme $ X ^ {1} $, then the sheaf of ideals $ f ^ { * } ( J) = J \cdot {\mathcal O} _ {X ^ {1} } $ on $ X ^ {1} $( defining the exceptional subscheme $ f ^ { - 1 } ( D ) $ on $ X ^ {1} $) is invertible. This means that $ f ^ { - 1 } ( D ) $ is a divisor on $ X ^ {1} $; in addition, $ f $ induces an isomorphism between $ X ^ {1} \setminus f ^ { - 1 } ( D ) $ and $ X \setminus D $. A monoidal transformation $ f : X ^ {1} \rightarrow X $ of a scheme $ X $ with centre $ D $ is characterized by the following universal property [1]: The sheaf of ideals $ f ^ { * } ( J ) $ is invertible and for any morphism $ g : X _ {1} \rightarrow X $ for which $ g ^ {*} ( J ) $ is invertible there is a unique morphism $ h : X _ {1} \rightarrow X ^ {1} $ such that $ g = f \circ h $.

A monoidal transformation of an algebraic or analytic space $ X $ with as centre a closed subspace $ D \subset X $ 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 $ D $ is non-singular and $ X $ is a normally flat scheme along $ D $. The latter means that all sheaves $ J ^ {n} / J ^ {n+} 1 $ are flat $ ( {\mathcal O} _ {X} / J ) $- 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 $ f : X _ {1} \rightarrow X $ is a monoidal transformation with a non-singular centre $ D \subset X $, then $ X _ {1} $ is again non-singular and the exceptional subspace $ f ^ { - 1 } ( D ) $ is canonically isomorphic to the projectivization of the conormal sheaf to $ D $ in $ X $. In the special case when $ D $ 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 $ K $- functor, and Chern classes) under admissible monoidal transformations see [2]–[5].

References

[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 MR199184 Zbl 0122.38603
[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 $\ell$-adique et fonctions $L$. SGA 5" , Lect. notes in math. , 589 , Springer (1977) MR491704
[4]	I. Porteous, "Blowing up Chern classes" Proc. Cambridge Philos. Soc. , 56 : 2 (1960) pp. 118–124 MR0121813 Zbl 0166.16701
[5]	Yu.I. Manin, "Lectures on the $K$-functor in algebraic geometry" Russian Math. Surveys , 24 (1969) pp. 1–89 Uspekhi Mat. Nauk , 24 : 5 (1969) pp. 3–86 MR265355

Comments

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

References

[a1]	H. Hopf, "Schlichte Abbildungen und lokale Modifikationen 4-dimensionaler komplexer Mannigfaltigkeiten" Comm. Math. Helv. , 29 (1954) pp. 132–156 MR0068008 Zbl 0064.41703