# Monoidal transformation

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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].

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