Monoidal transformation
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].
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 |
[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 ![]() ![]() |
[4] | I. Porteous, "Blowing up Chern classes" Proc. Cambridge Philos. Soc. , 56 : 2 (1960) pp. 118–124 |
[5] | Yu.I. Manin, "Lectures on the ![]() |
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 |
Monoidal transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Monoidal_transformation&oldid=13914