Blow-up algebra
Geometric description.
Associate to the punctured affine -space
over
or
, the submanifold
of
of points
, where
varies in
and
denotes the equivalence class of
in the projective
-dimensional space. The closure
of
is smooth and is called the blow-up of
with centre the origin. In the real case and for
it is equal to the Möbius strip. The mapping
induced by the projection
is an isomorphism over
; its fibre over
is
, the exceptional divisor of
.
The strict transform of a subvariety
of
is the closure of the inverse image
in
. For instance, if
is the cuspidal curve
in
parametrized by
, then
is given by
and hence is smooth. This forms the simplest example of resolution of singularities by a blow-up.
Higher-dimensional smooth centres in
are blown up by decomposing
locally along
into a Cartesian product
of submanifolds, where
is transversal to
with
a point. Then
is given locally as
, where
denotes the blow-up of
in
.
Algebraic description.
See also [a1]. Let be a Noetherian ring and let
be an ideal of
. Define the blow-up algebra (or Rees algebra) of
as the graded ring
(where
denotes the
th power of
,
). Then
is the blow-up of
with centre
and coincides with the above construction when
is the polynomial ring in
variables over
or
. Here,
denotes the algebraic variety or scheme given by all homogeneous prime ideals of
not containing the ideal
, and
is the affine variety or scheme of all prime ideals of
.
Local description.
Any generator system of
gives rise to a covering
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by affine charts, the quotients
being considered as elements of the localization of
at
(cf. Localization in a commutative algebra). In the
th chart
, the morphism
is induced by the inclusion
. For
an ideal of
contained in
, the strict transform of
is
. The exceptional divisor has the equation
. If the centre
given by the ideal
of
is smooth,
is generated by part of a regular parameter system of
and
is given by
for
,
, and by
for
or
.
Properties.
Different centres may induce the same blow-up. A composite of blow-ups is again a blow-up. Blowing up commutes with base change; the strict transform of a variety equals its blow-up in the given centre. The morphism is birational, proper and surjective (cf. Birational morphism; Proper morphism; Surjection). Any birational projective morphism of quasi-projective varieties (cf. Quasi-projective scheme) is the blowing up of a suitable centre. The singularities of varieties over a field of characteristic
can be resolved by a finite sequence of blow-ups of smooth centres [a2]. In positive characteristic, this has only been proven for dimension
[a3]. See [a4] for a survey on resolution of singularities, and [a5] for an account on the role of blow-up algebras in commutative algebra.
References
[a1] | R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001 |
[a2] | H. Hironaka, "Resolution of singularities of an algebraic variety over a field of characteristic zero" Ann. of Math. , 79 (1964) pp. 109–326 MR0199184 Zbl 0122.38603 |
[a3] | S. Abhyankar, "Resolution of singularities of embedded algebraic surfaces" , Acad. Press (1966) MR0217069 Zbl 0147.20504 |
[a4] | J. Lipman, "Introduction to resolution of singularities" , Proc. Symp. Pure Math. , 29 , Amer. Math. Soc. (1975) pp. 187–230 MR0389901 Zbl 0306.14007 |
[a5] | W. Vasconcelos, "Arithmetic of blowup algebras" , Lecture Notes Ser. , 195 , London Math. Soc. (1994) MR1275840 Zbl 0813.13008 |
Blow-up algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Blow-up_algebra&oldid=23768