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
 |
 |
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=46087