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 .
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 .
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 .
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.
|[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://www.encyclopediaofmath.org/index.php?title=Blow-up_algebra&oldid=23768