Blow-up algebra

Geometric description.

Associate to the punctured affine $ n $- space $ X _ {0} = \mathbf A ^ {n} \setminus \{ 0 \} $ over $ \mathbf R $ or $ \mathbf C $, the submanifold $ {\widetilde{X} } _ {0} $ of $ \mathbf A ^ {n} \times \mathbf P ^ {n - 1 } $ of points $ ( x, [ x ] ) $, where $ x $ varies in $ X _ {0} $ and $ [ x ] $ denotes the equivalence class of $ x $ in the projective $ ( n - 1 ) $- dimensional space. The closure $ {\widetilde{X} } $ of $ {\widetilde{X} } _ {0} $ is smooth and is called the blow-up of $ X = \mathbf A ^ {n} $ with centre the origin. In the real case and for $ n = 2 $ it is equal to the Möbius strip. The mapping $ \pi : { {\widetilde{X} } } \rightarrow X $ induced by the projection $ \mathbf A ^ {n} \times \mathbf P ^ {n - 1 } \rightarrow \mathbf A ^ {n} $ is an isomorphism over $ X _ {0} $; its fibre over $ 0 $ is $ \mathbf P ^ {n - 1 } $, the exceptional divisor of $ \pi $.

The strict transform $ Y ^ \prime $ of a subvariety $ Y $ of $ X $ is the closure of the inverse image $ \pi ^ {-1 } ( Y \setminus \{ 0 \} ) $ in $ {\widetilde{X} } $. For instance, if $ Y $ is the cuspidal curve $ x ^ {3} = y ^ {2} $ in $ \mathbf A ^ {2} $ parametrized by $ ( t ^ {2} ,t ^ {3} ) $, then $ Y ^ \prime $ is given by $ ( t ^ {2} ,t ^ {3} ,t ) $ and hence is smooth. This forms the simplest example of resolution of singularities by a blow-up.

Higher-dimensional smooth centres $ C $ in $ \mathbf A ^ {n} $ are blown up by decomposing $ \mathbf A ^ {n} $ locally along $ C $ into a Cartesian product $ X _ {1} \times X _ {2} $ of submanifolds, where $ X _ {1} $ is transversal to $ C $ with $ X _ {1} \cap C = \{ p \} $ a point. Then $ {\widetilde{X} } $ is given locally as $ { {X _ {1} } tilde } \times X _ {2} $, where $ { {X _ {1} } tilde } $ denotes the blow-up of $ X _ {1} $ in $ p $.

Algebraic description.

See also [a1]. Let $ A $ be a Noetherian ring and let $ I $ be an ideal of $ A $. Define the blow-up algebra (or Rees algebra) of $ I $ as the graded ring $ S = \oplus _ {k \geq 0 } I ^ {k} $( where $ I ^ {k} $ denotes the $ k $ th power of $ I $, $ I ^ {0} = A $). Then $ Bl _ {I} ( A ) = { \mathop{\rm Proj} } S $ is the blow-up of $ { \mathop{\rm Spec} } A $ with centre $ I $ and coincides with the above construction when $ A $ is the polynomial ring in $ n $ variables over $ \mathbf R $ or $ \mathbf C $. Here, $ { \mathop{\rm Proj} } S $ denotes the algebraic variety or scheme given by all homogeneous prime ideals of $ S $ not containing the ideal $ S _ {+} = \oplus _ {k > 0 } I ^ {k} $, and $ { \mathop{\rm Spec} } A $ is the affine variety or scheme of all prime ideals of $ A $.

Local description.

Any generator system $ x _ {1} \dots x _ {k} $ of $ I $ gives rise to a covering

$$ Bl _ {I} ( A ) = \cup _ {j = 1 } ^ { k } { \mathop{\rm Spec} } A [ {I / {x _ {j} } } ] = $$

$$ = \cup _ {j = 1 } ^ { k } { \mathop{\rm Spec} } A [ { {x _ {i} } / {x _ {j} } } , 1 \leq i \leq k ] $$

by $ k $ affine charts, the quotients $ { {x _ {i} } / {x _ {j} } } $ being considered as elements of the localization of $ A $ at $ x _ {j} $( cf. Localization in a commutative algebra). In the $ j $ th chart $ {\widetilde{X} } _ {j} $, the morphism $ \pi : { {\widetilde{X} } _ {j} } \rightarrow X $ is induced by the inclusion $ A \subset A [ {I / {x _ {j} } } ] $. For $ J $ an ideal of $ A $ contained in $ I $, the strict transform of $ J $ is $ J ^ \prime = \cup _ {n \geq 0 } x _ {j} ^ {- n } ( J \cap I ^ {n} ) {\widetilde{A} } _ {j} $. The exceptional divisor has the equation $ x _ {j} = 0 $. If the centre $ C $ given by the ideal $ I $ of $ A $ is smooth, $ I $ is generated by part of a regular parameter system of $ A $ and $ \pi : { {\widetilde{X} } _ {j} } \rightarrow X $ is given by $ x _ {i} \rightarrow x _ {i} x _ {j} $ for $ i \leq k $, $ i \neq j $, and by $ x _ {i} \rightarrow x _ {i} $ for $ i > k $ or $ i = j $.

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 $ \pi $ 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 $ 0 $ can be resolved by a finite sequence of blow-ups of smooth centres [a2]. In positive characteristic, this has only been proven for dimension $ \leq 3 $[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