# 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.

How to Cite This Entry:
Blow-up algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Blow-up_algebra&oldid=46087
This article was adapted from an original article by H. Hauser (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article