mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, specifically

algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

, an exceptional divisor for a regular map :

f: X \rightarrow Y

of varieties is a kind of 'large' subvariety of

X

which is 'crushed' by

f

, in a certain definite sense. More strictly, ''f'' has an associated exceptional locus which describes how it identifies nearby points in codimension one, and the exceptional divisor is an appropriate algebraic construction whose support is the exceptional locus. The same ideas can be found in the theory of holomorphic mappings of

complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a com ...

s. More precisely, suppose that :

f: X \rightarrow Y

is a

regular map of varieties In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regula ...

which is

birational In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational f ...

(that is, it is an isomorphism between open subsets of

X

and

Y

). A codimension-1 subvariety

Z \subset X

is said to be ''exceptional'' if

f(Z)

has codimension at least 2 as a subvariety of

Y

. One may then define the ''exceptional divisor'' of

f

to be :

\sum_i Z_i \in Div(X),

where the sum is over all exceptional subvarieties of

f

, and is an element of the group of Weil divisors on

X

. Consideration of exceptional divisors is crucial in

birational geometry In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational ...

: an elementary result (see for instance Shafarevich, II.4.4) shows (under suitable assumptions) that any birational regular map that is not an isomorphism has an exceptional divisor. A particularly important example is the

blowup ''Blowup'' (sometimes styled as ''Blow-up'' or ''Blow Up'') is a 1966 mystery drama thriller film directed by Michelangelo Antonioni and produced by Carlo Ponti. It was Antonioni's first entirely English-language film, and stars David Hemming ...

\sigma: \tilde \rightarrow X

of a subvariety :

W \subset X

: in this case the exceptional divisor is exactly the preimage of

W

References

*{{cite book , author=Shafarevich, Igor , title=Basic Algebraic Geometry, Vol. 1 , publisher=Springer-Verlag , year=1994 , isbn=3-540-54812-2 , url-access=registration , url=https://archive.org/details/basicalgebraicge00irsh Algebraic geometry Birational geometry