Birational geometry

Table of contents
- Blowing up (algebraic geometry) Birational geometry
  - Strict transform of an algebraic subvariety Blowing up (algebraic geometry)
  - Algebraic exceptional divisor Blowing up (algebraic geometry)

Blowing up (algebraic geometry, $Bl_{Z} X$ )

The projective morphism

Proj_{X} ⨁_{j \geq 0} I^{j} \to X

associated to the powers of the ideal sheaf of a closed centre. Off that centre it is an isomorphism. Blowing up a smooth point of an

n

-dimensional smooth variety replaces the point by a projective space

P^{n - 1}

, the algebraic exceptional divisor.

Strict transform of an algebraic subvariety ( $Y = \overline{π^{- 1} (Y ∖ Z)}$ )

For a blowup of an algebraic variety

π : X \to X

with centre

Z

, the strict transform of an algebraic subvariety of a subvariety

Y

not contained in

Z

is the closure of

π^{- 1} (Y ∖ Z)

. It excludes components supported entirely in the algebraic exceptional divisor.

Algebraic exceptional divisor ( $E$ )

The divisor contracted by a blowup of an algebraic variety to its centre. For the blowup of a smooth point on an

n

-dimensional smooth variety it is

P^{n - 1}

. Being a closed projective subvariety of positive dimension, it can obstruct affineness of a fibre containing it.

 Ancestors (5)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (1)

Birational geometry by

Wikipedia Bot  1

Birational geometry is a branch of algebraic geometry that studies the relationships between algebraic varieties through birational equivalences. These are equivalences that allow the objects in question to be related by rational maps, which can typically be viewed as fewer-dimensional representations of the varieties.

 Read the full article

  See all articles in the same topic Create my own version