Bézout's theorem

In algebraic geometry , the classical Bézout theorem describes the number of points of intersection of plane algebraic curves . It was formulated and proven by Étienne Bézout in the 18th century (within the framework of the more lax claims of the time).

statement

Let be an algebraically closed body and let and two projective plane curves in two-dimensional projective space without common components. Then: ${\displaystyle k}$ ${\displaystyle F}$ ${\displaystyle G}$ ${\displaystyle \mathbb {P} ^{2}(k)}$

${\displaystyle \sum _{P\in \mathbb {P} ^{2}(k)}I(P,F\cap G)=\deg F\cdot \deg G,}$

where denotes the number of cuts . ${\displaystyle I(P,F\cap G)}$

Inferences

• Two projective plane curves and always intersect in at least one point and at most in different points. ${\displaystyle F}$ ${\displaystyle G}$ ${\displaystyle \deg F\cdot \deg G}$
• The inequality holds for affine plane curves and without common components . ${\displaystyle F}$ ${\displaystyle G}$ ${\displaystyle \sum _{P\in k^{2}}I(P,F\cap G)\leq \deg F\cdot \deg G}$

generalization

A generalization for algebraic varieties is as follows:

Be , algebraic varieties of degree or in -dimensional projective space . Furthermore, be a variety of dimension . ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle \deg A=a}$ ${\displaystyle \deg B=b}$ ${\displaystyle n}$ ${\displaystyle \mathbb {P} ^{n}}$ ${\displaystyle A\cap B}$ ${\displaystyle \dim(A\cap B)=\dim A+\dim B-n}$

Then is . ${\displaystyle \deg A\cap B=ab}$

Web links

Wikiversity: A proof of the theorem in the even case  - course materials

literature

Klaus Hulek: Elementare Algebraische Geometrie , 1st edition, 2000, ISBN 978-3-528-03156-5 , pp. 145-146.