# 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 . ${\ displaystyle A}$ ${\ displaystyle B}$ ${\ displaystyle \ deg A = a}$ ${\ displaystyle \ deg B = b}$ ${\ displaystyle n}$ ${\ displaystyle \ mathbb {P} ^ {n}}$ ${\ displaystyle A \ cap B}$ ${\ displaystyle \ dim (A \ cap B) = \ dim A + \ dim Bn}$ Then is . ${\ displaystyle \ deg A \ cap B = from}$ ## 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.