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 ...

, the Chasles–Cayley–Brill formula, also known as the Cayley–Brill formula, states that a correspondence ''T'' of valence ''k'' from an algebraic curve ''C'' of

genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...

''g'' to itself has ''d'' + ''e'' + 2''kg'' united points, where ''d'' and ''e'' are the degrees of ''T'' and its inverse.

Michel Chasles Michel Floréal Chasles (; 15 November 1793 – 18 December 1880) was a French mathematician. Biography He was born at Épernon in France and studied at the École Polytechnique in Paris under Siméon Denis Poisson. In the War of the Sixth Coal ...

introduced the formula for genus ''g'' = 0,

Arthur Cayley Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific United Kingdom of Great Britain and Ireland, British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics. As a child, C ...

stated the general formula without proof, and

Alexander von Brill Alexander Wilhelm von Brill (20 September 1842 – 18 June 1935) was a German mathematician. Born in Darmstadt, Hesse, Brill was educated at the University of Giessen, where he earned his doctorate under supervision of Alfred Clebsch. He held a c ...

gave the first proof. The number of united points of the correspondence is the intersection number of the correspondence with the diagonal Δ of ''C''×''C''. The correspondence has valence ''k'' if and only if it is homologous to a linear combination ''a''(''C''×1) + ''b''(1×''C'') – ''k''Δ where Δ is the diagonal of ''C''×''C''. The Chasles–Cayley–Brill formula follows easily from this together with the fact that the self-intersection number of the diagonal is 2 – 2''g''.

References

* * {{DEFAULTSORT:Chasles-Cayley-Brill formula Algebraic curves Theorems in algebraic geometry