Chasles–Cayley–Brill formula

In algebraic geometry, 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 g to itself has d + e + 2kg united points, where d and e are the degrees of T and its inverse.

Michel Chasles introduced the formula for genus g = 0, Arthur Cayley stated the general formula without proof, and Alexander von Brill 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) – kΔ 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 – 2g.

References

Coolidge, Julian Lowell (1959) [1931], A treatise on algebraic plane curves, New York: Dover Publications, ISBN 978-0-486-49576-7, MR 0120551
Griffiths, Phillip; Harris, Joseph (1994), Principles of algebraic geometry, Wiley Classics Library, New York: John Wiley & Sons, ISBN 978-0-471-05059-9, MR 1288523

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