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.
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.
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- 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