Let \(f : X \to Y\) be a finite separable morphism of curves. Then \(\deg R\) is even.
\(R \sim K_X - f^* K_Y\), so \[ \deg R = (2 g_X - 2) - \deg(f)(2 g_Y - 2) , \] a difference of two even integers.
(Parity consequences)
The proof is Riemann–Hurwitz with the terms moved, and the resulting parity constraint gives a consistency check on any ramification count. A branching pattern with odd \(\deg R\) is impossible, so parity gives an immediate consistency check on ramification counts.
Used forward with tame simple ramification, where every \(e_p = 2\) and \(\deg R\) counts the branch points, it says such a cover has an even number of branch points. This is the reason a degree-\(2\) cover of \(\PP^1\) is branched at \(2g+2\) points rather than an arbitrary number, and the reason the Legendre normal form of an elliptic curve has exactly four branch points and not three. It also rules out possibilities directly: no simply branched cover of \(\PP^1\) has an odd branch count, whatever its degree.
The hypothesis that bites is separability, since the whole statement descends from \(K_X \sim f^* K_Y + R\), which needs the cotangent sequence to be short exact. In characteristic \(p\) there is no parity statement about an inseparable morphism because there is no \(R\); see PR-IV2INSEP.