Let \(f : X \to Y\) be a finite separable morphism of smooth projective curves of degree \(n\). Then \[ 2g_X - 2 = n(2g_Y - 2) + \deg R , \] where \(R = \sum_{p} \length(\Omega_{X/Y})_p\) is the ramification divisor. When every ramification is tame, \(\deg R = \sum_p (e_p - 1)\).
(Proof structure)
The relative cotangent sequence for \(X \to Y \to \Spec k\) is \[ f^* \Omega_{Y} \to \Omega_{X} \to \Omega_{X/Y} \to 0 . \] For a nonconstant separable map of smooth curves the left map is injective, so the sequence is short exact, and \(\Omega_{X/Y}\) is a torsion sheaf supported at the ramification points. Taking degrees gives \(\deg \Omega_X = n \deg \Omega_Y + \deg R\), and \(\deg \Omega_C = 2g_C - 2\) from Riemann–Roch.
Thus the proof reduces to injectivity of the map on differentials, exactness of the cotangent sequence, and taking degrees. Separability is what makes the left map injective, and in characteristic \(p\) the Frobenius \(\PP^1 \to \PP^1\) shows the statement fails without it — the map on differentials is zero.
(Using it)
The formula is used in both directions. Forward, with \(Y = \PP^1\): a degree-\(2\) cover of \(\PP^1\) branched at \(2g+2\) points has genus \(g\), which is every hyperelliptic curve. Backward: a curve admitting a map of known degree to a known curve has its genus pinned, and since \(\deg R \geq 0\), a nonconstant map \(X \to Y\) forces \(g_X \geq g_Y\) — no curve maps onto one of larger genus.
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