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  "body": "\n## Statement\n\nLet $X$ and $Y$ be compact connected [[Riemann surfaces|5K0N8QMV6Q]] of [[genus|NJ787GZS1N]] $g_X$ and $g_Y$.\nLet $f \\colon X \\to Y$ be a nonconstant holomorphic map of degree $n$.\nFor $P \\in X$, let $e_P \\ge 1$ be the ramification index of $f$ at $P$, so that near $P$ the map looks like $z \\mapsto z^{e_P}$ in local coordinates.\nThen $e_P = 1$ for all but finitely many $P$, and\n$$2 g_X - 2 = n\\,(2 g_Y - 2) + \\sum_{P \\in X} (e_P - 1).$$\n\n## Remarks\n\nTriangulate $Y$ so that every branch value is a vertex, and lift the triangulation to $X$ along the [[branched covering|YARENZRXTD]] $f$.\nEach edge and face has exactly $n$ preimages, and a vertex $y$ has $n - \\sum_{f(P) = y} (e_P - 1)$ preimages.\nComparing [[Euler characteristics|RNDDWDAFJ5]] with $\\chi = 2 - 2g$ gives the formula.\nThe same formula holds for a finite separable morphism of smooth projective curves over an algebraically closed field when every $e_P$ is prime to the characteristic.\nWith wild ramification the term $e_P - 1$ must be replaced by a larger number.\nExample: the map $z \\mapsto z^n$ on the Riemann sphere has degree $n$ and ramifies only at $0$ and $\\infty$ with $e = n$; the formula reads $-2 = n(-2) + 2(n - 1)$, which is true.\nExample: a double cover $X \\to \\mathbb{P}^1$ branched at $2g + 2$ points has $2 g_X - 2 = -4 + 2g + 2$, so $g_X = g$; this is how the genus of a hyperelliptic curve is found.\n\n## Explanations\n\n### L1 human\n\nA map between surfaces that wraps one surface $n$ times around another almost everywhere, except at finitely many branch points, links the numbers of holes of the two surfaces.\nWithout branch points, the Euler characteristic simply multiplies by $n$.\nEach branch point, where several sheets come together, makes a correction.\nThe formula records this exactly.\n\n### L2 human\n\nThe formula is the main tool to compute genera of curves given as covers, such as hyperelliptic curves, Fermat curves and modular curves.\nIt shows at once that there is no nonconstant map from a curve of genus $g$ to a curve of larger genus.\nIt also shows that an unramified cover of a genus-$1$ curve has genus $1$.\nApplied to the quotient map by a group of automorphisms, it gives [[Hurwitz's automorphisms theorem|3GNKSCQSCD]].\nIt is the one-dimensional case of the comparison of canonical divisors $K_X = f^* K_Y + R$, where $R$ is the ramification divisor.\n",
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