Formulate of Riemann-Hurwitz

In Mathematical, the formula of Riemann-Hurwitz , named in the honor of the Mathematician S Bernhard Riemann and Adolf Hurwitz, described the relations of the characteristic of Euler of two Surface S when is a coating ramified of the other. This, consequently, connects the Ramification with the algebraic Topologie in this case. It is a prototype of result for much of others, and is often applied in the theory of the surfaces of Riemann (which is its origin) and of the algebraic curved .

For a directional surface S , the characteristic of Euler $\ chi (S) \,$ is

$2 - 2g \,$

where G is the kind (the many holes ), since the numbers of Betti are 1,2 G , 1,0,0,…. In the case of a coating ( not ramified ) of surfaces

$\ pi: \ Rightarrow S \,$

who is surjective and of degree NR , we should have the formula

$\ chi () = NR \ chi (S) \,$ .

This, because each simplex of S should be covered by exactly NR in S ′ — at least if we use a sufficiently good Triangulation of S , as we had the right to make it since the characteristic of Euler is a topological Invariant. What the formula of Riemann-Hurwitz does, is to add a correction which takes account of the ramification ( sheets rejoignat ).

Near to a point P of S where E sheets meets, $e = e_P \,$ being called the index of ramification , we note the loss of $e - 1 \,$ copies of P above P (in $\ pi^ {- 1} (P) \,$ ). Consequently, we can envisage a “corrected” formula

$\ chi () = NR \ chi (S) - \ Sigma (e_P - 1) \,$

the sum being taken on all the P in S (almost all the P have $e_P = 1 \,$ , thus, this is completely sure). This is the formula of Riemann-Hurwitz , but for a particular case although important (in other words where there exists right a point where sheets above P meet, or in an equivalent way the local Monodromie is a circular Shift). In the most general case, the final sum must be replaced by the sum of terms

$e_P - c_P \,$

where $c_P \,$ is the number of points of S ′ above P , or an equivalent way the number of cycles of the local monodromy acting on the sheets.

To give an example: All elliptic Courbe (kind 1) applies towards the projective Droite (kind 0) like a double cover ( NR = 2), with a ramification at only four points, where E = 2. We can check that this is read then

0 = 2.2 - \ Sigma 1 \,

with the sum taken on the four values of P . This cover comes from the function EP of Weierstrass which is a Fonction méromorphe, with values considered as being in the Sphère of Riemann. The formula can also be used to check the value of the formula like the hyperelliptic curved .

Another example: the sphere of Riemann applies to itself by the function $z^n \,$ , which has the index of ramification N to 0, for all whole N > 1. It can only exist another ramification at the point ad infinitum. To balance the equation

$2 = N \ cdot 2 (N - 1) - (e_ {\ infty} - 1) \,$

we must have the index of ramification N ad infinitum, too.

The formula can be used to show theorems. For example, it shows immediately that a curve of kind 0 does not have a cover with NR > 1 which is not ramified everywhere: because that would give place to a characteristic of Euler > 2.

For a correspondence of curves, there exists a more general formula, the theorem of Zeuthen , which gives a correction of the ramification while stating at first approximation that the characteristics of Euler are in inverse proportion of the degrees of the correspondences.

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