By R. Narasimhan

Those notes shape the contents of a Nachdiplomvorlesung given on the Forschungs institut fur Mathematik of the Eidgenossische Technische Hochschule, Zurich from November, 1984 to February, 1985. Prof. okay. Chandrasekharan and Prof. Jurgen Moser have inspired me to write down them up for inclusion within the sequence, released via Birkhiiuser, of notes of those classes on the ETH. Dr. Albert Stadler produced exact notes of the 1st a part of this direction, and extremely intelligible class-room notes of the remainder. with no this paintings of Dr. Stadler, those notes do not need been written. whereas i've got replaced a few issues (such because the evidence of the Serre duality theorem, right here performed completely within the spirit of Serre's unique paper), the current notes stick with Dr. Stadler's relatively heavily. My unique target in giving the direction was once twofold. i needed to provide the elemental theorems concerning the Jacobian from Riemann's personal viewpoint. Given the Riemann-Roch theorem, if Riemann's equipment are expressed in sleek language, they range little or no (if in any respect) from the paintings of contemporary authors.

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**Additional resources for Compact Riemann Surfaces (Lectures in Mathematics. ETH Zurich)**

**Example text**

If g = 3, E = 0, N = 3 and . N(N - l)(n - 1) + NE= 3(g - 2) = 3 = g. Thus we have equality, and Noether's theorem follows from Castelnuovo's. It should be added that if (g 2': 3 and) X is hyperelliptic, the above result definitely fails. g. from the fact that J(~m is very ample for large rn, but the mapping 'PKx induced by J(x is not injective. Before proceeding further, we recall some facts about compact oriented surfaces. We shall not prove them here; proofs can be found in, for example [6]. The basic theorem about the classification of compact orientable surfaces is the following: A compact orientable Ceo surface X without boundary with a finite number of handles attached.

Do GENERAL POSITION THEOREM. Let X C ll:'n be a noncurve of degree d. Then, the set of hyperplanes with the following = {Xl, ... ,Xd}, then any set of n points Xi,,· .. , Xin are linearly not lie in a plane of dimension n - 2) is dense in (IP'n)*. In proving this result, we shall assume familiarity with basic algebraic geometry. We begin with the following. Lemma 2. : 3. ) There is a proper algebraic subset A of U such that, if H E U - A, then no three pO'ints of X n Hare colin ear (lie on a line).

K=l and hl(Dm) we have = 0 for m 2': 2g - 1, while hO(Do) = hO(Dm) = m - 9 1, hl(Do) = g. Thus, for m 2': 2g - 1, + 1. Moreover, hO(Dm) -1 = 'L,{, (hO(Dk) - hO(Dk-l)) is the number of k :::;m which occur as the order of pole at P of an f E qX) holomorphic on X - P. Thus, the number "gaps" is m - (hO(Dm) - 1) = g. N ate. We could have applied this argument with an arbitrary sequence H, P2, ... of points and Do = 0, Dk = 'L~ Pi for k > O. We find that there is a meromorphic function f with (j) 2': - D k but (j) D k-l for all k except for 9 exceptional values nl, ...