# An Introduction to Riemann Surfaces, Algebraic Curves and by Martin Schlichenmaier

By Martin Schlichenmaier

This ebook offers an creation to fashionable geometry. ranging from an trouble-free point the writer develops deep geometrical thoughts, enjoying a huge position these days in modern theoretical physics. He provides a variety of innovations and viewpoints, thereby displaying the relatives among the choice ways. on the finish of every bankruptcy feedback for additional interpreting are given to permit the reader to review the touched issues in higher element. This moment version of the publication comprises extra extra complicated geometric strategies: (1) the trendy language and glossy view of Algebraic Geometry and (2) replicate Symmetry. The publication grew out of lecture classes. The presentation variety is for this reason just like a lecture. Graduate scholars of theoretical and mathematical physics will have fun with this booklet as textbook. scholars of arithmetic who're searching for a quick creation to a number of the facets of contemporary geometry and their interaction also will locate it important. Researchers will esteem the booklet as trustworthy reference.

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**Extra info for An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces**

**Sample text**

3. Let X and X be Riemann surfaces. e. let x ∈ X, f (x) ∈ X , ϕ1 a coordinate around x, ϕ2 a coordinate around f (x), then we require ϕ2 ◦ f ◦ ϕ1 −1 : W ⊂ C → C to be holomorphic, where it is deﬁned (see Fig. 1). We call f an analytic isomorphism if f is bijective and f and f −1 are holomorphic (this is just a repetition of the deﬁnition in Sect. 1). One of the main problems in the theory is the following: Classify all Riemann surfaces up to analytic isomorphy in a “natural way”. The set of isomorphy classes should carry some geometric structure which exhibits the appearance of Riemann surfaces.

Z Hence there exists locally meromorphic functions ω = f (z)dz = f (z )dz h(z) = with f (z ) = f (z) f (z) g(z) which glue together to form a global function because we have f (z) f (z ) = , g (z ) g(z) z = z (z) on the overlap. Hence h · β = ω and (h) + (β) = (ω) resp. (β) ∼ (ω). 6 in Sect. 3. As a general fact linearly equivalent divisors D and D have isomorphic L(D) and L(D ). With the same technique we are able to show the following isomorphy. Let ω = 0 be a meromorphic diﬀerential. The vector space L((ω)) := {f ∈ M(X) | (f ) ≥ −(ω)} is isomorphic to Ω(X) := {global holomorphic diﬀerentials}.

If we consider another lattice Γ := n+m ω2 | n, m ∈ Z ω1 and the associated torus T we get a well-deﬁned map Φ : T → T , which is an analytic isomorphism z =z+ n+m ω2 ω1 → ω1 z + nω1 + mω2 = ω1 z = Φ(z). Essentially this is multiplication by ω1 . We see from the classiﬁcation viewpoint that it is enough to consider only lattices of the type Γ . Hence we assume for the following that Γ is already of this type. In Γ we are able to choose the generator τ := ω2 /ω1 such that its imaginary part is strictly positive.