Fix a compact connected Riemann surface Xof genus g. Riemann’s inequality gives a su cient condition to construct meromorphic functions with prescribed singularities. For the rest of this paper, we will make use of a preferred basepoint xo (or *) in Mg. Let SPr(X) denote the r-fold symmetric product of X (i.e. Divisors on a Riemann surface. arXiv:1606.06449v1 [math.CV] 21 Jun 2016 A TORELLI TYPE THEOREM FOR EXP-ALGEBRAIC CURVES INDRANIL BISWAS AND KINGSHOOK BISWAS Abstract. Riemann surface. Moreover, they provide explicit coordinates (Tyurin parameters) in an open subset of the moduli space of … We will also need to know how to nd a symplectic form on this space. Matrix divisors are introduced in the work by A.Weil (1938) which is considered as a starting point of the theory of holomorphic vector bundles on Riemann surfaces. A compact orientable 2-manifold is homeomorphic to either S2 or a connected sum of a finite number of 2-tori. Before proving the theorem, there are a number of prerequisites to be dealt with, such as the Riemann 2010 Mathematics Subject Classification: Primary: 14-XX [][] In algebraic geometry, the term divisor is used as a generalization of the concept of a divisor of an element of a commutative ring. Some authors call H a Riemann form, focus more on the Hermitian form H instead of Im(H). The compactification divisor, thus consists of a union of lower dimensional Teichmu¨ller spaces, each such space consists of noded Riemann surfaces, obtained by pinching nontrivial short closed geodesics on the surface ([3], [10]). Then for all n>>0, nDis linearly equivalent to an e ective divisor. A divisor D is a formal expression of the form D = ~ niPi i=l (We adopt this notation to emphasize the fact that this sum does not refer to addition and scalar multiplication in c.) The order of D at k The degree of Dis the integer Z n .. i=l 1 Because a meromorphic function on a compact Riemann surface S with f . De nition 1.1 (Divisor). Then O(X) ∼=C. Conclude that they gen-erate M X. §2. Chapter 5 - Geometric Structures on Riemann Surfaces: Time permitting, the student will finish the course with a study of differential forms, divisors, and line bundles on Riemann surfaces. Sheaves. The collection of points pwhere D(p) 6= 0 … The divisors of a Riemann surface are the elements of the free abelian group of points on the surface.. Equivalently, a divisor is a finite linear combination of points of the surface with integer coefficients. The set of all divisors on Xforms an abelian group denoted Div(X). (b) implies additivity when one of the divisors is prime. Suppose Dis a divisor on a compact Riemann surface X of genus g. Then H0(X,O D) and H1(X,O D) are finite dimensional vector space and dimH0(X,O D) −dimH1(X,O D) = 1 −g+ deg(D) (0.1) Corollary 0.6. For example, on every compact Riemann surface of genus there is a meromorphic function which realizes a branched covering with at … Introduction and notations Let V be a compact Riemann surface and V be the complement in V of a nonvoid finite subset S. Riemann Surfaces and Theta Functions MAST 661G / MAST 837J M. Bertolaz1 zDepartment of Mathematics and Statistics, Concordia University 1455 de Maisonneuve W., Montr eal, Qu ebec, Canada H3G 1M8 1bertola@mathstat.concordia.ca Compiled: August 13, 2010 Foundational results on divisors and compact Riemann surfaces are also stated and proved. It is mostly based on the exposition in [5]. The Riemann-Roch formula B. Let D be a divisor such that H D 0. Suppose is a compact Riemann surface of genus g>0 with canonical class K, and let Dbe a divisor on . 2.1. = É we need deg D =o LI X = % we need deg D= 0 & another condition 1.1 Definition of a Riemann surface and basic examples In its broadest sense a Riemann surface is a one dimensional complex manifold that locally looks like an open set of the complex plane, while its global topology can be quite di erent from the complex plane. Definition. We apply the Riemann{Roch theorem to nD(here Kstands for a divisor corre-sponding to … In 1913, Weyl published Die Idee der Riemannschen Fläche (The Concept of a Riemann Surface), which gave a unified treatment of Riemann surfaces.In it Weyl utilized point set topology, in order to make Riemann surface theory more rigorous, a model followed in later work on manifolds.He absorbed L. E. J. a compact Riemann surface are determined almost completely when the order of the function is given at each point of the surface; and that is done most conveniently in terms of divisors. Compact Riemann Surfaces and Algebraic Curves. Note that we will be The degree of the canonical bundle. Show that f is a holomorphic map from X to the Riemann sphere Y = Cˆ of degree 1. (6) Let D be a very ample divisor on X. 3 Divisors 12. Theorem 5.1.1. Denote by ˇ 1(X D;P) the quotient of the fundamental group of a Riemann surface Xwithout a divisor D, which is the largest quotient of ˇ 1(X D;P), whose elements can be distinguished by an iterated integral. Kummer under the name of "ideal divisor" in his studies on … In this theory matrix divisors play the role similar to the role of usual divisors in the theory of line bundles. The main reason why Riemann surfaces are interesting There is no general answer to this question. Recall that a divisor on X is a formal sum of points p in X with integer coe–cients, D = X p2X npp; n 2 Z: Also, any meromorphic function f : X ! LECTURE 12: THETA DIVISOR 12.1. Let Hbe an ample divisor on a surface X, and let Dbe a divisor such that D:H>0 and D2 >0. THE JACOBIAN OF A RIEMANN SURFACE DONU ARAPURA Fix a compact connected Riemann surface X of genus g. The set of divi-sors Div(X) forms an abelian group. De nition 1.1 A Riemann surface is a connected one-complex-dimensional analytic manifold, that is, a two-real dimensional connected manifold Rwith a complex structure on it. The simplest compact Riemann surface is Cb= C[1with charts U 1 = C and U 2 = Cbf 0gwith f 1(z) = zand f 2(z) = 1=z. The map f(z) = z (the identity map) defines a chart for C, and {f} is an atlas for C.The map g(z) = z * (the conjugate map) also defines a chart on C and {g} is an atlas for C.The charts f and g are not compatible, so this endows C with two distinct Riemann surface structures. Let Xbe a compact Riemann surface. 3). Clifford's theorem. We then show that there is an algebraic relationship between them and the topological genus gof the Riemann surface, namely that dimL( D) = degD+ 1 g+ dim (D): Riemann surface can be immersed in Euclidean 3-space as in the above with at most 4p + 1 punctures, where p is the genus of the Riemann surface; ii) any hyperelliptic Riemann surface of genus p can be so immersed with at most ... Then d is the degree of the polar divisor of /. This proves (b). DIVISORS 59 2.1.1. The complex plane C is the most basic Riemann surface. 6. The constant χ(0) is the holomorphic Euler characteristic of the trivial bundle, and is equal to 1 + p a, where p a is the arithmetic genus of … It should be broken up into multiple pieces, and the notation should be pulled out. Let now ˝ jl be a period matrix of the Riemann surface X of genus g. The function Abstract. Weierstrass Problem If compact-additional conditions are needed. 1Clearly m and H determines each other. Given a Riemann surface C, we call the abelian di erential a global (1;0)-form on C, i.e. We will freely use the fact that any line bundle on a compact Riemann surface actually comes from a divisor, which can be seen by Jacobi’s inversion theorem, and define the degree of simply as the degree of its Let C(X) denote the eld of meromorphic functions on X. ), is the divisor de ned by the order function (!) Raghavan Narasimhan. Symmetric Products of Riemann Surfaces This talk will introduce symmetric products of Riemann surfaces and explains how to visualize maps of discs into such spaces. Proposition 1.1.2. For a compact Riemann surface M we denote by L(M) the abelian group (Z-module) of isomorphism classes of holomorphic line bundles L!Mwhich admit Riemann Surfaces and Graphs 6. Recall the Riemann-Roch theorem for Riemann surfaces: Theorem 1. a Riemann surface Let ☐ = [np [p] divisor on . The equality c 1(O X(P)) = X is the combination of the basic calculations 80 and 81 given in detail at the end of these notes. In this section, we study the influence of the theta divisor on the Riemann surface X. Let Mbe a compact connected Riemann surface and let Dbe a divisor on M. Then: dim C H 0(M;D) dim C H1(M;D) = d g+ 1 d= degree (D) g= genus (M) HIRZEBRUCH-RIEMANN-ROCH Mnon-singular projective algebraic variety / C Ean algebraic vector bundle … Let X - Mg be a genus g compact oriented Riemann surface with g > 0, Mo-E)1 being the Riemann sphere. Riemann-Roch and Serre Duality 29 4.4. Stack Exchange network consists of 179 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.. Visit Stack Exchange First introduced by E.E. 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