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122-502: Initially proved as Riemann's inequality by Riemann (1857) , the theorem reached its definitive form for Riemann surfaces after work of Riemann 's short-lived student Gustav Roch  ( 1865 ). It was later generalized to algebraic curves , to higher-dimensional varieties and beyond. A Riemann surface X {\displaystyle X} is a topological space that is locally homeomorphic to an open subset of C {\displaystyle \mathbb {C} } ,

244-423: A compact Riemann surface X is the free abelian group on the points of X . Equivalently, a divisor on a compact Riemann surface X is a finite linear combination of points of X with integer coefficients. The degree of a divisor on X is the sum of its coefficients. For any nonzero meromorphic function f on X , one can define the order of vanishing of f at a point p in X , ord p ( f ). It

366-417: A manifold , no matter how distorted it is. In his dissertation, he established a geometric foundation for complex analysis through Riemann surfaces , through which multi-valued functions like the logarithm (with infinitely many sheets) or the square root (with two sheets) could become one-to-one functions . Complex functions are harmonic functions (that is, they satisfy Laplace's equation and thus

488-410: A Cartier divisor { ( U i , f i ) } {\displaystyle \{(U_{i},f_{i})\}} on an integral Noetherian scheme X determines a Weil divisor on X in a natural way, by applying div {\displaystyle \operatorname {div} } to the functions f i on the open sets U i . If X is normal, a Cartier divisor is determined by

610-633: A Fourier series representing a continuous, almost nowhere-differentiable function, a case not covered by Dirichlet. He also proved the Riemann–Lebesgue lemma : if a function is representable by a Fourier series, then the Fourier coefficients go to zero for large  n . Riemann's essay was also the starting point for Georg Cantor 's work with Fourier series, which was the impetus for set theory . He also worked with hypergeometric differential equations in 1857 using complex analytical methods and presented

732-475: A Noetherian ring, but it can fail in general (even for proper schemes over C ), which lessens the interest of Cartier divisors in full generality. Assume D is an effective Cartier divisor. Then there is a short exact sequence This sequence is derived from the short exact sequence relating the structure sheaves of X and D and the ideal sheaf of D . Because D is a Cartier divisor, O ( D ) {\displaystyle {\mathcal {O}}(D)}

854-499: A Weil divisor: The sheaf can be restricted to the regular locus, where it becomes free and so corresponds to a Cartier divisor (again, see below), and because the singular locus has codimension at least two, the closure of the Cartier divisor is a Weil divisor. The Weil divisor class group Cl( X ) is the quotient of Div( X ) by the subgroup of all principal Weil divisors. Two divisors are said to be linearly equivalent if their difference

976-470: A compact Riemann surface, the degree of a principal divisor is zero; that is, the number of zeros of a meromorphic function is equal to the number of poles, counted with multiplicity. As a result, the degree is well-defined on linear equivalence classes of divisors. Given a divisor D on a compact Riemann surface X , it is important to study the complex vector space of meromorphic functions on X with poles at most given by D , called H ( X , O ( D )) or

1098-524: A competition with Weierstrass since 1857 to solve the Jacobian inverse problems for abelian integrals, a generalization of elliptic integrals . Riemann used theta functions in several variables and reduced the problem to the determination of the zeros of these theta functions. Riemann also investigated period matrices and characterized them through the "Riemannian period relations" (symmetric, real part negative). By Ferdinand Georg Frobenius and Solomon Lefschetz

1220-415: A curve (the free abelian group generated by all divisors) is closely related to the group of fractional ideals for a Dedekind domain. An algebraic cycle is a higher codimension generalization of a divisor; by definition, a Weil divisor is a cycle of codimension 1. A Riemann surface is a 1-dimensional complex manifold , and so its codimension-1 submanifolds have dimension 0. The group of divisors on

1342-526: A curve has ( d  − 1)( d  − 2)/2 different singularities, it is a rational curve and, thus, admits a rational parameterization. The Riemann–Hurwitz formula concerning (ramified) maps between Riemann surfaces or algebraic curves is a consequence of the Riemann–Roch theorem. Bernhard Riemann Georg Friedrich Bernhard Riemann ( German: [ˈɡeːɔʁk ˈfʁiːdʁɪç ˈbɛʁnhaʁt ˈʁiːman] ; 17 September 1826 – 20 July 1866)

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1464-739: A fear of speaking in public. During 1840, Riemann went to Hanover to live with his grandmother and attend lyceum (middle school years), because such a type of school was not accessible from his home village. After the death of his grandmother in 1842, he transferred to the Johanneum Lüneburg , a high school in Lüneburg . There, Riemann studied the Bible intensively, but he was often distracted by mathematics. His teachers were amazed by his ability to perform complicated mathematical operations, in which he often outstripped his instructor's knowledge. In 1846, at

1586-473: A field. Let D be a Weil divisor. Then O ( D ) {\displaystyle {\mathcal {O}}(D)} is a rank one reflexive sheaf , and since O ( D ) {\displaystyle {\mathcal {O}}(D)} is defined as a subsheaf of M X , {\displaystyle {\mathcal {M}}_{X},} it is a fractional ideal sheaf (see below). Conversely, every rank one reflexive sheaf corresponds to

1708-401: A generalization of codimension -1 subvarieties of algebraic varieties . Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mumford ). Both are derived from the notion of divisibility in the integers and algebraic number fields . Globally, every codimension-1 subvariety of projective space is defined by

1830-526: A generalization of the theorem to algebraic curves . The theorem will be illustrated by picking a point P {\displaystyle P} on the surface in question and regarding the sequence of numbers i.e., the dimension of the space of functions that are holomorphic everywhere except at P {\displaystyle P} where the function is allowed to have a pole of order at most n {\displaystyle n} . For n = 0 {\displaystyle n=0} ,

1952-455: A hamlet of Verbania on Lake Maggiore ), where he was buried in the cemetery in Biganzolo (Verbania). Riemann was a dedicated Christian, the son of a Protestant minister, and saw his life as a mathematician as another way to serve God. During his life, he held closely to his Christian faith and considered it to be the most important aspect of his life. At the time of his death, he was reciting

2074-599: A meromorphic form on the Riemann sphere: it has a double pole at infinity, since Thus, its canonical divisor is K := div ⁡ ( ω ) = − 2 P {\displaystyle K:=\operatorname {div} (\omega )=-2P} (where P {\displaystyle P} is the point at infinity). Therefore, the theorem says that the sequence ℓ ( n ⋅ P ) {\displaystyle \ell (n\cdot P)} reads This sequence can also be read off from

2196-479: A monoid with product given as the reflexive hull of a tensor product. Then D ↦ O X ( D ) {\displaystyle D\mapsto {\mathcal {O}}_{X}(D)} defines a monoid isomorphism from the Weil divisor class group of X to the monoid of isomorphism classes of rank-one reflexive sheaves on X . Let X be a normal variety over a perfect field . The smooth locus U of X

2318-456: A natural, geometric treatment of complex analysis. His 1859 paper on the prime-counting function , containing the original statement of the Riemann hypothesis , is regarded as a foundational paper of analytic number theory . Through his pioneering contributions to differential geometry , Riemann laid the foundations of the mathematics of general relativity . He is considered by many to be one of

2440-478: A number (scalar), with the surfaces of constant positive or negative curvature being models of the non-Euclidean geometries . The Riemann metric is a collection of numbers at every point in space (i.e., a tensor ) which allows measurements of speed in any trajectory, whose integral gives the distance between the trajectory's endpoints. For example, Riemann found that in four spatial dimensions, one needs ten numbers at each point to describe distances and curvatures on

2562-463: A quotient g / h , where g and h are in O X , Z , {\displaystyle {\mathcal {O}}_{X,Z},} and the order of vanishing of f is defined to be ord Z ( g ) − ord Z ( h ) . With this definition, the order of vanishing is a function ord Z  : k ( X ) → Z . If X is normal , then the local ring O X , Z {\displaystyle {\mathcal {O}}_{X,Z}}

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2684-447: A result of this property, much of algebraic geometry studies an arbitrary variety by analysing its codimension-1 subvarieties and the corresponding line bundles . On singular varieties, this property can also fail, and so one has to distinguish between codimension-1 subvarieties and varieties which can locally be defined by one equation. The former are Weil divisors while the latter are Cartier divisors. Topologically, Weil divisors play

2806-437: A section of O X × . {\displaystyle {\mathcal {O}}_{X}^{\times }.} Cartier divisors also have a sheaf-theoretic description. A fractional ideal sheaf is a sub- O X {\displaystyle {\mathcal {O}}_{X}} -module of M X . {\displaystyle {\mathcal {M}}_{X}.} A fractional ideal sheaf J

2928-539: A short exact sequence A Cartier divisor on X is a global section of M X × / O X × . {\displaystyle {\mathcal {M}}_{X}^{\times }/{\mathcal {O}}_{X}^{\times }.} An equivalent description is that a Cartier divisor is a collection { ( U i , f i ) } , {\displaystyle \{(U_{i},f_{i})\},} where { U i } {\displaystyle \{U_{i}\}}

3050-547: A theta function lies. Through the summation of this approximation function over the non-trivial zeros on the line with real portion 1/2, he gave an exact, "explicit formula" for π ( x ) {\displaystyle \pi (x)} . Riemann knew of Pafnuty Chebyshev 's work on the Prime Number Theorem . He had visited Dirichlet in 1852. Riemann's works include: Divisor (algebraic geometry)#Weil divisors In algebraic geometry , divisors are

3172-612: A trivial bundle on each open set. For each U i , choose an isomorphism O U i → O ( D ) | U i . {\displaystyle {\mathcal {O}}_{U_{i}}\to {\mathcal {O}}(D)|_{U_{i}}.} The image of 1 ∈ Γ ( U i , O U i ) = Γ ( U i , O X ) {\displaystyle 1\in \Gamma (U_{i},{\mathcal {O}}_{U_{i}})=\Gamma (U_{i},{\mathcal {O}}_{X})} under this map

3294-413: Is invertible if, for each x in X , there exists an open neighborhood U of x on which the restriction of J to U is equal to O U ⋅ f , {\displaystyle {\mathcal {O}}_{U}\cdot f,} where f ∈ M X × ( U ) {\displaystyle f\in {\mathcal {M}}_{X}^{\times }(U)} and

3416-450: Is a discrete valuation ring , and the function ord Z is the corresponding valuation. For a non-zero rational function f on X , the principal Weil divisor associated to f is defined to be the Weil divisor It can be shown that this sum is locally finite and hence that it indeed defines a Weil divisor. The principal Weil divisor associated to f is also notated ( f ) . If f is a regular function, then its principal Weil divisor

3538-400: Is a more precise statement along these lines. On the other hand, the precise dimension of H ( X , O ( D )) for divisors D of low degree is subtle, and not completely determined by the degree of D . The distinctive features of a compact Riemann surface are reflected in these dimensions. One key divisor on a compact Riemann surface is the canonical divisor . To define it, one first defines

3660-466: Is a section of O ( D ) {\displaystyle {\mathcal {O}}(D)} on U i . Because O ( D ) {\displaystyle {\mathcal {O}}(D)} is defined to be a subsheaf of the sheaf of rational functions, the image of 1 may be identified with some rational function f i . The collection { ( U i , f i ) } {\displaystyle \{(U_{i},f_{i})\}}

3782-539: Is a two-dimensional lattice (a group isomorphic to Z 2 {\displaystyle \mathbb {Z} ^{2}} ). Its genus is one: its first singular homology group is freely generated by two loops, as shown in the illustration at the right. The standard complex coordinate z {\displaystyle z} on C {\displaystyle C} yields a one-form ω = d z {\displaystyle \omega =dz} on X {\displaystyle X} that

Riemann–Roch theorem - Misplaced Pages Continue

3904-437: Is always a line bundle. In general, however, a Weil divisor on a normal scheme need not be locally principal; see the examples of quadric cones above. Effective Cartier divisors are those which correspond to ideal sheaves. In fact, the theory of effective Cartier divisors can be developed without any reference to sheaves of rational functions or fractional ideal sheaves. Let X be a scheme. An effective Cartier divisor on X

4026-525: Is always true that at most points the sequence starts with g + 1 {\displaystyle g+1} ones and there are finitely many points with other sequences (see Weierstrass points ). Using the close correspondence between divisors and holomorphic line bundles on a Riemann surface, the theorem can also be stated in a different, yet equivalent way: let L be a holomorphic line bundle on X . Let H 0 ( X , L ) {\displaystyle H^{0}(X,L)} denote

4148-514: Is an integral closed subscheme Z of codimension 1 in X . A Weil divisor on X is a formal sum over the prime divisors Z of X , where the collection { Z : n Z ≠ 0 } {\displaystyle \{Z:n_{Z}\neq 0\}} is locally finite. If X is quasi-compact, local finiteness is equivalent to { Z : n Z ≠ 0 } {\displaystyle \{Z:n_{Z}\neq 0\}} being finite. The group of all Weil divisors

4270-432: Is an effective divisor and so f g {\displaystyle fg} is regular thanks to the normality of X . Conversely, if O ( D ) {\displaystyle {\mathcal {O}}(D)} is isomorphic to O X {\displaystyle {\mathcal {O}}_{X}} as an O X {\displaystyle {\mathcal {O}}_{X}} -module, then D

4392-416: Is an element of the free abelian group on the points of the surface. Equivalently, a divisor is a finite linear combination of points of the surface with integer coefficients. Any meromorphic function f {\displaystyle f} gives rise to a divisor denoted ( f ) {\displaystyle (f)} defined as where R ( f ) {\displaystyle R(f)}

4514-520: Is an increasing sequence. The Riemann sphere (also called complex projective line ) is simply connected and hence its first singular homology is zero. In particular its genus is zero. The sphere can be covered by two copies of C {\displaystyle \mathbb {C} } , with transition map being given by Therefore, the form ω = d z {\displaystyle \omega =dz} on one copy of C {\displaystyle \mathbb {C} } extends to

4636-425: Is an integer, negative if f has a pole at p . The divisor of a nonzero meromorphic function f on the compact Riemann surface X is defined as which is a finite sum. Divisors of the form ( f ) are also called principal divisors . Since ( fg ) = ( f ) + ( g ), the set of principal divisors is a subgroup of the group of divisors. Two divisors that differ by a principal divisor are called linearly equivalent . On

4758-520: Is an open cover of X , f i {\displaystyle X,f_{i}} is a section of M X × {\displaystyle {\mathcal {M}}_{X}^{\times }} on U i , {\displaystyle U_{i},} and f i = f j {\displaystyle f_{i}=f_{j}} on U i ∩ U j {\displaystyle U_{i}\cap U_{j}} up to multiplication by

4880-446: Is an open subset whose complement has codimension at least 2. Let j : U → X be the inclusion map, then the restriction homomorphism: is an isomorphism, since X − U has codimension at least 2 in X . For example, one can use this isomorphism to define the canonical divisor K X of X : it is the Weil divisor (up to linear equivalence) corresponding to the line bundle of differential forms of top degree on U . Equivalently,

5002-525: Is called the Hilbert polynomial of a genus g curve . Analyzing this equation further, the Euler characteristic reads as Since deg ⁡ ( ω C ⊗ n ) = n ( 2 g − 2 ) {\displaystyle \deg(\omega _{C}^{\otimes n})=n(2g-2)} for n ≥ 3 {\displaystyle n\geq 3} , since its degree

Riemann–Roch theorem - Misplaced Pages Continue

5124-450: Is denoted Div( X ) . A Weil divisor D is effective if all the coefficients are non-negative. One writes D ≥ D′ if the difference D − D′ is effective. For example, a divisor on an algebraic curve over a field is a formal sum of finitely many closed points. A divisor on Spec Z is a formal sum of prime numbers with integer coefficients and therefore corresponds to a non-zero fractional ideal in Q . A similar characterization

5246-483: Is effective, but in general this is not true. The additivity of the order of vanishing function implies that Consequently div is a homomorphism, and in particular its image is a subgroup of the group of all Weil divisors. Let X be a normal integral Noetherian scheme. Every Weil divisor D determines a coherent sheaf O X ( D ) {\displaystyle {\mathcal {O}}_{X}(D)} on X . Concretely it may be defined as subsheaf of

5368-549: Is everywhere holomorphic, i.e., has no poles at all. Therefore, K {\displaystyle K} , the divisor of ω {\displaystyle \omega } is zero. On this surface, this sequence is and this characterises the case g = 1 {\displaystyle g=1} . Indeed, for D = 0 {\displaystyle D=0} , ℓ ( K − D ) = ℓ ( 0 ) = 1 {\displaystyle \ell (K-D)=\ell (0)=1} , as

5490-495: Is generally considered while constructing the Hilbert scheme of curves (and the moduli space of algebraic curves ). This polynomial is H C ( t ) = ( 6 t − 1 ) ( g − 1 ) = 6 ( g − 1 ) t + ( 1 − g ) {\displaystyle {\begin{aligned}H_{C}(t)&=(6t-1)(g-1)\\&=6(g-1)t+(1-g)\end{aligned}}} and

5612-473: Is invertible. When this happens, O ( D ) {\displaystyle {\mathcal {O}}(D)} (with its embedding in M X ) is the line bundle associated to a Cartier divisor. More precisely, if O ( D ) {\displaystyle {\mathcal {O}}(D)} is invertible, then there exists an open cover { U i } such that O ( D ) {\displaystyle {\mathcal {O}}(D)} restricts to

5734-423: Is isomorphic to the affine n -space with the coordinates y i = x i / x 0 . Let Then ω is a rational differential form on U ; thus, it is a rational section of Ω P n n {\displaystyle \Omega _{\mathbf {P} ^{n}}^{n}} which has simple poles along Z i = { x i = 0}, i = 1, ..., n . Switching to a different affine chart changes only

5856-474: Is locally free, and hence tensoring that sequence by O ( D ) {\displaystyle {\mathcal {O}}(D)} yields another short exact sequence, the one above. When D is smooth, O D ( D ) {\displaystyle O_{D}(D)} is the normal bundle of D in X . A Weil divisor D is said to be Cartier if and only if the sheaf O ( D ) {\displaystyle {\mathcal {O}}(D)}

5978-1089: Is negative for all g ≥ 2 {\displaystyle g\geq 2} , implying it has no global sections, there is an embedding into some projective space from the global sections of ω C ⊗ n {\displaystyle \omega _{C}^{\otimes n}} . In particular, ω C ⊗ 3 {\displaystyle \omega _{C}^{\otimes 3}} gives an embedding into P N ≅ P ( H 0 ( C , ω C ⊗ 3 ) ) {\displaystyle \mathbb {P} ^{N}\cong \mathbb {P} (H^{0}(C,\omega _{C}^{\otimes 3}))} where N = 5 g − 5 − 1 = 5 g − 6 {\displaystyle N=5g-5-1=5g-6} since h 0 ( ω C ⊗ 3 ) = 6 g − 6 − g + 1 {\displaystyle h^{0}(\omega _{C}^{\otimes 3})=6g-6-g+1} . This

6100-443: Is negative, then we require that h {\displaystyle h} has a zero of at least that multiplicity at z {\displaystyle z} – if the coefficient in D {\displaystyle D} is positive, h {\displaystyle h} can have a pole of at most that order. The vector spaces for linearly equivalent divisors are naturally isomorphic through multiplication with

6222-487: Is non-zero, then the order of vanishing of f along Z , written ord Z ( f ) , is the length of O X , Z / ( f ) . {\displaystyle {\mathcal {O}}_{X,Z}/(f).} This length is finite, and it is additive with respect to multiplication, that is, ord Z ( fg ) = ord Z ( f ) + ord Z ( g ) . If k ( X ) is the field of rational functions on X , then any non-zero f ∈ k ( X ) may be written as

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6344-653: Is paralleled by the condition that the algebraic curve be complete , which is equivalent to being projective . Over a general field k , there is no good notion of singular (co)homology. The so-called geometric genus is defined as i.e., as the dimension of the space of globally defined (algebraic) one-forms (see Kähler differential ). Finally, meromorphic functions on a Riemann surface are locally represented as fractions of holomorphic functions. Hence they are replaced by rational functions which are locally fractions of regular functions . Thus, writing ℓ ( D ) {\displaystyle \ell (D)} for

6466-461: Is principal, so the divisor class group is the group of divisors modulo linear equivalence. For a variety X of dimension n over a field, the divisor class group is a Chow group ; namely, Cl( X ) is the Chow group CH n −1 ( X ) of ( n −1)-dimensional cycles. Let Z be a closed subset of X . If Z is irreducible of codimension one, then Cl( X − Z ) is isomorphic to the quotient group of Cl( X ) by

6588-813: Is principal. It follows that D is locally principal if and only if O ( D ) {\displaystyle {\mathcal {O}}(D)} is invertible; that is, a line bundle. If D is an effective divisor that corresponds to a subscheme of X (for example D can be a reduced divisor or a prime divisor), then the ideal sheaf of the subscheme D is equal to O ( − D ) . {\displaystyle {\mathcal {O}}(-D).} This leads to an often used short exact sequence, The sheaf cohomology of this sequence shows that H 1 ( X , O X ( − D ) ) {\displaystyle H^{1}(X,{\mathcal {O}}_{X}(-D))} contains information on whether regular functions on D are

6710-534: Is the Jacobian variety of the Riemann surface, an example of an abelian manifold. Many mathematicians such as Alfred Clebsch furthered Riemann's work on algebraic curves. These theories depended on the properties of a function defined on Riemann surfaces. For example, the Riemann–Roch theorem (Roch was a student of Riemann) says something about the number of linearly independent differentials (with known conditions on

6832-549: Is the divisor of a rational function on X . Two Cartier divisors are linearly equivalent if their difference is principal. Every line bundle L on an integral Noetherian scheme X is the class of some Cartier divisor. As a result, the exact sequence above identifies the Picard group of line bundles on an integral Noetherian scheme X with the group of Cartier divisors modulo linear equivalence. This holds more generally for reduced Noetherian schemes, or for quasi-projective schemes over

6954-422: Is the image of a nowhere vanishing rational function, its image in O ( D ) {\displaystyle {\mathcal {O}}(D)} vanishes along D because the transition functions vanish along D . When D is a smooth Cartier divisor, the cokernel of the above inclusion may be identified; see #Cartier divisors below. Assume that X is a normal integral separated scheme of finite type over

7076-519: Is the set of all zeroes and poles of f {\displaystyle f} , and s ν {\displaystyle s_{\nu }} is given by The set R ( f ) {\displaystyle R(f)} is known to be finite; this is a consequence of X {\displaystyle X} being compact and the fact that the zeros of a (non-zero) holomorphic function do not have an accumulation point . Therefore, ( f ) {\displaystyle (f)}

7198-419: Is then a Cartier divisor. This is well-defined because the only choices involved were of the covering and of the isomorphism, neither of which change the Cartier divisor. This Cartier divisor may be used to produce a sheaf, which for distinction we will notate L ( D ). There is an isomorphism of O ( D ) {\displaystyle {\mathcal {O}}(D)} with L ( D ) defined by working on

7320-411: Is thought of as a correction term (also called index of speciality) so the theorem may be roughly paraphrased by saying Because it is the dimension of a vector space, the correction term ℓ ( K − D ) {\displaystyle \ell (K-D)} is always non-negative, so that This is called Riemann's inequality . Roch's part of the statement is the description of

7442-476: Is true for divisors on Spec ⁡ O K , {\displaystyle \operatorname {Spec} {\mathcal {O}}_{K},} where K is a number field. If Z ⊂ X is a prime divisor, then the local ring O X , Z {\displaystyle {\mathcal {O}}_{X,Z}} has Krull dimension one. If f ∈ O X , Z {\displaystyle f\in {\mathcal {O}}_{X,Z}}

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7564-430: Is useful in the construction of the moduli space of algebraic curves because it can be used as the projective space to construct the Hilbert scheme with Hilbert polynomial H C ( t ) {\displaystyle H_{C}(t)} . An irreducible plane algebraic curve of degree d has ( d  − 1)( d  − 2)/2 −  g singularities, when properly counted. It follows that, if

7686-428: Is well-defined. Any divisor of this form is called a principal divisor . Two divisors that differ by a principal divisor are called linearly equivalent . The divisor of a meromorphic 1-form is defined similarly. A divisor of a global meromorphic 1-form is called the canonical divisor (usually denoted K {\displaystyle K} ). Any two meromorphic 1-forms will yield linearly equivalent divisors, so

7808-481: Is whether the canonical divisor has negative degree (so X has genus zero), zero degree (genus one), or positive degree (genus at least 2). For example, this determines whether X has a Kähler metric with positive curvature , zero curvature, or negative curvature. The canonical divisor has negative degree if and only if X is isomorphic to the Riemann sphere CP . Let X be an integral locally Noetherian scheme . A prime divisor or irreducible divisor on X

7930-413: The C {\displaystyle \mathbb {C} } -dimension of the first singular homology group H 1 ( X , C ) {\displaystyle H_{1}(X,\mathbb {C} )} with complex coefficients. The genus classifies compact Riemann surfaces up to homeomorphism , i.e., two such surfaces are homeomorphic if and only if their genus is the same. Therefore,

8052-578: The Cauchy–Riemann equations ) on these surfaces and are described by the location of their singularities and the topology of the surfaces. The topological "genus" of the Riemann surfaces is given by g = w / 2 − n + 1 {\displaystyle g=w/2-n+1} , where the surface has n {\displaystyle n} leaves coming together at w {\displaystyle w} branch points. For g > 1 {\displaystyle g>1}

8174-563: The Dirichlet principle . Karl Weierstrass found a gap in the proof: Riemann had not noticed that his working assumption (that the minimum existed) might not work; the function space might not be complete, and therefore the existence of a minimum was not guaranteed. Through the work of David Hilbert in the Calculus of Variations, the Dirichlet principle was finally established. Otherwise, Weierstrass

8296-877: The Lord's Prayer with his wife and died before they finished saying the prayer. Meanwhile, in Göttingen his housekeeper discarded some of the papers in his office, including much unpublished work. Riemann refused to publish incomplete work, and some deep insights may have been lost. Riemann's tombstone in Biganzolo (Italy) refers to Romans 8:28 : Georg Friedrich Bernhard Riemann Professor in Göttingen born in Breselenz, 17 September 1826 died in Selasca, 20 July 1866 Riemann's published works opened up research areas combining analysis with geometry. These would subsequently become major parts of

8418-595: The method of least squares ). Gauss recommended that Riemann give up his theological work and enter the mathematical field; after getting his father's approval, Riemann transferred to the University of Berlin in 1847. During his time of study, Carl Gustav Jacob Jacobi , Peter Gustav Lejeune Dirichlet , Jakob Steiner , and Gotthold Eisenstein were teaching. He stayed in Berlin for two years and returned to Göttingen in 1849. Riemann held his first lectures in 1854, which founded

8540-419: The space of sections of the line bundle associated to D . The degree of D says a lot about the dimension of this vector space. For example, if D has negative degree, then this vector space is zero (because a meromorphic function cannot have more zeros than poles). If D has positive degree, then the dimension of H ( X , O ( mD )) grows linearly in m for m sufficiently large. The Riemann–Roch theorem

8662-401: The zeta function that now bears his name, establishing its importance for understanding the distribution of prime numbers . The Riemann hypothesis was one of a series of conjectures he made about the function's properties. In Riemann's work, there are many more interesting developments. He proved the functional equation for the zeta function (already known to Leonhard Euler ), behind which

8784-761: The Cartier divisor is denoted D , then the corresponding fractional ideal sheaf is denoted O ( D ) {\displaystyle {\mathcal {O}}(D)} or L ( D ). By the exact sequence above, there is an exact sequence of sheaf cohomology groups: A Cartier divisor is said to be principal if it is in the image of the homomorphism H 0 ( X , M X × ) → H 0 ( X , M X × / O X × ) , {\displaystyle H^{0}(X,{\mathcal {M}}_{X}^{\times })\to H^{0}(X,{\mathcal {M}}_{X}^{\times }/{\mathcal {O}}_{X}^{\times }),} that is, if it

8906-476: The Riemann surface has ( 3 g − 3 ) {\displaystyle (3g-3)} parameters (the " moduli "). His contributions to this area are numerous. The famous Riemann mapping theorem says that a simply connected domain in the complex plane is "biholomorphically equivalent" (i.e. there is a bijection between them that is holomorphic with a holomorphic inverse) to either C {\displaystyle \mathbb {C} } or to

9028-628: The Riemann–Roch formula reads Giving the degree 1 {\displaystyle 1} Hilbert polynomial of ω C {\displaystyle \omega _{C}} Because the tri-canonical sheaf ω C ⊗ 3 {\displaystyle \omega _{C}^{\otimes 3}} is used to embed the curve, the Hilbert polynomial H C ( t ) = H ω C ⊗ 3 ( t ) {\displaystyle H_{C}(t)=H_{\omega _{C}^{\otimes 3}}(t)}

9150-410: The Riemann–Roch theorem, the surface X {\displaystyle X} is always assumed to be compact . Colloquially speaking, the genus g {\displaystyle g} of a Riemann surface is its number of handles ; for example the genus of the Riemann surface shown at the right is three. More precisely, the genus is defined as half of the first Betti number , i.e., half of

9272-552: The University of Göttingen), he was promoted to head the mathematics department at the University of Göttingen. He was also the first to suggest using dimensions higher than merely three or four in order to describe physical reality. In 1862 he married Elise Koch; their daughter Ida Schilling was born on 22 December 1862. Riemann fled Göttingen when the armies of Hanover and Prussia clashed there in 1866. He died of tuberculosis during his third journey to Italy in Selasca (now

9394-410: The above formulation of the Riemann–Roch theorem for divisors on Riemann surfaces has an analogue in algebraic geometry . The analogue of a Riemann surface is a non-singular algebraic curve C over a field k . The difference in terminology (curve vs. surface) is because the dimension of a Riemann surface as a real manifold is two, but one as a complex manifold. The compactness of a Riemann surface

9516-424: The age of 19, he started studying philology and Christian theology in order to become a pastor and help with his family's finances. During the spring of 1846, his father, after gathering enough money, sent Riemann to the University of Göttingen , where he planned to study towards a degree in theology . However, once there, he began studying mathematics under Carl Friedrich Gauss (specifically his lectures on

9638-456: The associated Weil divisor, and a Weil divisor is Cartier if and only if it is locally principal. A Noetherian scheme X is called factorial if all local rings of X are unique factorization domains . (Some authors say "locally factorial".) In particular, every regular scheme is factorial. On a factorial scheme X , every Weil divisor D is locally principal, and so O ( D ) {\displaystyle {\mathcal {O}}(D)}

9760-476: The canonical bundle K {\displaystyle K} has h 0 ( X , K ) = g {\displaystyle h^{0}(X,K)=g} , applying Riemann–Roch to L = K {\displaystyle L=K} gives which can be rewritten as hence the degree of the canonical bundle is deg ⁡ ( K ) = 2 g − 2 {\displaystyle \deg(K)=2g-2} . Every item in

9882-427: The canonical divisor is uniquely determined up to linear equivalence (hence "the" canonical divisor). The symbol deg ⁡ ( D ) {\displaystyle \deg(D)} denotes the degree (occasionally also called index) of the divisor D {\displaystyle D} , i.e. the sum of the coefficients occurring in D {\displaystyle D} . It can be shown that

10004-481: The canonical sheaf ω C {\displaystyle \omega _{C}} has degree 2 g − 2 {\displaystyle 2g-2} , which gives an ample line bundle for genus g ≥ 2 {\displaystyle g\geq 2} . If we set ω C ( n ) = ω C ⊗ n {\displaystyle \omega _{C}(n)=\omega _{C}^{\otimes n}} then

10126-654: The class of Z . If Z has codimension at least 2 in X , then the restriction Cl( X ) → Cl( X − Z ) is an isomorphism. (These facts are special cases of the localization sequence for Chow groups.) On a normal integral Noetherian scheme X , two Weil divisors D , E are linearly equivalent if and only if O ( D ) {\displaystyle {\mathcal {O}}(D)} and O ( E ) {\displaystyle {\mathcal {O}}(E)} are isomorphic as O X {\displaystyle {\mathcal {O}}_{X}} -modules. Isomorphism classes of reflexive sheaves on X form

10248-455: The dimension (over k ) of the space of rational functions on the curve whose poles at every point are not worse than the corresponding coefficient in D , the very same formula as above holds: where C is a projective non-singular algebraic curve over an algebraically closed field k . In fact, the same formula holds for projective curves over any field, except that the degree of a divisor needs to take into account multiplicities coming from

10370-450: The divisor of a global meromorphic function always has degree 0, so the degree of a divisor depends only on its linear equivalence class. The number ℓ ( D ) {\displaystyle \ell (D)} is the quantity that is of primary interest: the dimension (over C {\displaystyle \mathbb {C} } ) of the vector space of meromorphic functions h {\displaystyle h} on

10492-509: The divisor of a nonzero meromorphic 1-form along the lines above. Since the space of meromorphic 1-forms is a 1-dimensional vector space over the field of meromorphic functions, any two nonzero meromorphic 1-forms yield linearly equivalent divisors. Any divisor in this linear equivalence class is called the canonical divisor of X , K X . The genus g of X can be read from the canonical divisor: namely, K X has degree 2 g − 2. The key trichotomy among compact Riemann surfaces X

10614-400: The field of Riemannian geometry and thereby set the stage for Albert Einstein 's general theory of relativity . In 1857, there was an attempt to promote Riemann to extraordinary professor status at the University of Göttingen . Although this attempt failed, it did result in Riemann finally being granted a regular salary. In 1859, following the death of Dirichlet (who held Gauss 's chair at

10736-821: The field of real analysis , he discovered the Riemann integral in his habilitation . Among other things, he showed that every piecewise continuous function is integrable. Similarly, the Stieltjes integral goes back to the Göttinger mathematician, and so they are named together the Riemann–Stieltjes integral . In his habilitation work on Fourier series , where he followed the work of his teacher Dirichlet, he showed that Riemann-integrable functions are "representable" by Fourier series. Dirichlet has shown this for continuous, piecewise-differentiable functions (thus with countably many non-differentiable points). Riemann gave an example of

10858-405: The foundations of geometry. Over many months, Riemann developed his theory of higher dimensions and delivered his lecture at Göttingen on 10 June 1854, entitled Ueber die Hypothesen, welche der Geometrie zu Grunde liegen . It was not published until twelve years later in 1868 by Dedekind, two years after his death. Its early reception appears to have been slow, but it is now recognized as one of

10980-418: The functions are thus required to be entire , i.e., holomorphic on the whole surface X {\displaystyle X} . By Liouville's theorem , such a function is necessarily constant. Therefore, ℓ ( 0 ) = 1 {\displaystyle \ell (0)=1} . In general, the sequence ℓ ( n ⋅ P ) {\displaystyle \ell (n\cdot P)}

11102-436: The genus is an important topological invariant of a Riemann surface. On the other hand, Hodge theory shows that the genus coincides with the C {\displaystyle \mathbb {C} } -dimension of the space of holomorphic one-forms on X {\displaystyle X} , so the genus also encodes complex-analytic information about the Riemann surface. A divisor D {\displaystyle D}

11224-477: The global meromorphic function (which is well-defined up to a scalar). The Riemann–Roch theorem for a compact Riemann surface of genus g {\displaystyle g} with canonical divisor K {\displaystyle K} states Typically, the number ℓ ( D ) {\displaystyle \ell (D)} is the one of interest, while ℓ ( K − D ) {\displaystyle \ell (K-D)}

11346-645: The greatest mathematicians of all time. Riemann was born on 17 September 1826 in Breselenz , a village near Dannenberg in the Kingdom of Hanover . His father, Friedrich Bernhard Riemann, was a poor Lutheran pastor in Breselenz who fought in the Napoleonic Wars . His mother, Charlotte Ebell, died in 1846. Riemann was the second of six children. Riemann exhibited exceptional mathematical talent, such as calculation abilities, from an early age but suffered from timidity and

11468-441: The important consequences of Riemann–Roch is it gives a formula for computing the Hilbert polynomial of line bundles on a curve. If a line bundle L {\displaystyle {\mathcal {L}}} is ample, then the Hilbert polynomial will give the first degree L ⊗ n {\displaystyle {\mathcal {L}}^{\otimes n}} giving an embedding into projective space. For example,

11590-436: The interior of the unit circle. The generalization of the theorem to Riemann surfaces is the famous uniformization theorem , which was proved in the 19th century by Henri Poincaré and Felix Klein . Here, too, rigorous proofs were first given after the development of richer mathematical tools (in this case, topology). For the proof of the existence of functions on Riemann surfaces, he used a minimality condition, which he called

11712-489: The most important works in geometry. The subject founded by this work is Riemannian geometry . Riemann found the correct way to extend into n dimensions the differential geometry of surfaces, which Gauss himself proved in his theorema egregium . The fundamental objects are called the Riemannian metric and the Riemann curvature tensor . For the surface (two-dimensional) case, the curvature at each point can be reduced to

11834-402: The open cover { U i }. The key fact to check here is that the transition functions of O ( D ) {\displaystyle {\mathcal {O}}(D)} and L ( D ) are compatible, and this amounts to the fact that these functions all have the form f i / f j . {\displaystyle f_{i}/f_{j}.} In the opposite direction,

11956-433: The possible difference between the sides of the inequality. On a general Riemann surface of genus g {\displaystyle g} , K {\displaystyle K} has degree 2 g − 2 {\displaystyle 2g-2} , independently of the meromorphic form chosen to represent the divisor. This follows from putting D = K {\displaystyle D=K} in

12078-440: The possible extensions of the base field and the residue fields of the points supporting the divisor. Finally, for a proper curve over an Artinian ring , the Euler characteristic of the line bundle associated to a divisor is given by the degree of the divisor (appropriately defined) plus the Euler characteristic of the structural sheaf O {\displaystyle {\mathcal {O}}} . The smoothness assumption in

12200-444: The product is taken in M X . {\displaystyle {\mathcal {M}}_{X}.} Each Cartier divisor defines an invertible fractional ideal sheaf using the description of the Cartier divisor as a collection { ( U i , f i ) } , {\displaystyle \{(U_{i},f_{i})\},} and conversely, invertible fractional ideal sheaves define Cartier divisors. If

12322-416: The restrictions of regular functions on X . There is also an inclusion of sheaves This furnishes a canonical element of Γ ( X , O X ( D ) ) , {\displaystyle \Gamma (X,{\mathcal {O}}_{X}(D)),} namely, the image of the global section 1. This is called the canonical section and may be denoted s D . While the canonical section

12444-416: The role of homology classes, while Cartier divisors represent cohomology classes. On a smooth variety (or more generally a regular scheme ), a result analogous to Poincaré duality says that Weil and Cartier divisors are the same. The name "divisor" goes back to the work of Dedekind and Weber , who showed the relevance of Dedekind domains to the study of algebraic curves . The group of divisors on

12566-437: The sequence mentioned above is It is shown from this that the ? term of degree 2 is either 1 or 2, depending on the point. It can be proven that in any genus 2 curve there are exactly six points whose sequences are 1, 1, 2, 2, ... and the rest of the points have the generic sequence 1, 1, 1, 2, ... In particular, a genus 2 curve is a hyperelliptic curve . For g > 2 {\displaystyle g>2} it

12688-414: The set of complex numbers . In addition, the transition maps between these open subsets are required to be holomorphic . The latter condition allows one to transfer the notions and methods of complex analysis dealing with holomorphic and meromorphic functions on C {\displaystyle \mathbb {C} } to the surface X {\displaystyle X} . For the purposes of

12810-456: The sheaf O ( K X ) {\displaystyle {\mathcal {O}}(K_{X})} on X is the direct image sheaf j ∗ Ω U n , {\displaystyle j_{*}\Omega _{U}^{n},} where n is the dimension of X . Example : Let X = P be the projective n -space with the homogeneous coordinates x 0 , ..., x n . Let U = { x 0 ≠ 0}. Then U

12932-517: The sheaf of rational functions That is, a nonzero rational function f is a section of O X ( D ) {\displaystyle {\mathcal {O}}_{X}(D)} over U if and only if for any prime divisor Z intersecting U , where n Z is the coefficient of Z in D . If D is principal, so D is the divisor of a rational function g , then there is an isomorphism since div ⁡ ( f g ) {\displaystyle \operatorname {div} (fg)}

13054-489: The sign of ω and so we see ω has a simple pole along Z 0 as well. Thus, the divisor of ω is and its divisor class is where [ H ] = [ Z i ], i = 0, ..., n . (See also the Euler sequence .) Let X be an integral Noetherian scheme. Then X has a sheaf of rational functions M X . {\displaystyle {\mathcal {M}}_{X}.} All regular functions are rational functions, which leads to

13176-418: The solutions through the behaviour of closed paths about singularities (described by the monodromy matrix ). The proof of the existence of such differential equations by previously known monodromy matrices is one of the Hilbert problems. Riemann made some famous contributions to modern analytic number theory . In a single short paper , the only one he published on the subject of number theory, he investigated

13298-537: The space of holomorphic sections of L . This space will be finite-dimensional; its dimension is denoted h 0 ( X , L ) {\displaystyle h^{0}(X,L)} . Let K denote the canonical bundle on X . Then, the Riemann–Roch theorem states that The theorem of the previous section is the special case of when L is a point bundle . The theorem can be applied to show that there are g linearly independent holomorphic sections of K , or one-forms on X , as follows. Taking L to be

13420-437: The surface, such that all the coefficients of ( h ) + D {\displaystyle (h)+D} are non-negative. Intuitively, we can think of this as being all meromorphic functions whose poles at every point are no worse than the corresponding coefficient in D {\displaystyle D} ; if the coefficient in D {\displaystyle D} at z {\displaystyle z}

13542-475: The theorem can be relaxed, as well: for a (projective) curve over an algebraically closed field, all of whose local rings are Gorenstein rings , the same statement as above holds, provided that the geometric genus as defined above is replaced by the arithmetic genus g a , defined as (For smooth curves, the geometric genus agrees with the arithmetic one.) The theorem has also been extended to general singular curves (and higher-dimensional varieties). One of

13664-399: The theorem. In particular, as long as D {\displaystyle D} has degree at least 2 g − 1 {\displaystyle 2g-1} , the correction term is 0, so that The theorem will now be illustrated for surfaces of low genus. There are also a number other closely related theorems: an equivalent formulation of this theorem using line bundles and

13786-411: The theories of Riemannian geometry , algebraic geometry , and complex manifold theory. The theory of Riemann surfaces was elaborated by Felix Klein and particularly Adolf Hurwitz . This area of mathematics is part of the foundation of topology and is still being applied in novel ways to mathematical physics . In 1853, Gauss asked Riemann, his student, to prepare a Habilitationsschrift on

13908-419: The theory of partial fractions . Conversely if this sequence starts this way, then g {\displaystyle g} must be zero. The next case is a Riemann surface of genus g = 1 {\displaystyle g=1} , such as a torus C / Λ {\displaystyle \mathbb {C} /\Lambda } , where Λ {\displaystyle \Lambda }

14030-497: The trivial bundle, h 0 ( X , L ) = 1 {\displaystyle h^{0}(X,L)=1} since the only holomorphic functions on X are constants. The degree of L is zero, and L − 1 {\displaystyle L^{-1}} is the trivial bundle. Thus, Therefore, h 0 ( X , K ) = g {\displaystyle h^{0}(X,K)=g} , proving that there are g holomorphic one-forms. Since

14152-402: The validity of this relation is equivalent with the embedding of C n / Ω {\displaystyle \mathbb {C} ^{n}/\Omega } (where Ω {\displaystyle \Omega } is the lattice of the period matrix) in a projective space by means of theta functions. For certain values of n {\displaystyle n} , this

14274-445: The vanishing of one homogeneous polynomial ; by contrast, a codimension- r subvariety need not be definable by only r equations when r is greater than 1. (That is, not every subvariety of projective space is a complete intersection .) Locally, every codimension-1 subvariety of a smooth variety can be defined by one equation in a neighborhood of each point. Again, the analogous statement fails for higher-codimension subvarieties. As

14396-457: The zeros and poles) of a Riemann surface. According to Detlef Laugwitz , automorphic functions appeared for the first time in an essay about the Laplace equation on electrically charged cylinders. Riemann however used such functions for conformal maps (such as mapping topological triangles to the circle) in his 1859 lecture on hypergeometric functions or in his treatise on minimal surfaces . In

14518-407: Was a German mathematician who made profound contributions to analysis , number theory , and differential geometry . In the field of real analysis , he is mostly known for the first rigorous formulation of the integral, the Riemann integral , and his work on Fourier series . His contributions to complex analysis include most notably the introduction of Riemann surfaces , breaking new ground in

14640-466: Was mentioned above. For D = n ⋅ P {\displaystyle D=n\cdot P} with n > 0 {\displaystyle n>0} , the degree of K − D {\displaystyle K-D} is strictly negative, so that the correction term is 0. The sequence of dimensions can also be derived from the theory of elliptic functions . For g = 2 {\displaystyle g=2} ,

14762-543: Was successful. An anecdote from Arnold Sommerfeld shows the difficulties which contemporary mathematicians had with Riemann's new ideas. In 1870, Weierstrass had taken Riemann's dissertation with him on a holiday to Rigi and complained that it was hard to understand. The physicist Hermann von Helmholtz assisted him in the work overnight and returned with the comment that it was "natural" and "very understandable". Other highlights include his work on abelian functions and theta functions on Riemann surfaces. Riemann had been in

14884-410: Was very impressed with Riemann, especially with his theory of abelian functions . When Riemann's work appeared, Weierstrass withdrew his paper from Crelle's Journal and did not publish it. They had a good understanding when Riemann visited him in Berlin in 1859. Weierstrass encouraged his student Hermann Amandus Schwarz to find alternatives to the Dirichlet principle in complex analysis, in which he

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