Riemann–Roch theorem

Riemann–Roch theorem Riemann–Roch theorem Field Algebraic geometry and complex analysis First proof by Gustav Roch First proof in 1865 Generalizations Atiyah–Singer index theorem Grothendieck–Riemann–Roch theorem Hirzebruch–Riemann–Roch theorem Riemann–Roch theorem for surfaces Riemann–Roch-type theorem Consequences Clifford's theorem on special divisors Riemann–Hurwitz formula The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus g, in a way that can be carried over into purely algebraic settings.

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.

Contenuti 1 Preliminary notions 2 Enunciato del teorema 2.1 Esempi 2.1.1 Genus zero 2.1.2 Genus one 2.1.3 Genus two and beyond 2.2 Riemann–Roch for line bundles 2.3 Degree of canonical bundle 2.4 Riemann–Roch theorem for algebraic curves 3 Applicazioni 3.1 Hilbert polynomial 3.2 Pluricanonical embedding 3.3 Genus of plane curves with singularities 3.4 Riemann–Hurwitz formula 3.5 Clifford's theorem on special divisors 4 Prova 4.1 Proof for algebraic curves 4.2 Proof for compact Riemann surfaces 5 Generalizations of the Riemann–Roch theorem 6 Guarda anche 7 Appunti 8 References Preliminary notions A Riemann surface of genus 3.

A Riemann surface {stile di visualizzazione X} is a topological space that is locally homeomorphic to an open subset of {displaystyle mathbb {C} } , the set of complex numbers. Inoltre, 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 {displaystyle mathbb {C} } to the surface {stile di visualizzazione X} . For the purposes of the Riemann–Roch theorem, the surface {stile di visualizzazione X} is always assumed to be compact. Colloquially speaking, the genus {stile di visualizzazione g} of a Riemann surface is its number of handles; for example the genus of the Riemann surface shown at the right is three. Più precisamente, the genus is defined as half of the first Betti number, cioè., half of the {displaystyle mathbb {C} } -dimension of the first singular homology group {stile di visualizzazione H_{1}(X,mathbb {C} )} with complex coefficients. The genus classifies compact Riemann surfaces up to homeomorphism, cioè., two such surfaces are homeomorphic if and only if their genus is the same. Perciò, the genus is an important topological invariant of a Riemann surface. D'altro canto, Hodge theory shows that the genus coincides with the {displaystyle mathbb {C} } -dimension of the space of holomorphic one-forms on {stile di visualizzazione X} , so the genus also encodes complex-analytic information about the Riemann surface.[1] A divisor {stile di visualizzazione D} is an element of the free abelian group on the points of the surface. Equivalentemente, a divisor is a finite linear combination of points of the surface with integer coefficients.

Any meromorphic function {stile di visualizzazione f} gives rise to a divisor denoted {stile di visualizzazione (f)} definito come {stile di visualizzazione (f):=somma _{z_{non }in r(f)}S_{non }z_{non }} dove {stile di visualizzazione R(f)} is the set of all zeroes and poles of {stile di visualizzazione f} , e {stile di visualizzazione s_{non }} è dato da {stile di visualizzazione s_{non }:={inizio{casi}a&{testo{Se }}z_{non }{testo{ is a zero of order }}a\-a&{testo{Se }}z_{non }{testo{ is a pole of order }}a.end{casi}}} Il set {stile di visualizzazione R(f)} is known to be finite; this is a consequence of {stile di visualizzazione X} being compact and the fact that the zeros of a (diverso da zero) holomorphic function do not have an accumulation point. Perciò, {stile di visualizzazione (f)} 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 {stile di visualizzazione K} ). Any two meromorphic 1-forms will yield linearly equivalent divisors, so the canonical divisor is uniquely determined up to linear equivalence (quindi "il" canonical divisor).

The symbol {displaystyle deg(D)} denotes the degree (occasionally also called index) of the divisor {stile di visualizzazione D} , cioè. the sum of the coefficients occurring in {stile di visualizzazione D} . It can be shown that the divisor of a global meromorphic function always has degree 0, so the degree of a divisor depends only on its linear equivalence class.

Il numero {stile di visualizzazione ell (D)} is the quantity that is of primary interest: the dimension (Sopra {displaystyle mathbb {C} } ) of the vector space of meromorphic functions {stile di visualizzazione h} on the surface, such that all the coefficients of {stile di visualizzazione (h)+D} are non-negative. Intuitivamente, we can think of this as being all meromorphic functions whose poles at every point are no worse than the corresponding coefficient in {stile di visualizzazione D} ; if the coefficient in {stile di visualizzazione D} a {stile di visualizzazione con} is negative, then we require that {stile di visualizzazione h} has a zero of at least that multiplicity at {stile di visualizzazione con} – if the coefficient in {stile di visualizzazione D} is positive, {stile di visualizzazione h} can have a pole of at most that order. The vector spaces for linearly equivalent divisors are naturally isomorphic through multiplication with the global meromorphic function (which is well-defined up to a scalar).

Statement of the theorem The Riemann–Roch theorem for a compact Riemann surface of genus {stile di visualizzazione g} with canonical divisor {stile di visualizzazione K} states {stile di visualizzazione ell (D)-ell (K-D)=deg(D)-g+1.} Typically, il numero {stile di visualizzazione ell (D)} is the one of interest, mentre {stile di visualizzazione ell (K-D)} is thought of as a correction term (also called index of speciality[2][3]) so the theorem may be roughly paraphrased by saying dimension − correction = degree − genus + 1.

Because it is the dimension of a vector space, the correction term {stile di visualizzazione ell (K-D)} is always non-negative, affinché {stile di visualizzazione ell (D)geq deg(D)-g+1.} This is called Riemann's inequality. Roch's part of the statement is the description of the possible difference between the sides of the inequality. On a general Riemann surface of genus {stile di visualizzazione g} , {stile di visualizzazione K} has degree {displaystyle 2g-2} , independently of the meromorphic form chosen to represent the divisor. This follows from putting {displaystyle D=K} in the theorem. In particolare, as long as {stile di visualizzazione D} has degree at least {displaystyle 2g-1} , the correction term is 0, affinché {stile di visualizzazione ell (D)=deg(D)-g+1.} 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 a generalization of the theorem to algebraic curves.

Examples The theorem will be illustrated by picking a point {stile di visualizzazione P} on the surface in question and regarding the sequence of numbers {stile di visualizzazione ell (ncdot P),ngq 0} cioè., the dimension of the space of functions that are holomorphic everywhere except at {stile di visualizzazione P} where the function is allowed to have a pole of order at most {stile di visualizzazione n} . Per {displaystyle n=0} , the functions are thus required to be entire, cioè., holomorphic on the whole surface {stile di visualizzazione X} . By Liouville's theorem, such a function is necessarily constant. Perciò, {stile di visualizzazione ell (0)=1} . In generale, la sequenza {stile di visualizzazione ell (ncdot P)} is an increasing sequence.

Genus zero 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 {displaystyle mathbb {C} } , with transition map being given by {displaystyle mathbb {C} ^{volte }ni zmapsto {frac {1}{z}}in matematica bb {C} ^{volte }.} Perciò, the form {displaystyle omega =dz} on one copy of {displaystyle mathbb {C} } extends to a meromorphic form on the Riemann sphere: it has a double pole at infinity, da {displaystyle dleft({frac {1}{z}}Giusto)=-{frac {1}{z^{2}}},dz.} così, its divisor {stile di visualizzazione K:=nome operatore {div} (omega )=-2P} (dove {stile di visualizzazione P} is the point at infinity).

Perciò, the theorem says that the sequence {stile di visualizzazione ell (ncdot P)} reads 1, 2, 3, ... .

This sequence can also be read off from the theory of partial fractions. Conversely if this sequence starts this way, poi {stile di visualizzazione g} deve essere zero.

Genus one A torus.

The next case is a Riemann surface of genus {displaystyle g=1} , such as a torus {displaystyle mathbb {C} /Lambda } , dove {displaystyle Lambda } is a two-dimensional lattice (a group isomorphic to {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 {stile di visualizzazione con} Su {stile di visualizzazione C} yields a one-form {displaystyle omega =dz} Su {stile di visualizzazione X} that is everywhere holomorphic, cioè., has no poles at all. Perciò, {stile di visualizzazione K} , the divisor of {stile di visualizzazione omega } è zero.

On this surface, this sequence is 1, 1, 2, 3, 4, 5 ... ; and this characterises the case {displaystyle g=1} . Infatti, per {displaystyle D=0} , {stile di visualizzazione ell (K-D)=ell (0)=1} , as was mentioned above. Per {displaystyle D=ncdot P} insieme a {displaystyle n>0} , the degree of {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.

Genus two and beyond For {displaystyle g=2} , the sequence mentioned above is 1, 1, ?, 2, 3, ... .

It is shown from this that the ? term of degree 2 is either 1 o 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 particolare, a genus 2 curve is a hyperelliptic curve. Per {displaystyle g>2} it is always true that at most points the sequence starts with {displaystyle g+1} ones and there are finitely many points with other sequences (see Weierstrass points).

Riemann–Roch for line bundles 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. Permettere {stile di visualizzazione H^{0}(X,l)} denote the space of holomorphic sections of L. This space will be finite-dimensional; its dimension is denoted {stile di visualizzazione h^{0}(X,l)} . Let K denote the canonical bundle on X. Quindi, the Riemann–Roch theorem states that {stile di visualizzazione h^{0}(X,l)-h^{0}(X,L^{-1}otimes K)=deg(l)+1-g.} 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 the trivial bundle, {stile di visualizzazione h^{0}(X,l)=1} since the only holomorphic functions on X are constants. The degree of L is zero, e {stile di visualizzazione L^{-1}} is the trivial bundle. così, {displaystyle 1-h^{0}(X,K)=1-g.} Perciò, {stile di visualizzazione h^{0}(X,K)=g} , proving that there are g holomorphic one-forms.

Degree of canonical bundle Since the canonical bundle {stile di visualizzazione K} ha {stile di visualizzazione h^{0}(X,K)=g} , applying Riemann–Roch to {displaystyle L=K} dà {stile di visualizzazione h^{0}(X,K)-h^{0}(X,K^{-1}otimes K)=deg(K)+1-g} which can be rewritten as {displaystyle g-1=deg(K)+1-g} hence the degree of the canonical bundle is {displaystyle deg(K)=2g-2} .

Riemann–Roch theorem for algebraic curves Every item in 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 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 {stile di visualizzazione g(C):=dim _{K}Gamma (C,Omega _{C}^{1})} cioè., as the dimension of the space of globally defined (algebraic) one-forms (see Kähler differential). Infine, 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. così, writing {stile di visualizzazione ell (D)} for the dimension (oltre 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: {stile di visualizzazione ell (D)-ell (K-D)=deg(D)-g+1.} where C is a projective non-singular algebraic curve over an algebraically closed field k. Infatti, 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 the possible extensions of the base field and the residue fields of the points supporting the divisor.[4] Infine, 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 {stile di visualizzazione {matematico {o}}} .[5] The smoothness assumption in the theorem can be relaxed, anche: 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 ga, definito come {stile di visualizzazione g_{un}:=dim _{K}H^{1}(C,{matematico {o}}_{C}).} [6] (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).[7] Applications Hilbert polynomial One of 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 {stile di visualizzazione {matematico {l}}} is ample, then the Hilbert polynomial will give the first degree {stile di visualizzazione {matematico {l}}^{otimes n}} giving an embedding into projective space. Per esempio, the canonical sheaf {stile di visualizzazione omega _{C}} has degree {displaystyle 2g-2} , which gives an ample line bundle for genus {displaystyle ggeq 2} .[8] If we set {stile di visualizzazione omega _{C}(n)=omega _{C}^{otimes n}} then the Riemann–Roch formula reads {stile di visualizzazione {inizio{allineato}chi (omega _{C}(n))&=deg(omega _{C}^{otimes n})-g+1\&=n(2g-2)-g+1\&=2ng-2n-g+1\&=(2n-1)(g-1)fine{allineato}}} Giving the degree {stile di visualizzazione 1} Hilbert polynomial of {stile di visualizzazione omega _{C}} {stile di visualizzazione H_{omega _{C}}(t)=2(g-1)t-g+1} Because the tri-canonical sheaf {stile di visualizzazione omega _{C}^{a volte 3}} is used to embed the curve, the Hilbert polynomial {stile di visualizzazione H_{C}(t)=H_{omega _{C}^{a volte 3}}(t)} is generally considered while constructing the Hilbert scheme of curves (and the moduli space of algebraic curves). This polynomial is {stile di visualizzazione {inizio{allineato}H_{C}(t)&=(6t-1)(g-1)\&=6(g-1)t+(1-g)fine{allineato}}} and is called the Hilbert polynomial of an genus g curve.

Pluricanonical embedding Analyzing this equation further, the Euler characteristic reads as {stile di visualizzazione {inizio{allineato}chi (omega _{C}^{otimes n})&=h^{0}sinistra(C,omega _{C}^{otimes n}Giusto)-h^{0}sinistra(C,omega _{C}otimes left(omega _{C}^{otimes n}Giusto)^{v }Giusto)\&=h^{0}sinistra(C,omega _{C}^{otimes n}Giusto)-h^{0}sinistra(C,sinistra(omega _{C}^{a volte (n-1)}Giusto)^{v }Giusto)fine{allineato}}} Da {displaystyle deg(omega _{C}^{otimes n})=n(2g-2)} {stile di visualizzazione h^{0}sinistra(C,sinistra(omega _{C}^{a volte (n-1)}Giusto)^{v }Giusto)=0} per {stile di visualizzazione ngeq 3} , since its degree is negative for all {displaystyle ggeq 2} , implying it has no global sections, there is an embedding into some projective space from the global sections of {stile di visualizzazione omega _{C}^{otimes n}} . In particolare, {stile di visualizzazione omega _{C}^{a volte 3}} gives an embedding into {displaystyle mathbb {P} ^{N}cong mathbb {P} (H^{0}(C,omega _{C}^{a volte 3}))} dove {displaystyle N=5g-5-1=5g-6} da {stile di visualizzazione h^{0}(omega _{C}^{a volte 3})=6g-6-g+1} . This 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 {stile di visualizzazione H_{C}(t)} .[9] Genus of plane curves with singularities An irreducible plane algebraic curve of degree d has (d − 1)(d − 2)/2 − g singularities, when properly counted. Ne consegue che, if a curve has (d − 1)(d − 2)/2 different singularities, it is a rational curve and, così, admits a rational parameterization.

Riemann–Hurwitz formula The Riemann–Hurwitz formula concerning (ramified) maps between Riemann surfaces or algebraic curves is a consequence of the Riemann–Roch theorem.

Clifford's theorem on special divisors Clifford's theorem on special divisors is also a consequence of the Riemann–Roch theorem. It states that for a special divisor (cioè., tale che {stile di visualizzazione ell (K-D)>0} ) soddisfacente {stile di visualizzazione ell (D)>0,} the following inequality holds:[10] {stile di visualizzazione ell (D)leq {frac {deg D}{2}}+1.} Proof Proof for algebraic curves The statement for algebraic curves can be proved using Serre duality. The integer {stile di visualizzazione ell (D)} is the dimension of the space of global sections of the line bundle {stile di visualizzazione {matematico {l}}(D)} associated to D (cfr. Cartier divisor). In terms of sheaf cohomology, we therefore have {stile di visualizzazione ell (D)= matematica {dim} H^{0}(X,{matematico {l}}(D))} , and likewise {stile di visualizzazione ell ({matematico {K}}_{X}-D)=dim H^{0}(X,omega _{X}a volte {matematico {l}}(D)^{v })} . But Serre duality for non-singular projective varieties in the particular case of a curve states that {stile di visualizzazione H^{0}(X,omega _{X}a volte {matematico {l}}(D)^{v })} is isomorphic to the dual {stile di visualizzazione H^{1}(X,{matematico {l}}(D))^{v }} . The left hand side thus equals the Euler characteristic of the divisor D. When D = 0, we find the Euler characteristic for the structure sheaf is {displaystyle 1-g} per definizione. To prove the theorem for general divisor, one can then proceed by adding points one by one to the divisor and ensure that the Euler characteristic transforms accordingly to the right hand side.

Proof for compact Riemann surfaces The theorem for compact Riemann surfaces can be deduced from the algebraic version using Chow's Theorem and the GAGA principle: infatti, every compact Riemann surface is defined by algebraic equations in some complex projective space. (Chow's Theorem says that any closed analytic subvariety of projective space is defined by algebraic equations, and the GAGA principle says that sheaf cohomology of an algebraic variety is the same as the sheaf cohomology of the analytic variety defined by the same equations).

One may avoid the use of Chow's theorem by arguing identically to the proof in the case of algebraic curves, but replacing {stile di visualizzazione {matematico {l}}(D)} with the sheaf {stile di visualizzazione {matematico {o}}_{D}} of meromorphic functions h such that all coefficients of the divisor {stile di visualizzazione (h)+D} are nonnegative. Here the fact that the Euler characteristic transforms as desired when one adds a point to the divisor can be read off from the long exact sequence induced by the short exact sequence {displaystyle 0to {matematico {o}}_{D}a {matematico {o}}_{D+P}a matematicabb {C} _{P}a 0} dove {displaystyle mathbb {C} _{P}} is the skyscraper sheaf at P, and the map {stile di visualizzazione {matematico {o}}_{D+P}a matematicabb {C} _{P}} returns the {displaystyle -k-1} th Laurent coefficient, dove {displaystyle k=D(P)} .[11] Generalizations of the Riemann–Roch theorem See also: Riemann–Roch-type theorem The Riemann–Roch theorem for curves was proved for Riemann surfaces by Riemann and Roch in the 1850s and for algebraic curves by Friedrich Karl Schmidt in 1931 as he was working on perfect fields of finite characteristic. As stated by Peter Roquette,[12] The first main achievement of F. K. Schmidt is the discovery that the classical theorem of Riemann–Roch on compact Riemann surfaces can be transferred to function fields with finite base field. In realtà, his proof of the Riemann–Roch theorem works for arbitrary perfect base fields, not necessarily finite.

It is foundational in the sense that the subsequent theory for curves tries to refine the information it yields (for example in the Brill–Noether theory).

There are versions in higher dimensions (for the appropriate notion of divisor, or line bundle). Their general formulation depends on splitting the theorem into two parts. One, which would now be called Serre duality, interprets the {stile di visualizzazione ell (K-D)} term as a dimension of a first sheaf cohomology group; insieme a {stile di visualizzazione ell (D)} the dimension of a zeroth cohomology group, or space of sections, the left-hand side of the theorem becomes an Euler characteristic, and the right-hand side a computation of it as a degree corrected according to the topology of the Riemann surface.

In algebraic geometry of dimension two such a formula was found by the geometers of the Italian school; a Riemann–Roch theorem for surfaces was proved (there are several versions, with the first possibly being due to Max Noether).

An n-dimensional generalisation, the Hirzebruch–Riemann–Roch theorem, was found and proved by Friedrich Hirzebruch, as an application of characteristic classes in algebraic topology; he was much influenced by the work of Kunihiko Kodaira. At about the same time Jean-Pierre Serre was giving the general form of Serre duality, as we now know it.

Alexander Grothendieck proved a far-reaching generalization in 1957, now known as the Grothendieck–Riemann–Roch theorem. His work reinterprets Riemann–Roch not as a theorem about a variety, but about a morphism between two varieties. The details of the proofs were published by Armand Borel and Jean-Pierre Serre in 1958.[13] Dopo, Grothendieck and his collaborators simplified and generalized the proof.[14] Finally a general version was found in algebraic topology, anche. These developments were essentially all carried out between 1950 e 1960. After that the Atiyah–Singer index theorem opened another route to generalization. Di conseguenza, the Euler characteristic of a coherent sheaf is reasonably computable. For just one summand within the alternating sum, further arguments such as vanishing theorems must be used.

See also Arakelov theory Grothendieck–Riemann–Roch theorem Hirzebruch–Riemann–Roch theorem Kawasaki's Riemann–Roch formula Hilbert polynomial Moduli of algebraic curves Notes ^ Griffith, Harris, p. 116, 117 ^ Stichtenoth p.22 ^ Mukai pp.295–297 ^ Liu, Qing (2002), Algebraic Geometry and Arithmetic Curves, la stampa dell'università di Oxford, ISBN 978-0-19-850284-5, Sezione 7.3 ^ * Altman, Allen; Cleiman, Stefano (1970), Introduction to Grothendieck duality theory, Appunti delle lezioni in matematica, vol. 146, Berlino, New York: Springer-Verlag, Theorem VIII.1.4., p. 164 ^ Hartshorn, Robin (1986), "Generalized divisors on Gorenstein curves and a theorem of Noether", Journal of Mathematics of Kyoto University, 26 (3): 375–386, doi:10.1215/kjm/1250520873, ISSN 0023-608X ^ Baum, Paolo; Fulton, William; MacPherson, Roberto (1975), "Riemann–Roch for singular varieties", Publications Mathématiques de l'IHÉS, 45 (45): 101–145, doi:10.1007/BF02684299, ISSN 1618-1913, S2CID 83458307 ^ Note the moduli of elliptic curves can be constructed independently, see https://arxiv.org/abs/0812.1803, and there is only one smooth curve of genus 0, {displaystyle mathbb {P} ^{1}} , which can be found using deformation theory. See https://arxiv.org/abs/math/0507286 ^ Deligne, P.; Mumford, D. (1969). "Irreducibility of the space of curves of given genus". IHES. 36: 75–110. CiteSeerX doi:10.1007/BF02684599. S2CID 16482150. ^ Fulton, William (1989), Algebraic curves (PDF), Advanced Book Classics, Addison-Wesley, ISBN 978-0-201-51010-2, p. 109 ^ Forster, Ottone (1981), Lectures on Riemann Surfaces, Springer Nature, ISBN 978-1-4612-5963-3, Sezione 16 ^ "Manuscripts". ^ A. Borel and J.-P. Stretto. Toro. soc. Matematica. Francia 86 (1958), 97-136. ^ SGA 6, Springer-Verlag (1971). References Borel, Armand & Serre, JeanPierre (1958), Le théorème de Riemann–Roch, d'après Grothendieck, Bull.S.M.F. 86 (1958), 97–136. Griffith, Phillip; Harris, Joseph (1994), Principles of algebraic geometry, Wiley Classics Library, New York: John Wiley & Sons, doi:10.1002/9781118032527, ISBN 978-0-471-05059-9, SIG 1288523 Grothendieck, Alessandro, et al. (1966/67), Théorie des Intersections et Théorème de Riemann–Roch (SGA 6), LNM 225, Springer-Verlag, 1971. Fulton, William (1974). Curve algebriche (PDF). Serie di appunti di lezione di matematica. W.A. Beniamino. ISBN 0-8053-3080-1. Jost, Jurgen (2006). Compact Riemann Surfaces. Berlino, New York: Springer-Verlag. ISBN 978-3-540-33065-3. See pages 208–219 for the proof in the complex situation. Note that Jost uses slightly different notation. Hartshorne, Robin (1977). Geometria algebrica. Berlino, New York: Springer-Verlag. ISBN 978-0-387-90244-9. SIG 0463157. OCLC 13348052., contains the statement for curves over an algebraically closed field. See section IV.1. "Riemann–Roch theorem", Enciclopedia della matematica, EMS Press, 2001 [1994] Hirzebruch, Friedrich (1995). Topological methods in algebraic geometry. I classici in matematica. Berlino, New York: Springer-Verlag. ISBN 978-3-540-58663-0. SIG 1335917.. A good general modern reference. Shigeru Mukai (2003). An Introduction to Invariants and Moduli. Studi Cambridge in matematica avanzata. vol. 81. William Oxbury (trans.). New York: Cambridge University Press. ISBN 0-521-80906-1. Vector bundles on Compact Riemann Surfaces, M. S. Narasimhan, pp. 5–6. Riemann, Bernhard (1857). "Theorie der Abel'schen Functionen". Diario di matematica pura e applicata. 1857 (54): 115–155. doi:10.1515/crll.1857.54.115. hdl:2027/coo.31924060183864. S2CID 16593204. Roch, Gustav (1865). "Ueber die Anzahl der willkurlichen Constanten in algebraischen Functionen". Diario di matematica pura e applicata. 1865 (64): 372–376. doi:10.1515/crll.1865.64.372. S2CID 120178388. Schmidt, Friedrich Karl (1931), "Analytische Zahlentheorie in Körpern der Charakteristik p", Giornale di matematica, 33: 1–32, doi:10.1007/BF01174341, Zbl 0001.05401, archiviato dall'originale in poi 2017-12-22, recuperato 2020-05-16 Stichtenoth, Henning (1993). Algebraic Function Fields and Codes. Springer-Verlag. ISBN 3-540-56489-6. Misha Kapovich, The Riemann–Roch Theorem (lecture note) an elementary introduction J. Gray, The Riemann–Roch theorem and Geometry, 1854–1914. Is there a Riemann–Roch for smooth projective curves over an arbitrary field? on MathOverflow show vte Topics in algebraic curves show vte Bernhard Riemann Categories: Theorems in algebraic geometryGeometry of divisorsTopological methods of algebraic geometryTheorems in complex analysisBernhard Riemann

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