Riemann-Roch theorem

In algebraic geometry the Riemann-Roch theorem states that if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} is a smooth algebraic curve, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{L}} is an invertible sheaf on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} then the the following properties hold:

• The Euler characteristic of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{L}} is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h^0(\mathcal{L})-h^1(\mathcal{L})=deg(\mathcal{L})-(g-1)}
• There is a canonical isomorphism Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H^0(L)(K_C\otimes\mathcal{L}^\vee)\cong H^1(\mathcal{L})}

Some examples and applications

The examples we give arise from considering complete linear systems on curves.

• Any curve Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} of genus 0 is isomorphic to the projective line: Indeed if p is a point on the curve then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h^0(p)-0=1-(0-1)=2} ; hence the map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C\to\mathbb{P}H^0(O_C(p))} is a degree 1 map, or an isomorphism.
• Any curve of genus 1 is a double cover of a projective line: Indeed if p is a point on the curve then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h^0(2p)-0=2-(1-1)=2} ; hence the map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C\to\mathbb{P}H^0(O_C(2p))} is a degree 2 map.
• Any curve of genus 2 is a double cover of a projective line: Indeed the degree of the canonical class Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K_C} is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2g-2} and therefore Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h^0(K_C)-h^0(O_C)=2-(2-1)=1} ; since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h^0(O_C)=1} the map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C\to\mathbb{P}H^0(K_C)} is a degree 2 map.
• The Riemann-Hurwitz formula.

Geometric Riemann-Roch

Some linear systems on a smooth cannonicaly embedded genus 3 curve

From the statement of the theorem one sees that an effective divisor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D} of degree Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} on a curve Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} satisfies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h^0(D)>d-(g-1)} if and only if there is an effective divisor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D'} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D+D'\sim K_C} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Pic(C)} . In this case there is a natural isomorphism

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{[H]\in|K_C|, H\cdot C=D'\}\cong\mathbb{P}H^0(D)} , where we identify Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} with it's image in the dual canonical system Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |K_C|^*} .

As an example we consider effective divisors of degrees Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2,3} on a non hyperelliptic curve Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} of genus 3. The degree of the canonical class is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\mbox{genus}(c)-2=4} , whereas Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h^0(K_C)=2\mbox{genus}(C)-2-(\mbox{genus}(C)-1)+h^0(0)=g} . Hence the canonical image of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} is a smooth plane quartic. We now idenitfy Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} with it's image in the dual canonical system. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p,q} be two points on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} then there are exactly two points Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r,s} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C\cap\overline{pq}=\{p,q,r,s\}} , where we intersect with multiplicities, and if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p=q} we consider the tangent line Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_p C} instead of the line Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{pq}} . Hence there is a natural isomorphism between Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{P}h^0(O_C(p+q))} and the unique point in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |K_C|} representing the line Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{pq}} . There is also a natural ismorphism between Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{P}(O_C(p+q+r))} and the points in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |K_C|} representing lines through the points Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s} .

Generalizations

The generalizations of th Riemann-Roch theorem come in two flavors: One direction views Riemann Roch theorem as a tool to study any linear system on a any curve. Clifford's theorem gives a better bound on the dimension of special linear systems on curve. If the assumption on the curve is relaxed to be a generic curve, then the Brill-Noether Theorem and the Petri Theorem give good descriptions of the geometry of the linear system.

Another direction of generalization, with more far-reaching consequences, is to view Riemann roch as a tool to compute the Euler characteristic of a vector bundle on a Variety. The first generalizations in this direction go back to the beginning of the 20th century with Riemann-Roch for surfaces and Noether's formula on surfaces. The next step, taken during the 1960s, is the Hirzebruch-Riemann-Roch theorem, which analyze the Euler characteristic of the canonical bundle of an arbitrary. The final step in the algebro-geometric setting is the Grothendieck-Riemann-Roch theorem, which analyzes the behaviour of the Euler characteristic of vector bundles under pullbacks; e.g. the Riemann-Roch theorem can be deduced from the Grothendieck-Riemann Roch theorem by projecting a curve to a point. In the analytic setting Grothendieck Riemann Roch had one more generalization: the Atiya-Singer index theorem.

Proofs

• The sheaf theoretic proof: Using modern tools, the theorem is an immediate consequence of Serre's duality, and the fact that if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D,D'} are divisors on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi(O_C(D+D'))=\chi(O_C(D))+\chi(O_C(D'))} .
• The analytic proof was chronologically the first one given - one analyzes the relation between meromorphic functions on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} with prescribed poles, and holomorphic differentials on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} with prescribed zeros over the same points (see Griffiths and Harris).
• The Italian proof follows from immersing the curve in the projective plane, and from explicit work with the adjunction formula (see ACGH)
• Weil's algebraic proof over function fields. (see Rosen)

References

• E. Arabarello M. Cornalba P. Griffiths and J. Harris
• P. Grifiths and J. Harris Principles of Algebraic geometry Chapter 2.3
• M. Rosen Number theory in Function Fields Chapter 6
• W. Fulton Intersection Theory