Riemann-Roch theorem

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In algebraic geometry the Riemann-Roch theorem states that if  is a smooth algebraic curve, and  is an invertible sheaf on  then the the following properties hold:

Some examples and applications

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

• Any curve  of genus 0 is isomorphic to the projective line: Indeed if p is a point on the curve then ; hence the map  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 ; hence the map  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  is  and therefore ; since  the map  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  of degree  on a curve  satisfies  if and only if there is an effective divisor  such that  in . In this case there is a natural isomorphism

, where we identify  with it's image in the dual canonical system .

As an example we consider effective divisors of degrees  on a non hyperelliptic curve  of genus 3. The degree of the canonical class is , whereas . Hence the canonical image of  is a smooth plane quartic. We now idenitfy  with it's image in the dual canonical system. Let  be two points on  then there are exactly two points  such that , where we intersect with multiplicities, and if  we consider the tangent line  instead of the line . Hence there is a natural isomorphism between  and the unique point in  representing the line . There is also a natural ismorphism between  and the points in  representing lines through the points .

History and proofs

In 1857 Riemann proved the inequality ; the proof used function theoretic methods on a gluing diagram of a 4g-gon. In 1865 Riemann's student Roch identified the difference in the inequality as the dimension of the cohomology group of differentials with prescribed zeros given by . In the second half of the 19th century, and in the beginning of the 20th century, a huge effort went into making the proof more geometric, and later more algebraic. (i.e. to get a proof over function fields which is independent of the geometric model). The problem in "geometrizing" the theorem is that in the 19th century, curves would always arise as embedded curves in some ambient projective space, and one had to find birationaly invariant definitions for the cohomology groups involved in the theorem. The problem in "algebraizing" the theorem is that one has to define differentials over function fields. In subsequent years the analytic, geometric and algebraic proofs have all been considerably simplified - we list references below. The current "standard" proof is the sheaf theoretic one: using modern tools, the theorem is an immediate consequence of Serre's duality, and the fact that if  are divisors on  then .

Generalizations

The generalizations of the Riemann-Roch theorem come in two flavors: One direction views the 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 Atiyah-Singer index theorem.

References

• J. Gray The Riemann-Roch theorem and Geometry Documenta Mathematica Extra Vo. ICM 1998 III 811-822, available online at [1] (history)
• E. Arabarello M. Cornalba P. Griffiths and J. Harris Geometry of Algebraic Curves: Volume 1 ISBN 0387909974 (classical geometric proof, many applications and examples)
• P. Grifiths and J. Harris Principles of Algebraic geometry ISBN 0-471-05059-8 Chapter 2.3 (analytic approach and sheaf theoretic approach, many applications and examples)
• M. Rosen Number theory in Function Fields ISBN 0-387-95335-3 Chapter 6 (algebraic approach)
• W. Fulton Intersection Theory ISBN 0387985492 Chapters 15 and 18 (sheaf theoretic approach and generalizations)