Riemann-Roch theorem

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Revision as of 17:02, 10 March 2007 by imported>David Lehavi (wrote about generalizations. sketched proofs and references)
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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:

  • The Euler characteristic of is given by
  • There is a canonical isomorphism

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 .

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 Rieman roch as a toll 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 is the Hirzebruch-Riemann-Roch theorem, which analyze the Euler characteristic of the canonical bundle. 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.
  • The analytic proof (see Griffiths and Harris)
  • Weil's proof (see Rosen)
  • The Italyan proof (see ACGH)

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