Riemann-Roch notes / XI

A working checklist

A practical use of the theorem is mostly a discipline of signs: compute the degree, inspect the correction, then substitute.

A Riemann-Roch computation should keep the signs visible. Most mistakes come from confusing \(K-D\) with \(D-K\), or from forgetting that positive coefficients in \(D\) allow poles in \(L(D)\).

A reliable order

Given a divisor \(D\) on a compact Riemann surface \(X\):

  1. Write down the genus \(g\).
  2. Compute \(\deg(D)\).
  3. Use \(\deg(K)=2g-2\).
  4. Compute \(\deg(K-D)=2g-2-\deg(D)\).
  5. Decide whether \(\ell(K-D)\) vanishes.
  6. Apply
\[\ell(D)=\deg(D)+1-g+\ell(K-D).\]

Fast vanishing test

If \(\deg(K-D)<0\), then \(\ell(K-D)=0\). Equivalently, if \(\deg(D)>2g-2\), then

\(\ell(D)=\deg(D)+1-g.\)

Two reference computations

On \(\mathbb P^1\), with \(D=n\infty\) and \(n\ge 0\),

\[g=0, \qquad K\sim -2\infty, \qquad \ell(D)=n+1.\]

On an elliptic curve, with \(D\) effective of degree \(d>0\),

\[g=1, \qquad K\sim 0, \qquad \ell(D)=d.\]

These two cases are worth checking before any more complicated example.

A third check is useful because it is the first one not covered by genus zero or one. Let

\[C:y^2=x^5-x+1\]

be a genus two hyperelliptic curve with point \(O\) at infinity. For \(D=3O\),

\[\deg(D)=3, \qquad 2g-2=2,\]

so the correction term vanishes. Therefore

\[\ell(3O)=3+1-2=2.\]

The explicit basis is \(\{1,x\}\), since \(x\) has pole order \(2\) at \(O\) while \(y\) has pole order \(5\).

What to record in a computation

For a SageMath notebook, generated figure, or hand calculation, record the following fields explicitly:

Field What should be written
curve equation, projective model, or lattice model
genus the integer \(g\)
divisor points and multiplicities
degree \(\deg(D)\)
canonical data \(K\) or at least \(\deg(K)\)
correction \(\ell(K-D)\)
check both sides of Riemann-Roch

The final line should be an equality, not a paragraph:

\[\ell(D)-\ell(K-D)=\deg(D)+1-g.\]

If the equality fails, the bug is almost always a sign, a degree, or a mistaken canonical divisor.