Lagmental Vicfred

The Canonical Divisor Has Degree Two g Minus Two by Vicfred

tags: mathematics, algebraic curves, divisors, Riemann--Roch

Zeros and poles of a rational differential define a divisor whose degree is determined by the genus. A small computation will anchor the general statement before the abstraction takes over.

Statement

On a smooth projective curve \(C\), each closed point \(p\) defines a valuation \(\operatorname{ord}_p\). Divisors combine these local orders into a global bookkeeping device.

$$ 0\ne\omega\in\Omega_{k(C)/k} $$

There are two layers here: the object \(\mathsf D\) and the law \(\mathsf C\). Writing them separately makes the direction of \(\Longrightarrow\) visible and keeps an accidental converse from slipping in.

$$ K_C=\operatorname{div}(\omega),\qquad\deg K_C=2g-2 $$

Worked algebra

Now evaluate one representative case. The result should agree with the structural law above, but it is obtained without assuming the conclusion.

$$ \begin{array}{c|ccc}C&\mathbf P^1&\text{elliptic curve}&g\ge2\\\hline\deg K_C&-2&0&2g-2>0\end{array} $$

Conceptual compression

The invariant statement is the one that does not depend on a convenient choice of coordinates, representatives, basis, or enumeration.

$$ \begin{aligned} \mathsf{D}\;&:\quad 0\ne\omega\in\Omega_{k(C)/k},\\[5pt] \mathsf{C}\;&:\quad K_C=\operatorname{div}(\omega),\qquad\deg K_C=2g-2. \end{aligned} $$

Caveat

A divisor being degree zero does not make it principal. The difference is measured by the Picard group and, in degree zero, by the Jacobian.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] K_C=\operatorname{div}(\omega),\qquad\deg K_C=2g-2 \end{gathered}} $$

The result is compact enough to reuse without pretending that the caveat has disappeared. The worked line remains the quickest consistency check.

This article was posted on Wed 08 August 2018. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.