Lagmental Vicfred

The Picard Group Splits Divisors by Degree by Vicfred

Last updated: Tue 21 November 2023

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

Divisor classes of degree zero form Pic zero, while each degree gives a torsor when a rational point is available. The example is deliberately concrete; it is a test of the statement, not a substitute for it.

Notation

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.

$$ \operatorname{Pic}(C)=\operatorname{Div}(C)/\operatorname{Prin}(C) $$

The definition determines which expressions are legal; only then does the identity become meaningful. An equality in \(\mathcal A\) may change ambient meaning, so I keep \(\mathsf D\) separate from \(\mathsf C\).

$$ \deg:\operatorname{Pic}(C)\to\mathbf Z,\qquad\ker(\deg)=\operatorname{Pic}^0(C) $$

Stress the formula

The computation below is not a second theorem. It is a checksum for the definitions and a place to inspect the difficult LaTeX at full size.

$$ p_0\in C(k)\quad\Longrightarrow\quad\operatorname{Pic}^d(C)\xrightarrow{\sim}\operatorname{Pic}^0(C),\quad[D]\mapsto[D-dp_0] $$

Interpretation

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

$$ \begin{aligned} \mathsf{D}\;&:\quad \operatorname{Pic}(C)=\operatorname{Div}(C)/\operatorname{Prin}(C),\\[5pt] \mathsf{C}\;&:\quad \deg:\operatorname{Pic}(C)\to\mathbf Z,\qquad\ker(\deg)=\operatorname{Pic}^0(C). \end{aligned} $$

Limit of the argument

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.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \deg:\operatorname{Pic}(C)\to\mathbf Z,\qquad\ker(\deg)=\operatorname{Pic}^0(C) \end{gathered}} $$

A symbolic summary is trustworthy only because the example and limitation remain visible beside it. The box compresses the conclusion without hiding its origin.

This article was posted on Sat 18 May 2019. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.