On the Goppa morphism

Abstract

We study the Goppa construction of linear codes from algebraic curves as a morphism of moduli stacks. For integers $g,n,d$ with $n>d>2g-2$ and $k:=1-g+d$, let ${\mathfrak{LS}}_{g,n,d}$ be the stack of rank-one level structures $(X,p_1,\dots ,p_n,L,{\hat{I}}_1^3,\dots ,{\hat{I}}_n^3)$, where $X$ is a smooth genus-$g$ curve with $n$ marked points, $L$ a degree-$d$ line bundle, and ${\hat{I}}_i^3$ a trivialization of $L$ at $p_i$. We construct the Goppa morphism ${\mathrm{Goppa}}_{g,n,d}:{\mathfrak{LS}}_{g,n,d}\to \mathrm{Gr}(k,n)$. We prove that, if $n>d>2g-1$, the extended morphism $\hat{I}{¦}_{g,n,d}:{\mathfrak{LS}}_{g,n,d}\to \mathrm{Gr}(k,n)\times {\mathfrak{M}}_{g,n}$ is an immersion of stacks, and that ${\mathrm{Goppa}}_{g,n,d}$ is universally injective if $n/2>d>2g+1$. If $n>d>2g+1$, we identify the fiber over a non-degenerate code $C$ with the moduli stack of $n$-pointed smooth genus-$g$ curves of degree $d$ in ${\mathbb{P}}_C$ whose marked points lie at the distinguished points determined by the coordinate projections of $C$, recovering the classical incidence problem of curves of fixed degree and genus through assigned points. For a fixed $n$-pointed curve $(X,D)$, $D=p_1+\cdots +p_n$, with $n=2(1-g+d)$, we show that the self-dual level structures form the fixed-point subscheme of a natural involution on ${\mathfrak{LS}}_{X,D,d}$, isomorphic to the $2$-torsion subscheme of ${\mathfrak{LS}}_{X,D,0}$ whenever it has a $\mathbb{K}$-rational point. In genus zero we identify ${\mathfrak{LS}}_{0,n,d}$ with ${\mathbb{G}}_m^{n-1}\times {\mathfrak{M}}_{0,n}$ and prove that, for $2\le d\le n-3$, the morphism ${\mathrm{Goppa}}_{0,n,d}$ is an immersion. Its restriction to each $\hat{I}»\in {\mathbb{G}}_m^{n-1}$ is then a map ${\mathfrak{M}}_{0,n}\hookrightarrow \mathrm{Gr}(k,n)$, giving a canonical ${\mathbb{G}}_m^{n-1}$-family of immersions of ${\mathfrak{M}}_{0,n}$ into the Grassmannian.