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 $(X,p_1,\dots,p_n,L,\gamma_1,\dots,\gamma_n)$, where $X$ is a smooth genus-$g$ curve with $n$ marked points, $L$ a degree-$d$ line bundle, and $\gamma_i$ a trivialization of $L$ at $p_i$. We construct the Goppa morphism $\operatorname{Goppa}_{g,n,d}:\mathfrak{LS}_{g,n,d}\to\operatorname{Gr}(k,n)$. We prove that, if $n>d>2g-1$, the extended morphism $\Phi_{g,n,d}:\mathfrak{LS}_{g,n,d}\to\operatorname{Gr}(k,n)\times\mathfrak{M}_{g,n}$ is an immersion of stacks, and that $\operatorname{Goppa}_{g,n,d}$ 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 $(X,D)$, $D=p_1+\dots+p_n$, 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 $2\leq d\leq n-3$, the morphism $\operatorname{Goppa}_{0,n,d}$ is an immersion. Its restriction to each $\lambda\in\mathbb{G}_m^{n-1}$ is then a map $\mathfrak{M}_{0,n}\hookrightarrow\operatorname{Gr}(k,n)$, giving a canonical $\mathbb{G}_m^{n-1}$-family of immersions of $\mathfrak{M}_{0,n}$ into the Grassmannian.