On the Goppa morphism

Abstract

We develop a moduli-theoretic framework, over an arbitrary base field, for the classical Goppa construction of algebraic-geometric codes. We consider the moduli stack ${\mathcal{LS}}_{g,n,d}$ of line bundles of degree $d$ on $n$-pointed smooth projective curves of genus $g$, trivialized along the marked points, as the space of Goppa structures and show that evaluation of global sections defines a natural Goppa morphism ${\mathrm{Goppa}}_{g,n,d}:{\mathcal{LS}}_{g,n,d}⟶\mathrm{Gr}(k,n)$, $k=1-g+d$, together with an extended morphism to $\mathrm{Gr}(k,n)\times {\mathcal{M}}_{g,n}$. Under explicit numerical hypotheses, we prove that the extended Goppa morphism is an immersion. The pullbacks of the Plücker coordinates are identified with sections of the theta line bundle on the relative Picard scheme, and the locus of non-degenerate codes for which a subset $I$ fails to be an information set is precisely the corresponding theta locus. We identify the fibers of the Goppa morphism over non-degenerate codes $C$ with moduli spaces of non-degenerate $n$-pointed genus-$g$ curves of degree $d$ in the projective space $\mathbb{P}C$ canonically determined by $C$. This yields a Hilbert-incidence description of the Goppa construction and, in a natural range, implies that ${\mathrm{Goppa}}_{g,n,d}$ is schematic and quasi-projective and that ${\mathcal{LS}}_{g,n,d}$ is a quasi-projective scheme. In genus zero, we further obtain, for $2\le d\le n-3$, a canonical ${\mathbb{G}}_m^{n-1}$-family of immersions ${\mathcal{M}}_{0,n}\hookrightarrow \mathrm{Gr}(d+1,n)$, which connects the Goppa morphism with the classical geometry of ${\mathcal{M}}_{0,n}$.