Pointed curve with formal coordinates

Assume full AC. A pointed curve with formal coordinates is a smooth projective complex curve $X$ with specified presentation as defined above, an integer $n\ge 0$, an ordered tuple of pairwise distinct points $(p_1,\dots ,p_n)$, and for every $i$ an isomorphism of complete local rings $t_i:{\widehat{\mathcal{O}}}_{X,p_i}\stackrel{\sim }{\to }\mathbb{C}\mathrm{⟦}t\mathrm{⟧}$. The genus $g(X)$ is the genus of the underlying compact connected Riemann surface, using the Euler-characteristic convention of the imported genus definition. The full AC assumption is carried because that supplier's stated surface-classification proof uses it; this definition itself makes no further selection.

For a family, the marked points are pairwise disjoint holomorphic sections of a family as defined in the smooth-projective-curve definition, and each $t_i$ is the relative formal coordinate ${\tau }_i:{\widehat{\mathcal{O}}}_{\mathcal{X},s_i(S)}\stackrel{\sim }{\to }{\mathcal{O}}_S\mathrm{⟦}{t}_i\mathrm{⟧}$ of that supplier, preserving the section ideal as specified there. For an individual marked point, put ${\widehat{\mathfrak{m}}}_{p_i}:=\ker ({\widehat{\mathcal{O}}}_{X,p_i}\to \mathbb{C})$ and ${\xi }_i:=t_i^{-1}(t)\in {\widehat{\mathfrak{m}}}_{p_i}$. The data retain the full formal coordinate isomorphisms; they do not pass to only the first-order class ${\xi }_i<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>{\widehat{\mathfrak{m}}}_{p_i}^2$, a nonzero element of the cotangent space ${\widehat{\mathfrak{m}}}_{p_i}\mathrm{/}{\widehat{\mathfrak{m}}}_{p_i}^2\cong {\mathfrak{m}}_{p_i}\mathrm{/}{\mathfrak{m}}_{p_i}^2$. The identification is induced by completion and follows by truncating a local chart parameter at degree two. Replacing the full coordinate by this class requires a separately specified descent action of the remaining coordinate changes, which this item does not assert. An analytic coordinate germ means a holomorphic chart coordinate $z$ centered at $p$. Taylor expansion in $z$ induces a continuous isomorphism ${\tau }_z:{\widehat{\mathcal{O}}}_{X,p}\stackrel{\sim }{\to }\mathbb{C}\mathrm{⟦}t\mathrm{⟧}$ sending $z$ to $t$; its restriction to holomorphic germs has convergent Taylor series. An arbitrary formal coordinate isomorphism need not be of this form, and formal series need not converge. No stability condition is imposed here: stable $(g,n)$, meaning $2g-2+n>0$, is required only when a later moduli or sewing theorem explicitly assumes it. No moduli stack or boundary-stratum theorem is included.

Riemann surfaces and holomorphic atlases

Genus and Euler characteristic of a compact Riemann surface

The Axiom of Choice

Smooth projective complex curve with formal-coordinate bundle

Formal-coordinate change group Aut O

A formal-coordinate fibre is an Aut O torsor