Compact analytic stable-domain orbifold and actual universal plumbing

We need only the following elementary real-algebraic fact: a set specified by finitely many polynomial equalities, strict inequalities and finite quantifiers is a finite Boolean combination of polynomial sign conditions; if its projection on the first coordinate contains (0,ε), one can, after shrinking ε, select its other coordinates as real-analytic functions algebraic over R(t). Here real and imaginary coordinates of complex varieties are treated as separate real coordinates. We give the elimination/selection construction because an unspecified semialgebraic quotient theorem would conceal the needed compactness step.

Given a finite family of polynomials in $(x_1,\dots ,x_m)$, view them as polynomials in $x_m$. In the projection family include every coefficient, and all coefficients of all subresultant remainders for every pair of nonzero reducta and every reductum with its derivative. A reductum means each polynomial obtained by deleting a finite number of its highest powers; this ensures degree drops are included rather than assuming a generic leading coefficient. Repeat in the m−1 variables. The resulting lists are finite. For $m=1$, the finitely many real roots divide the line into points and open intervals. Inductively suppose the projection variables have been partitioned into connected cylindrical cells on which every projection polynomial has a constant sign.

Over one such cell, either a polynomial is identically zero or its degree is fixed with nonzero leading coefficient. The subresultant determinant identities are just Gaussian elimination of the coefficient rows of $P,Q,x_{m}P,x_{m}Q,\dots$: the first nonzero subresultant determines their monic gcd and its degree, with rational coefficient functions whose denominators never vanish on the cell. Thus each polynomial's squarefree part and each pairwise common factor have constant degrees there. The squarefree parts have simple roots; the ordinary implicit-function theorem and the nonzero leading coefficient show their real roots continue locally, cannot escape at an interior point, and cannot cross unless their common-factor pattern changes. Increasing order therefore gives finitely many globally continuous, locally analytic real root functions on the cell. This argument works on a lower-dimensional cell in its existing analytic parameter coordinates. Sorting the union of these roots partitions its cylinder into root sections and open sectors. On each lifted cell every original polynomial has constant sign. Identically zero specializations are omitted there, using their already-fixed coefficient-zero pattern; no well-orientedness or non-nullification hypothesis is assumed.

These cells themselves have finite polynomial sign descriptions. For a squarefree polynomial use its Euclidean Sturm remainders with the derivative. The number of distinct real roots below a nonroot a is the difference of endpoint sign-variation counts. The Euclidean divisions have precisely the subresultant denominators already fixed as nonzero; multiply inequalities by their fixed signs. At a root take the one-sided variation, omitting zeros. Consequently the condition that $x_m$ is the j-th ordered root, or lies between the j-th and (j+1)-st roots, is a finite Boolean sign condition. The usual Sturm count follows directly because the signs in a remainder pair change oppositely at an interior remainder zero, whereas at a simple root of P the variation drops by one; evaluating at ±∞ starts the count. Repeated factors were removed by the gcd calculation. This proves the claimed finite sign descriptions without using quantifier elimination to define the roots.

Projection now eliminates one existential quantifier: truth is constant on every lifted cell, and its projection is its already-described base cell. A finite union of these projections gives exactly the existential set. Negation gives universal quantifiers; induct on their number. This proves the finite elimination assertion in 1.1. For selection, perform the same decomposition in the order $t,x_1,\dots ,x_m$. Finitely many cells cover the stipulated set. One projects onto an interval (0,ε') after shrinking, since their projections cover all sufficiently small $t>0$. Choose one full cylindrical chain over that interval. At each root section take the root; in a bounded sector take its midpoint; in a sector unbounded on one side take the finite endpoint plus or minus one; for the whole line take zero. Recursively these are real analytic and algebraic over the field generated by the earlier selections, hence algebraic over R(t). The selected point stays in the chosen cell. No assertion about arbitrary smooth or definable selection is needed.

Let M be the point-target stable-curve stack from [F1]. Choose a quasi-compact étale surjective atlas $U\to M$, with U a finite disjoint union of affine finite-type complex schemes; such a finite selection exists because M is of finite type. Put $R=U{\times }_{M}U$. The finite diagonal makes R a scheme finite over U×U: base change of the diagonal is finite over an affine scheme, hence affine. Its two maps to U are étale. A finite complex algebraic map is analytically proper directly. Choose finitely many integral generators for its affine algebra; each satisfies a monic equation over the base. On a compact base set its coefficients are bounded, so the elementary monic root bound bounds every generator. Its preimage is a closed subset of a compact base set times these bounded closed coordinate balls, hence compact. Covering a compact base by finitely many affine neighborhoods gives the same conclusion. Thus $R(C)\to U(C)\times U(C)$ is a proper analytic relation.

The quotient map p:U(C)→Q is open, since the saturation of an open set is t(s^−1 of that set) and s,t are local biholomorphisms. The relation image is closed by 6.1. Disjoint inequivalent orbits therefore have disjoint saturated neighborhoods: otherwise points in successively smaller coordinate neighborhoods of two fixed representatives would be related, contradicting closedness. More explicitly, first choose product neighborhoods avoiding the closed relation, then shrink each within that neighborhood and use the openness of p; the images are disjoint precisely because no pair in their product is related. This proves Hausdorffness. Countable bases on the finite affine atlas push to a countable base on Q.

Each isomorphism class has finitely many representatives in U: its atlas fiber is an étale quasi-compact finite-type algebraic space over C, hence a finite discrete set. Representability here follows from the atlas; zero-dimensional affine étale charts and quasi-compactness give the finite-point assertion. Write U as a finite disjoint union of closed affine embeddings $U_i\subset C^{(N_i)}$, and put $\rho (u)=1+\mathrm{\mid }u{\mathrm{\mid }}^2$ on each. Then ρ is continuous and proper on this finite disjoint union. Define $h(p(u))=\min \rho (v):p(v)=p(u)\mathrm{.}$ The minimum exists. Its pullback to U is semialgebraic by 5.1 applied to $(\mathrm{\exists }v)[(u,v)\in imageRand\rho (v)=r$ $and(\mathrm{\forall }w)((u,w)\in imageRimplies\rho (w)\ge r)]\mathrm{.}$ The relation image itself has a sign description by projecting the affine equations of R, using the same elimination. Finite stabilizer multiplicities do not affect this set-theoretic minimum.

The pullback h∘p is continuous. For upper semicontinuity, choose a minimizing v for u and one isomorphism arrow $u\to v$. Since its source is étale, nearby u' lift to nearby arrows whose targets tend to v; their ρ values bound h(p(u')) from above. For lower semicontinuity, if $u_j\to u$ and $h(p(u_j))$ stays bounded, choose minimizing $v_j$. Properness of ρ gives a subsequence $v_j\to v$. Closedness of the relation gives $p(v)=p(u)$, so $h(p(u))\le \rho (v)=limh(p(u_j))$ along a subsequence realizing the liminf. An unbounded liminf needs no argument. This proves continuity on U, hence on the open quotient Q. Moreover $h^{-1}([1,A])=p(\rho \le A)$ is compact, being a continuous image of a compact set. Thus h is a proper continuous exhaustion of Q.

Suppose h were unbounded. The semialgebraic set of pairs (t,u) with $t>0$ and $h(p(u))=1\mathrm{/}t$ has projection containing (0,ε): an unbounded semialgebraic subset of R contains a final interval, by the one-dimensional decomposition in 2.1–5.1. Selection in 5.1 gives u(t), $0<t<\epsilon$, with each real and imaginary coordinate algebraic over R(t), and $h(p(u(t)))=1\mathrm{/}t$. The finitely many coordinates therefore generate a finite function-field extension K/C(t). All affine atlas equations hold over this field because they vanish on the real interval. The selected real branch defines a place of the smooth projective normalization of this algebraic curve over $t=0$. The finite-normalization theorem constructs this normalization, and regular complex curve charts identify its local uniformizer with an analytic disk coordinate. Thus u is a genuine algebraic generic object with a specified punctured analytic branch, not an arbitrary nonconvergent formal arc.

Apply the actual proper-stack valuative existence to this object and this DVR. It extends over a dominating DVR D; enlarging and completing the DVR is harmless. Lift its special fiber to the étale atlas U, allowing a residue extension. Étale formal lifting gives compatible atlas lifts over $D\mathrm{/}{m}_D^a$: in a local Jacobian presentation the invertible derivative uniquely solves each successive linear correction; their inverse limit is a formal atlas point v together with the object isomorphism. One may use a complete dominating henselian extension to realize this inverse limit. This step does not assert that its chosen formal coefficients are automatically convergent.

Convergence here follows from the GENERIC quasi-finiteness, not formal algebraization. The atlas fiber $U{\times }_M\mathrm{Spec}K$ is zero-dimensional and finite type. The generic atlas point of the lift therefore has residue field L finite over K. Every coordinate of v, and the needed finite tuple describing its isomorphism to u, is algebraic over K and hence over C(t). Restrict the dominating valuation to L. On the smooth projective normalization of L it is a point over $t=0$; its residue field is C. Since v's affine coordinates have nonnegative valuation, they are regular at that point. They define an actual algebraic map from a neighborhood of that smooth curve point into U. In its analytic disk coordinate s the resulting v(s) is holomorphic and bounded near $s=0$. All atlas equations hold because they hold generically. The generic isomorphism to u is an actual isomorphism for every sufficiently small nonzero s, after removing the finite exceptional points of its algebraic domain of definition.

The chosen place dominates the selected branch. The finite analytic curve map is locally $t=s^{m}a(s)$, a(0)≠0; take a holomorphic m-th root of the nonvanishing unit a on a disk and absorb it in s. Along $t>0$ choose one root s(t) tending to zero in this branch. Then $p(u(t))=p(v(s(t)))$, so $1\mathrm{/}t=h(p(u(t)))\le \rho (v(s(t)))\mathrm{.}$ The right side stays bounded, a contradiction. Hence h is bounded, and $Q=h^{-1}([1,A])$ is compact for a sufficiently large A. Being Hausdorff, second countable and locally compact, Q is metrizable: construct a countable collection of compactly supported continuous functions separating points using coordinate balls and finite quotient averaging, and map into $[0,1{]}^N$. Compactness makes this continuous separating map a homeomorphism onto its metrizable image. This also avoids a separate metrization theorem.

At a point u, the arrows u→u form its finite group Γ. Each arrow has, by étaleness, a local holomorphic germ on U near u. Choose common neighborhoods for these finitely many germs; the composition laws hold after shrinking because they hold on the actual groupoid. Properness of R→U×U prevents any other arrows from relating arbitrarily nearby points: such arrows would have a convergent subsequence over (u,u), landing on one of the already included isotropy arrows and hence on its local branch. Shrink again, and intersect the finitely many Γ-translates to obtain an invariant neighborhood. Its restricted groupoid is Γ acting on that neighborhood. If M is smooth, U is smooth and this is the required complex orbifold chart; an ineffective Γ action remains part of the orbifold. Its local universal family is the restriction of the actual algebraic universal domain curve, now viewed analytically.

Work on a smooth analytic atlas chart of the actual flat universal curve supplied by [F1] and [F5], and choose a projective local embedding of its fiber. At a nodal point of the central fiber, its ideal has $N-2$ generators with independent linear terms in a smooth ambient $N$-fold; after lifting them to the flat family, the inverse theorem eliminates them and leaves a smooth relative two-dimensional chart with coordinates $z,w$ over the smooth base coordinates $y$. The remaining ideal is finitely generated, and its reduction to the central fiber is the plane-node ideal $(zw)$. Flatness of ${\mathcal{O}}_C$ over the base makes tensoring $0\to I\to \mathcal{O}\to {\mathcal{O}}_C\to 0$ with the residue field left-exact, so $I\mathrm{/}\mathfrak{m}I$ is exactly $(zw)$. Choose $F\in I$ reducing to $zw$. Then $I=(F)+\mathfrak{m}I$. For the finite module $I\mathrm{/}(F)$, choose generators $g_i$ and write $g_i={\sum }_j{a}_{ij}{g}_j$ with $a_{ij}\in \mathfrak{m}$; the determinant of $I-(a_{ij})$ is a unit, so every $g_i=0$. Thus $I=(F)$ and the actual family has a hypersurface equation $F(z,w,y)=0$ with $F(z,w,0)=zw$. Define the local first-order smoothing space by $T_e^1:=\mathcal{O}\mathrm{/}(F,F_z,F_w){\mathrm{\mid }}_{y=0}\cong \mathbb{C}$; this quotient is the constant perturbation of the nodal equation modulo infinitesimal changes of $z,w$. By [F2], the local-to-global deformation map onto ${⨁}_{e=1}^d{T}_e^1$ is surjective and the corresponding obstruction group ${\mathrm{Ext}}^2$ vanishes.

Make a linear change in $z,w$ so $F_{zz}(0,0,0)\mathrm{\ne }0$; this is possible because the central Hessian of $zw$ is nondegenerate. The implicit theorem solves $F_z=0$ as $z=a(w,y)$. Taylor's formula gives $F(z,w,y)=G(w,y)+(z-a(w,y){)}^{2}A(z-a(w,y),w,y)$, where $A$ is holomorphic and $A(0,0,0)\mathrm{\ne }0$. A holomorphic square root of $A$ exists after shrinking; the coordinate $Z=(z-a)\sqrt{A}$ makes the equation $Z^2+G(w,y)=0$. The Schur complement of the nondegenerate Hessian shows $G_{ww}(0,0)\mathrm{\ne }0$. Solve $G_w=0$ as $w=b(y)$, write $G(w,y)=c(y)+(w-b(y){)}^{2}B(w-b(y),y)$ with $B$ nonvanishing, and take its local holomorphic square root. The resulting coordinates $Z,W$ give $Z^2+W^2+c(y)=0$, hence, after $U=Z+iW$ and $V=Z-iW$, $UV=q(y)$ with $q(y)=-c(y)$. Differentiating at $y=0$ shows $dq$ is the local $T^1$ smoothing class, because $F_z=F_w=0$ at the node. By [F2] the $d$ classes at the $d$ disjoint nodes are independent on the smooth atlas; therefore $(q_1,\dots ,q_d)$ extend to local base coordinates $(q_1,\dots ,q_d,y^{\mathrm{\prime }})$ by the inverse theorem. This proves the parameterized normal form $zw=q$ at every node, including simultaneous independent smoothing parameters. The remaining thick part is a smooth proper family with boundary after deleting fixed smaller nodal disks; integrating smooth horizontal lifts on its compact closure gives a smooth local trivialization. These charts have precisely the end geometry used in the weighted gluing construction.

On a compact Riemann surface Σ let L be a holomorphic line bundle, with its actual ∂bar operator. The adopted closed-surface real CR theorem applies to this arbitrary complex bundle, not only to tangent bundles of maps. Since the operator is complex linear, divide its real index by two to obtain ${\dim }_C\ker {\bar{\mathrm{\partial }}}_L-{\dim }_C\mathrm{coker}{\bar{\mathrm{\partial }}}_L=degL+1-g\mathrm{.}$ Its cokernel is smooth by the actual adjoint theorem. In a holomorphic coordinate z and a holomorphic frame with Hermitian square norm h, integration by parts identifies the adjoint equation for α=a dbar z with ∂_z(ha)=0, after the positive source density factors have canceled. Thus α↦h bar a dz is an antilinear identification of this adjoint kernel with holomorphic sections of K⊗L*. On overlaps the transformation is exactly that of K⊗L*, so this is a global identification. This proves the needed analytic duality as an actual equation. The operator's closed range and bounded complement inverse are part of the adopted Fredholm supplier.

A nonzero holomorphic section of a line bundle has isolated zeros of positive integer order, hence finitely many zeros. Its nowhere-zero normalization trivializes the bundle off small disks around those zeros. The boundary transition at a zero of order m winds m times. Stokes on the complement, or the explicit line-bundle clutching definition of c1, gives $\mathrm{deg}L=\sum m$. In particular a negative-degree line bundle has no nonzero holomorphic section. The same clutching calculation and the tangent Euler-class supplier give $\mathrm{deg}K=2g-2$. Therefore the dual kernel in 17.1 vanishes whenever $\mathrm{deg}L>2g-2$, and ${\bar{\mathrm{\partial }}}_L$ is onto.

For an analytic divisor D, construct O(D) directly by the transition $z^{(-m)}$ at each support point of multiplicity m; its holomorphic sections are meromorphic functions with the allowed poles, and its degree is $\sum m$. For a finite prescribed jet scheme Z of length ℓ supported at smooth points, assume $\mathrm{deg}(D-Z)=\mathrm{deg}D-\mathrm{\ell }>2g-2$. Then the exact sequence $O(D-Z)\to O(D)\to O(D){\mathrm{\mid }}_Z$ has the usual evaluation map. Its surjectivity follows without assuming a sheaf-cohomology theorem: extend the desired finite jets smoothly in coordinate disks, take ∂bar of that extension (which vanishes near the jets), and solve it in O(D−Z) using 18.1 under this degree hypothesis. Subtract the solution. Thus if deg $D\ge 2g+1$, global sections separate any two points and the first tangent jet at each point, and have no common zero. A finite basis defines an injective holomorphic immersion $\mathrm{\Sigma }\to {\mathbb{P}}^N$. Compactness makes it an embedding.

Choose a nonconstant meromorphic function f on $\mathrm{\Sigma }$ from this linear system. Its map to P¹ is finite: local nonconstant holomorphic coordinates are $z\mapsto z^m$ times a unit, critical points are isolated and finite, and compactness makes every regular fiber a finite set of fixed cardinality d. For any meromorphic g, the polynomial $P_g(z,T)={\prod }_{p\in f^{-1}(z)}(T-g(p))$ has single-valued holomorphic coefficients away from finitely many critical values and images of poles. Near each omitted value, a local expression $f-z_0=w^m$ times a nonzero unit shows that these coefficients have at most a finite-order pole. They extend meromorphically, and meromorphic functions on P¹ are rational: subtract their finitely many principal parts and use the maximum principle for the remaining holomorphic function. Thus every g used here is algebraic over C(f).

Choose finitely many meromorphic ratios separating the points of one regular fiber of f, and take a generic linear combination g separating all d points there. Its d branches remain distinct near that fiber. The covering $\mathrm{\Sigma }$ minus finitely many $points\to P^1$ minus finitely many points is connected, since deleting finitely many points from a connected surface does not disconnect it. Hence its monodromy is transitive. If $P_g$ factored over C(z), its roots would partition into two nonempty monodromy-invariant sets, a contradiction. Thus C(f,g) has degree d over C(f). Every other meromorphic ratio h lies in this field: interpolate its values on the d branches as a polynomial of degree $<d$ in g. The Vandermonde formulas give coefficients single-valued and meromorphic in z, with only finite-order poles, hence rational. This also proves that the chosen finite ratios generate the actual meromorphic function field, rather than merely unrelated algebraic extensions.

Form the projective algebraic plane curve defined by the irreducible relation $P_g(f,g)=0$ and normalize it by the actual finite-normalization theorem. Its smooth proper algebraic normalization has function field C(f,g). The meromorphic pair (f,g) lifts holomorphically from $\mathrm{\Sigma }$ to this normalization: every integral affine coordinate satisfies a monic equation with holomorphic coefficients, so the elementary root bound makes its meromorphic pullback bounded at the punctures and removable there. Away from finitely many points the lift is one-to-one and onto, since g distinguishes the d sheets of f. A nonconstant holomorphic map of compact smooth curves which is generically degree one is a biholomorphism: its local branching multiplicity would otherwise give more than one generic sheet. Thus every compact Riemann surface has an algebraic model. Biholomorphisms are algebraic because they induce the corresponding meromorphic function-field isomorphisms and hence the unique isomorphisms of the normalizations. Markings are closed C-points. Identifying each prescribed pair of distinct branch points by the subring of functions with equal residues produces the algebraic node Cz,w/(zw). The finite normalization/gluing construction in the adopted stable-domain supplier therefore algebraizes every analytic nodal marked curve and every isomorphism between them.

On an analytic stable nodal marked curve put $A={\omega }_C(p_1+\cdots +p_n)$ and $L=A^{1}0$. On a normalization component of genus h with b node branches and m original markings, set $a=2h-2+b+m>0$. Then deg $L=10a$. Even after prescribing values at all b node branches and four further jet lengths, one has $degL-b-4>2h-2,$ because the difference from 2h−2 is $9a+m-4\ge 5$. By 18.1, 19.1 the component sections therefore prescribe all these values and jets independently. Impose equality of the two values at each node in the relative-dualizing frame. The resulting global sections have $H^1=0$, separate every pair of points, distinguish the two branches at a node, and give an immersion on each branch. Choosing two sections with respective branch jets (z,0) and (0,w) gives the local node embedding. This proves very ampleness and vanishing directly on the nodal analytic curve. The normalization exact sequence used here is explicit: a tuple of sections descends exactly when its branch values match, and the quotient is the direct sum of one fiber for every node. Surjectivity onto that quotient follows from the same component interpolation. It is not an unproved singular Serre-duality input.

Apply this to a local analytic PLUMBING family constructed from smooth normalized component families and specified product node disks. Such component families may be chosen holomorphic: choose a complex basis of the tangent ∂bar cokernel with marked-point constraints, take its small linear Beltrami family ${\mu }_t$, and use the actual Cauchy-transform Neumann construction already given in the adopted smooth-domain slice lemma. The coefficients ${\mu }_t$ are holomorphic in the finite complex parameters. The same Neumann series is holomorphic in them. To check the transitions in the base parameter as well, let $f_t$ and $g_t$ be two resulting local coordinates. Both satisfy $w_{\bar{z}}={\mu }_t{w}_z$ and are holomorphic in t at fixed old z. At fixed $f_t$ the derivative in bar t of $g_t$ is $g_{(t,z)}$ times $(z_{\bar{t}}+{\mu }_t{\bar{z}}_{\bar{t}})$, which is zero by differentiating the identity fixing $f_t$. The overlap is therefore holomorphic both in its fiber coordinate and in t. These actual joint holomorphic transitions give the holomorphic family. Product-coordinate node disks and the now proved parameterized normal form in step 16.1 give $zw=q$. The parameter space is a polydisk in (q,t), of complex dimension 3g−3+n. Infinitesimal automorphisms vanish by stability, the marked tangent-line index gives the component dimensions, and one complex parameter is added for each node.

In this family $L=A^{1}0$ has a smooth holomorphic frame on every node fixture, namely $(dz\mathrm{/}z{)}^{1}0=(dw\mathrm{/}w{)}^{1}0$, and ordinary smooth frames away from nodes. At the central fiber its matched ∂bar operator is onto by 23.1. The actual weighted linear construction in [F4] applies to this LINEAR equation: the principal part is ${\mathrm{\partial }}_s+i{\mathrm{\partial }}_t$, the bundle coefficients are smooth and supported or decaying on the ends, and the shared node constants are the two identified fiber values. There is no nonlinear forcing. It gives uniformly bounded right inverses $Q_{(q,t)}$ and a continuous basis of holomorphic sections $s_j(q,t)=(I-Q_{q,t}{D}_{q,t})(s_j(0,0){)}_q\mathrm{.}$ These form a basis because their fixed core coordinate matrix stays invertible and the Fredholm index is constant. They converge uniformly on the compact cores and at node values, and in every fixed interior derivative order. The same assertion applies to every fixed power $L^r$, with its own finite basis. This is a continuous basis; it is not falsely called a holomorphic parameter basis at $q=0$.

The section maps remain embeddings. Away from nodes, point/jet separation persists on finite compact covers. On a node fixture a holomorphic coefficient has the convergent Laurent form $c_0+{\sum }_{m\ge 1}{c}_m{z}^m+{\sum }_{m\ge 1}{d}_m{w}^m,zw=q\mathrm{.}$ Cauchy integrals on the two fixed outer circles bound $\mathrm{\mid }{c}_m\mathrm{\mid }$ and $\mathrm{\mid }{d}_m\mathrm{\mid }$ by C $r^{(-m)}$. The two-circle Laurent decomposition is exact, with $w=q\mathrm{/}z$; the constant is counted once. Convergence of the core boundary data therefore gives convergence, including all z,w derivatives, on a smaller bidisk. The limiting sections which distinguish z and w have an invertible two-coordinate differential. Their extensions as functions on that bidisk remain local embeddings by the ordinary holomorphic inverse theorem, uniformly for small q. Compactness separates this node neighborhood from remote image pieces. Thus the continuous section bases give fiberwise projective embeddings of the entire family.

They also give a CONTINUOUS map to the actual finite Hilbert/frame presentation. To see the finite coefficients, take the finite degree r used in its Grassmannian Hilbert presentation at the central Hilbert point. The adopted Hilbert construction gives its finite relation equations and the required fixed Hilbert polynomial. Holomorphic sections of $L^r$ have a continuous basis by 25.1. Choose finitely many core evaluation points making the evaluation matrix injective on this central finite-dimensional section space, and retain its invertible minor nearby. Evaluate the finitely many degree-r monomials in the embedding sections there. Solving by that same minor gives their coefficient matrix continuously, hence the kernel/quotient Grassmann coordinates continuously. Higher finite defining equations hold fiberwise because every fiber has already been algebraized by 22.1 and the embedding is the complete L-linear system. This proves continuity into the Hilbert scheme with its actual scheme topology, not merely Hausdorff convergence of projective subsets. Marked coordinates and the L-frame data are continuous as well.

On the open set where every $q_e\mathrm{\ne }0$, this classifying map to the moduli STACK is holomorphic. Here is the finite-dimensional holomorphic-basis argument. A smooth proper analytic curve family is smoothly locally trivial by horizontal lifts on a compact fiber. The vertical ∂bar kernels of L have constant dimension and zero cokernel by 23.1; ordinary elliptic complement inverses therefore give a smooth finite kernel bundle. The total complex structure induces a (0,1) connection on it: lift a base (0,1) vector, apply ∂bar to a vertical holomorphic section, and take its kernel part. Differentiating the vertical equation shows it stays in the kernel; changing the lift changes it by the same vertical equation and hence not at all. Integrability of the total complex structure gives connection curvature zero. In a smooth frame write it ∂bar+A. On a small polydisk solve $\bar{\mathrm{\partial }}G=-AG$ with G invertible. Explicitly, solve its first coordinate by the Cauchy–Green integral equation $G=I-T_1(A_{1}G)$, which is a contraction after shrinking its radius; differentiate to obtain the smooth solution. Gauge transformation kills $A_1$, and curvature zero makes the remaining coefficients holomorphic in that first coordinate. Repeat in the remaining coordinates, whose Cauchy transforms preserve that holomorphy. This gives a local holomorphic frame of kernel sections. Such a frame yields a holomorphic projective embedding and a holomorphic Hilbert map. Different frames differ by a holomorphic GL matrix and give the same holomorphic stack map.

Compare with the compact algebraic stack and its finite étale atlas U. The continuous Hilbert/frame map in 27.1 pulls back the algebraic étale map U×_M H→H. At its central point the analytic inverse theorem provides a local sheet, so after shrinking the parameter polydisk the classifying map lifts CONTINUOUSLY into U. Off the nodal divisors this lift is holomorphic: use a local holomorphic section basis from 28.1, and the étale sheet over its holomorphic Hilbert map. Its choice agrees with the continuous lift because the choice of sheet is locally discrete. Changes of the possibly nonholomorphic continuous frame in 27.1 change the Hilbert/frame representative but not the underlying chosen étale moduli lift.

The continuous lift $B\to U$ is holomorphic across $q_e=0$ as well. Choose an affine coordinate neighborhood of its central value, containing its nearby continuous image. Every coordinate is continuous on the polydisk and holomorphic off the union of its coordinate divisors. The one-variable removable-singularity/Cauchy-integral formula extends it holomorphically across the first divisor while keeping the other variables as parameters; iterate across the finitely many divisors. The affine equations hold by continuity. Thus the resulting moduli lift is holomorphic without requiring differentiability of the weighted gluing map in q at zero.

This map has an invertible derivative. On the normalized components its derivative is the chosen complement to the image of ∂bar on vector fields vanishing at every mark and node branch. The nodal hypersurface $xy=q$ from step 16.1 has local first-order smoothing class $q$. The normalization local-to-global calculation gives $0\to \oplus H^1(T_{C_v}(-specialpoints))\to Ex{t}^1(L_C,O_C)$ $\to {\oplus }_{nodes}C\to 0,$ with the marked convention, and $Ex{t}^2=0$. This exact sequence follows from the actual two-row Čech calculation already used in the adopted algebraic domain proof: the local node Ext¹ is a one-dimensional skyscraper, and its H¹ and the curve tangent H² vanish. Analytic component tangent representatives have exactly these classes by the explicit coordinate differentiation. Consequently our parameters map isomorphically onto the tangent space of the smooth algebraic étale chart. If $3g-3+n=0$, both base germs are points and the inverse assertion is immediate. Otherwise the holomorphic inverse theorem makes the lift a local biholomorphism.

The original plumbing family is actually identified with the pulled-back universal curve, not only fiberwise assigned to the same points. Away from the divisors the just-constructed holomorphic Hilbert classifying map gives the actual family isomorphism. Its continuously chosen projective frames and node coordinates extend it continuously to the central fiber, where it is the chosen isomorphism. On source and target charts this is a continuous map holomorphic off the discriminant; its coordinate functions extend holomorphically by the same removable argument as in 30.1. The total plumbing fixture zw=q is smooth (q is an independent base coordinate), so its coordinate functions satisfy this ordinary several-variable argument. The inverse extends in the same way after the local base biholomorphism in 31.1; both composites are identities on the dense smooth-fiber locus and hence everywhere. This proves an isomorphism of the actual families. Finite automorphism germs extend by this same universal comparison, and the proper finite analytic groupoid constructed in steps 6.1–14.1 supplies the Γ quotient. Thus these are genuine local universal plumbing charts for the analytic stable domains in question.

The component algebraization and family comparison identify the compact algebraic quotient with the space of ALL analytic stable domains in its plumbing topology. The actual algebraic forgetting construction in [F1] is given by the full section algebra of the retained log-dualizing power; analytifying its finite chart equations gives a continuous forgetting/stabilization map. The local family comparisons commute with it by their actual universal isomorphisms. This proves all assertions without importing projectivity of a coarse space or analytic compactness as an extra premise.