Proper algebraic stack of stable maps with finite stabilizers

Part 1: Data and actual inputs. Fix a projective embedding $V\subset P_k^r$ and $A=O_V(1)$, g,n≥0 and the target's algebraic degree functional β on Pic(V), with $d=\beta (A)$. Families have proper flat finitely presented geometrically connected nodal genus-g domains, ordered distinct smooth markings, and a map f to V. Stability is exactly $2h-2+s_E>0$ on each target-constant normalized geometric component E of genus h, where $s_E$ counts node branches and markings. Nonreduced automorphism schemes are permitted. Put

$L={\omega }_{C\mathrm{/}S}(\mathrm{\Sigma }{p}_i)\otimes f\ast A^3$, $M=L^4$, $D=4(2g-2+n+3d)$, $s=D+1-g$.

Empty unstable degree-zero cases are dealt with in part 8; in all nonempty stable cases $D>0$ and $s>0$.

The actual published fixed-polynomial thm-hilbert-scheme-represents-projective-flat-families represents families on ALL test schemes, including non-Noetherian ones; its universal family is flat and finitely presented. We use it for a fixed projective Segre embedding over k. The actual published lem-proper-flat-fp-cohomology-perfect-complex applies over any ring to a finitely presented sheaf flat over that ring and gives a bounded nonnegative finite-projective complex computing every base change. Local finite-free models and lem-cohomology-base-change-finite-free-criterion give the matrix arguments used below. thm-h0-structure-sheaf-proper-curve, thm-serre-duality-curves-line-bundles, thm-full-riemann-roch-divisor, and negative-degree vanishing supply the field-curve inputs with their actual choice assumptions. Finite morphism/base-change, thm-proper-quasi-finite-is-finite, cor-quasi-finite-locus-open-finite-type-algebra, the Jacobian/étale criterion and Nakayama are used only after the relevant finiteness hypotheses are checked. Faithfully flat module/algebra descent gives effective descent of vector bundles and embedded homogeneous ideals. The existing lem-noetherian-approximation-proper-fp-flat-sheaf and finite-presentation spreading interfaces allow the proper flat finite-presentation data, including $O_C$, to be handled on a Noetherian stage when a local construction requires a coherent kernel. The all-field/nonaffine quotient argument below is explicit; it does not use a general-linear quotient theorem restricted to C or to affine parameter schemes.

The local nodal-locus/dualizing construction in part 3 expands the internal nodal-open calculation. The cohomology proofs in part 2 expand the internal cohomology calculation; the actual frame enhancement corrects the unenhanced Picard/PGL shortcut discussed in the internal framing construction part 1.

Part 2: Uniform complete embedding and the universal H⁰ facts. On a normalized geometric component E, let h be its genus, v its number of node branches, $s_E$ its node branches plus markings, and $d_E={\mathrm{deg}}_E$ f*A. Then $q_E={\mathrm{deg}}_E$ $L=2h-2+s_E+3d_E>0$. For $d_E=0$ this is stability. For $d_E>0$ it is at least $-2+3=1$. The power A³ is essential. Moreover ${\mathrm{deg}}_E$ $M=4q_E\ge 2h+v+1$.

For $h=0,d_E=0$ use $s_E\ge 3$ and $4s_E-8\ge s_E+1\ge v+1$. For $h=0,d_E>0$, $q_E\ge s_E+1$. For $h=1,d_E=0$ use $s_E\ge 1$ and $4s_E\ge s_E+3\ge v+3$; for $h=1,d_E>0$, $q_E\ge s_E+3$. For h≥2 the difference is $6h-9+4s_E-v+12d_E>0$. These bounds do not depend on the number or incidence pattern of components.

Let $B_E$ be the reduced branch divisor on E. For an effective divisor Q on E of degree at most two, $\mathrm{deg}(M{\mathrm{\mid }}_E(-B_E-Q))\ge 2h-1>2h-2$.

Serre duality and negative-degree vanishing give $H^1=0$. The restriction sequence therefore allows arbitrary values and jets along $B_E+Q$. Taking $Q=0$ also gives $H^1(E,M{\mathrm{\mid }}_E)=0$ and arbitrary branch values. The normalization sequence $0\to M\to {\nu }_{\ast }{\nu }^{\ast }M\to {⨁}_{\text{nodes }z}{M}_z\to 0$

takes differences of the two branch values. Its global map to node fibres is surjective, so $H^1(C,M)=0$. To separate a length-two subscheme of C, at smooth points prescribe its at most two values/jets. At a node $k[[x,y]]\mathrm{/}(xy)$, a length-two quotient prescribes the common constant and one linear combination of the two branch derivatives. Prescribe the two derivatives independently and then take that combination. This costs at most two additional conditions on any normalized component, including a self-node, and the preceding interpolation supplies them together with compatible values at every other branch pair. The complete linear system of M consequently separates every length-two subscheme and is very ample. Component Riemann–Roch followed by subtraction of one for each node gives deg $M=D$, $\chi (M)=D+1-g=s$, $h^0(M)=s$.

Here is the family passage on arbitrary test bases, rather than only on geometric points. For π:C→S proper flat finitely presented and an invertible M with the preceding fibre H¹-vanishing, take a local nonnegative finite-free complex K computing M. At a base point the degree-one comparison is surjective since its target is zero. The finite-free criterion gives base change and finite generation of H¹ near that point, which is zero after shrinking by Nakayama. It is therefore projective. Part 3 of the criterion at $q=1$ makes the degree-zero comparison surjective; the criterion at $q=0$, with $K^{-1}=0$, gives $E={\pi }_{\ast }M$ finite locally free of rank s, with every base change. All shrinkings are actual open shrinkings, since the invertible matrix minors and their finitely many inverse entries descend from the local ring. Thus E is locally free on S, not merely on its localizations.

Evaluation π*E→M is surjective on geometric fibres and hence at every stalk by Nakayama. It defines j:C→P(E), with quotient-line convention. The morphism is proper because π is proper and P(E) separated; it is finitely presented by its graph and the finite-presentation projective-bundle diagonal. Its fibres over geometric points of P(E) are finite since each $j_t$ is a closed immersion, so it is quasi-finite and therefore finite. Finite pushforward commutes with base change. The finitely generated cokernel of $O_{P(E)}\to j_{\ast }{O}_C$ has zero base-fibre restriction everywhere; Nakayama makes it zero. Hence j is a closed immersion. All constructions commute with arbitrary base change.

We will also need the universal identity ${\pi }_{\ast }{O}_C=O_S$. Use the same nonnegative complex for $O_C$. On each field fibre $H^0=\kappa (t)$ by geometric connectedness and reducedness. The unit section lifts its basis 1, so the $q=0$ comparison is surjective. The finite-free criterion with $K^{-1}=0$ makes H⁰ finite locally free and universally base-change compatible. It has rank one; its unit map from $O_S$ is an isomorphism on every residue fibre and hence an isomorphism of line bundles. This proof uses the actual cohomology comparison, not fibrewise dimension alone.

Part 3: The finite-type prestable Hilbert/marking locus. Start with the projective Hilbert scheme Q of subschemes $Z\subset P^{s-1}\times V$ with Segre Hilbert polynomial P(t)=(D+d)t+1−g.

Its fibres have dimension one since this polynomial is linear with positive leading coefficient. The universal family is proper flat and finitely presented. Embedded points or zero-dimensional components have not yet been allowed to pass the nodal test.

We describe the nodal open explicitly. At a geometric node of a fibre in an affine N-space embedding, the fibre ideal in the smooth ambient has N−1 minimal generators. The completed node presentation shows that N−2 can be selected with independent linear terms and the last equation on the resulting smooth surface has a square-free quadratic term. These are finite linear/quadratic jet conditions, detected by minors and faithful flatness of completion, not an attempted algebraization of arbitrary formal coordinates. Flatness of the family gives the exact fibre sequence $0\to I\otimes \kappa (t)\to O_{\mathrm{a}\mathrm{m}\mathrm{b}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{t}}\otimes \kappa (t)\to O_Z\otimes \kappa (t)\to 0$. Lift these generators to I; Nakayama kills their coherent quotient after shrinking. The invertible (N−2)-minor gives a smooth relative surface, with étale coordinates x,y, and the last equation is $F=0$.

Shrink to where its binary Hessian determinant is invertible. Each geometric point there is either smooth, if one first derivative is nonzero, or has a nondegenerate quadratic term. Over an algebraically closed field this quadratic is linearly equivalent to xy, including characteristic two: for ax²+bxy+cy² the determinant is 4ac−b², which in characteristic two is −b², exactly the distinct-tangent condition. To remove higher terms of degree m≥3, write the unwanted term as $x{A}_{m-1}+y{B}_{m-1}$ and substitute $x-B_{m-1}$, $y-A_{m-1}$. This kills degree m without altering lower degrees; iteration gives the invertible formal change to xy. No division by two occurs. Smooth one-dimensional points have ordinary smooth relative charts; smooth zero-dimensional points are excluded. These charts give an open U of the total universal curve consisting precisely of smooth one-dimensional or nodal fibre points. Properness makes the image of Z\U closed, and removing it gives the open on which every geometric fibre is a pure reduced nodal curve.

The last equation is a relative regular hypersurface equation. Its smooth ambient algebra P and P/(F) are base-flat. The image ideal (F) is base-flat from their exact sequence; hence the kernel of multiplication P→(F) remains injective after fibre tensoring. Fibre multiplication by the nonzero plane-curve equation is injective. Nakayama kills the coherent kernel near the point. This justifies lci adjunction: the local relative dualizing line is the hypersurface adjunction line, with residue generator dx∧dy/F; on a geometric split node its restrictions are $dx\mathrm{/}x=-dy\mathrm{/}y$. Determinants of the conormal exact sequences compare two embeddings, and composition of these determinant identifications gives the cocycle. They glue the relative line ω and commute with every base change by the one-equation Koszul resolution. This constructs the ω used above and below, including its normalized-component degree 2h−2+v. For a family over an arbitrary affine test base, apply the stated approximation interface to $O_C$. It gives a proper finitely presented $X_i$ and a base-flat finitely presented $F_i$ whose pullback is $O_C$. Descend the isomorphism $F_i\otimes A\cong O_{X_i}\otimes A$, its inverse and their equations to a later finitely generated stage, using finite-presentation sheaf spreading. At that stage $F_i\cong O_{X_i}$; hence $X_i$ itself is flat, as required. This supplies a proper flat finitely presented Noetherian model of the actual family. The base morphism factors through that stage's nodal open because all its geometric base-changed fibres are nodal. Pull back its adjunction line; comparison on local smooth embeddings gives independence of the stage and the canonical base-change isomorphisms. Work locally on a general base and glue by the same comparisons. Thus the dualizing line used in the all-test embedding has actually been supplied there as well.

On this nodal locus the nonnegative cohomology complex for $O_Z$ gives $h^0(Z_t,O)=rank$ K⁰−rank(d⁰_t). Matrix minors make this upper semicontinuous. For a nonempty reduced proper geometric curve, H⁰ is one copy of its field for each connected component. Hence h⁰≤1 is exactly geometric connectedness and is open; nonemptiness is forced by P. The fixed constant term then makes the arithmetic genus g.

Projection $e:Z\to P^{s-1}$ is proper. Its quasi-finite locus is open in Z. At a base point with $e_t$ a closed immersion, every point of that fibre belongs to this locus; the image of its closed complement under proper Z→Q is closed. Remove that image. Projection is now proper quasi-finite, hence finite. Finite pushforward and base change identify its fibre algebra. The coherent cokernel of $O_{P^{s-1}}\to e_{\ast }{O}_Z$ vanishes on the prescribed fibre; remove its support image under the proper projective-space projection. The algebra map is then surjective and e is a closed immersion. This proves openness of the closed-immersion locus on Q.

Impose deg $f\ast A=d$. For any target line bundle B the pullback degree equals $\chi (f\ast B)-\chi (O_Z)$, locally constant by the alternating ranks of the universal cohomology complex. Thus this condition is open and closed, and the Segre degree then implies deg $e\ast O(1)=D$. Add n markings by taking the n-fold fibre product of Z over the remaining Hilbert base and restricting to the open where all sections are smooth and pairwise disjoint. Denote the resulting separated finite-type scheme by B. Its universal curve has all the asserted properties; a Hilbert point with markings is exactly its represented family over every test scheme.

Finally impose the entire functional β. Each target line bundle gives a locally constant integer degree function on B. Since B is noetherian and finite type it has finitely many connected components, each open and closed. Each of those functions is constant on each component. An arbitrary, even infinite, list of these degree equalities therefore selects a union of a subset of the finitely many components. It is an open-and-closed finite-type subscheme. This is the algebraic class convention on arbitrary fields; it is not replaced by a single A-degree condition.

Part 4: Retaining α and imposing complete frames on all tests. On the universal curve over B form $M=(\omega (\mathrm{\Sigma }{p}_i)\otimes f\ast A^3{)}^4$ and $N=M\otimes e\ast O(-1)$. Represent sections of N without assuming $H^1(N)=0$. On an affine chart of B choose its actual nonnegative finite-free cohomology complex K. For every A-algebra A', $H^0(N_{A^{\mathrm{\prime }}})=\ker (K^0\otimes A^{\mathrm{\prime }}\to K^1\otimes A^{\mathrm{\prime }})$.

This kernel is represented by the affine linear scheme Spec Sym(coker((K¹)∨→(K⁰)∨)). Its universal section defines a homomorphism e*O(1)→M. Its noninvertibility locus is closed on the pulled-back curve; its proper image in the section scheme is closed. Removing that image represents EXACTLY the isomorphisms

α:e*O(1) ≅ M.

This works on all tests, since a line homomorphism is invertible precisely where it is nonzero on every residue fibre. Constructions on affine charts agree by their represented section functor and glue to a finite-type B-scheme I. It is separated, being an open subscheme of a scheme affine over separated B. Its invertibility open need not be affine; no such affineness is assumed.

Every geometric map over I is stable. Indeed e is a closed immersion and α identifies its ample line e*O(1) with M. On a contracted normalized component E this gives $4(2h-2+s_E)>0$, exactly the stability inequality. Thus no separate stability-openness theorem is hidden here. part 2 now applies: $E={\pi }_{\ast }M$ is a rank-s vector bundle with arbitrary base change. Projective coordinates and α induce $O_I^s\to E$. Restrict to the open where its determinant is invertible, and call this separated finite-type scheme H.

The functor of H is precisely stable maps together with a FULL FRAME $\tau :O^s\cong {\pi }_{\ast }M$. A frame gives evaluation $O_C^s\to M$, hence the complete-linear-system closed embedding $e:C\to P^{s-1}$ and the tautological α:e*O(1)≅M. Its graph in $P^{s-1}\times V$ has polynomial P and determines the Hilbert point and markings. Conversely an H-point recovers the frame by restricting coordinates and using α. The two operations are inverse on every test: they recover the identical quotient-line map, α and ordered sections. Functoriality of ω, cohomology base change and evaluation makes them compatible with all test changes, including nilpotent ones.

Changing the frame defines a $G{L}_s$-action on H. The scalar λI fixes the embedded curve but multiplies α by λ (up to the chosen left/right action convention). Therefore α cannot be discarded, and the frame group is $G{L}_s$. A framed map has no nonidentity automorphism preserving its frame: that automorphism acts identically on the complete embedding, so it is the identity on C; its canonical action on M is then the identity as well.

Part 5: Effective descent, the quotient atlas and relative Isom. The unframed map functor is a groupoid-valued fppf stack. Here is its effective descent, including the curves. From fppf descent data on stable maps, the functorial rank-s bundles $E={\pi }_{\ast }M$ descend by faithfully flat module descent. On the cover, the evaluation embeddings place the curves in P(E)×V. Their quasicoherent homogeneous ideals descend degree by degree by faithfully flat module/algebra descent; finite presentation, their products and their equations descend as finite data. Taking relative Proj recovers a closed finitely presented curve in this descended projective bundle, with the original flat family after pullback. Properness, flatness, nodal fibres, connectedness and the component degree conditions are preserved/detected after the cover. The markings descend as morphisms; the projection to V supplies the descended map. The adjunction ω, α and evaluation identifications descend by their canonical construction. This gives the actual object and the uniqueness of its descent isomorphisms, rather than naming an unproved polarized-scheme descent theorem.

Forget the frame. Over a specified family C/T and a specified identification with its image in the unframed stack, the fibre of H is exactly $Isom(O_T^s,E)$, its ordinary frame-bundle $G{L}_s$-torsor. It is smooth and surjective since E is locally free and $G{L}_s$ is det≠0 in affine s²-space over any field. Conversely, from a $G{L}_s$-torsor P→T and an equivariant map P→H, the preceding embedded-ideal descent gives the corresponding family over T; its frame torsor is P. These inverse constructions identify the stable-map stack with $[H\mathrm{/}G{L}_s]$ on all test schemes. The morphism $H\to [H\mathrm{/}G{L}_s]$ is a representable smooth surjective atlas, not a statement that underlying stable maps have trivial automorphism groups.

For two map families over an arbitrary T, frame both $E_1$ and $E_2$ on their product frame torsor T'→T. They give two morphisms $x_1,x_2:T^{\mathrm{\prime }}\to H$. An isomorphism of the unframed maps is equivalent to a matrix $g\in G{L}_s$ carrying $x_1$ to $x_2$: transport the second frame along that isomorphism and compare it with the first; conversely the frame comparison and the represented embedded data recover the map isomorphism. Thus their Isom functor over T' is the transporter

$\{g\in G{L}_{s,T^{\mathrm{\prime }}}:g\cdot x_1=x_2\}$.

It is a CLOSED subscheme of $G{L}_{s,T^{\mathrm{\prime }}}$, the pullback of the diagonal of the separated finite-type scheme H along $g\mapsto (g\cdot x_1,x_2)$. It is affine and finitely presented over T'. The natural frame-change identifications satisfy the fppf cocycle; affine algebra descent therefore gives an affine finitely presented T-scheme representing Isom. This proves the diagonal representable affine of finite presentation. It applies to arbitrary, non-Noetherian bases by the all-test construction. Together with the finite-type atlas it proves the all-field finite-type Artin assertion. It does not assert that relative inertia is finite before the complementary valuative argument.

Part 6: Optional complex integral homology class: the local family argument. When $k=C$ and the target convention fixes β∈H₂(V(C),Z), impose that condition on the finite-type Hilbert parameter using the integral fundamental class, not just the degrees of algebraic line bundles. We give the local argument for constancy; it also retains possible torsion in H₂.

First consider a nodal family pulled back to a smooth complex algebraic curve, near a closed base point $t=0$. A persistent node has analytic local form $xy=0$ over the disk. At a nonpersistent node its local form is $xy=a(t)$. This analytic form follows from the holomorphic implicit theorem applied to the two first derivatives (their Hessian is invertible), translation to their unique critical point, and the two-variable holomorphic completing-square argument with parameters. More explicitly, after translating the unique critical point to the origin and making one quadratic diagonal coefficient nonzero, solve $F_x=0$ for $x=b(y,t)$. Taylor division gives $F=F(b(y,t),y,t)+(x-b(y,t){)}^{2}u(x,y,t)$, with u a unit. Its analytic square root changes the last term to X². The remaining series is −a(t)+y²v(y,t), with v a unit by the nondegenerate Hessian; its square root changes it to −a(t)+Y². A linear change of X,Y gives xy−a(t). These roots exist analytically by the implicit theorem at a nonzero complex square root, not by a merely formal infinite substitution. If a is not identically zero, write $a(t)=u(t)t^m$, m≥1, and absorb the analytic unit by a local coordinate change. Thus the model is $xy=t^m$. The preceding use of analytic square roots concerns only this complex homology paragraph, not the all-characteristic nodal construction in part 3.

Choose a small radial real base interval 0≤t≤ε and small node polydisks. Off these polydisks the family is a proper smooth family of compact oriented real surfaces with boundary. A metric and partition of unity give a horizontal lift of the interval vector field: invert the surjective differential on each chart and patch the right inverses. Compactness permits its flow over the interval. It gives the integral cylindrical bordism of the complements, respecting their boundary collars. Persistent nodes are treated by their two normalization branches, each a product disk over the interval.

At a nonpersistent node the local bordism is completely explicit. For suitable small δ, take the oriented domain Q×S¹, where Q consists of 0≤r,s≤δ and $rs\le {\epsilon }^m$, and map it to the family by (r,s,θ) ↦ ($x=r$ exp(iθ), $y=s$ exp(−iθ), $t=(rs{)}^{1\mathrm{/}m}$).

The two axis faces give the two central disks, with their complex orientations. The face $rs={\epsilon }^m$ is the nearby annulus. The faces $r=\delta$ and $s=\delta$ are the boundary collars glued to the complement's cylinders. The circle over $r=s=0$ collapses to a point and contributes no three- or two-dimensional boundary. Triangulate the planar domain Q with its curved boundary, cross it with a finite oriented circle triangulation and use the prism subdivisions; pushing this finite relative three-chain forward gives boundary equal to the nearby annulus minus the two central disks plus the stated collar faces. This is also a direct cone/cylinder description of the node surgery; it does not require making an arbitrary algebraic map globally simplicial. Gluing the finitely many local chains to the complement's cylinder produces an integral three-chain in the total family with boundary $[C_{\epsilon }]-[C_0]$. Applying the continuous target map makes $f_{\ast }[C_{\epsilon }]-f_{\ast }[C_0]$ an integral boundary in V. The multiplicities on the two central branches are one in this parametrization. Hence no argument that only detects rational homology or Chern-character pairings has been used.

This disk argument gives constancy across specialization in any irreducible finite-type parameter base. To see the full component assertion, on each irreducible base take a dense smooth open where the number of persistent nodes and components is constant; after an étale cover these nodes and branches split. Their partial normalization is a smooth proper family. Horizontal vector fields as above show class constancy on its connected smooth open; the node identifications do not change the sum of component fundamental classes. For a point on its complement, use the actual published lem-complex-algebraic-boundary-points-admit-normalized-curve-arcs: there is an algebraic curve through it meeting that open, and a normalized disk with puncture in the open. The disk argument proves the same class at that point. Thus the class is constant on each irreducible component. Components meeting at a point have the same class; the finite incidence graph then gives constancy on every connected component. The existence of the dense constant-node open follows directly from the finite-presentation singular locus and its two étale branch charts: remove images of nonfinite or nonétale loci and component-incidence changes; generic branches separate after a finite extension, which spreads over a dense étale cover in characteristic zero. This is only a generic-open reduction for the complex class argument.

The parameter scheme has finitely many connected components. A specified integral homology class therefore selects a union of finitely many of them, open and closed. Repeat that restriction before or after the α/frame construction. This preserves the optional complex class convention of the target; it does not identify homology classes with Pic-degree functionals.

Part 7: Setup. Assume AC and DC as in the existing pinned cohomology suppliers. Let S be locally Noetherian, let (C/S;p₁,…,pₙ₊₁) be a stable pointed nodal family, and assume $2g-2+n>0$. Put $L={\omega }_C\mathrm{/}S(p_1+\dots +p_n)$, $M=L^4$ and $R={\oplus }_{m\ge 0}\pi \ast M^m$.

Then $C^{\mathrm{\prime }}=Pro{j}_S$ R is a stable n-pointed family and there is a canonical proper contraction q:C→C', carrying the first n markings to disjoint smooth markings. On each geometric fibre q contracts exactly the unstable component created by forgetting pₙ₊₁, if there is one. The construction commutes with every base change, including non-Noetherian base changes of this family, and $q_{\ast }{O}_C=O_C^{\mathrm{\prime }}$ universally. Its proof uses no stable reduction or Knudsen contraction theorem.

The relative dualizing sheaf here is the one already used in the stable-family definition. Its local nodal calculation is the hypersurface calculation for $xy=a$: adjunction gives an invertible dualizing module generated by the residue class of dx∧dy/(xy−a), restricting on the two branches to dx/x and −dy/y. This formula commutes with base change. Thus tensoring by the smooth Cartier marking divisors and taking L and M commute with base change. Part 3 supplies the family-level nodal charts and Gorenstein dualizing line, including arbitrary base change; these are the interfaces used in this forgetting construction.

Part 8: Fibre geometry and cohomology. On a normalized geometric component E of genus h, let $s_E$ count the retained markings and node branches. Then deg $L\mathrm{\mid }E=2h-2+s_E$. Before forgetting the degree was positive; forgetting reduces it by one only on the component containing pₙ₊₁. Consequently all degrees are nonnegative and at most one component has degree zero.

A zero component must be smooth rational, with exactly two retained special points. It is either an unmarked bridge with two node branches, or a tail with one retained mark and one node. The apparent isolated rational two-marked curve and isolated one-marked genus-one curve are excluded by $2g-2+n>0$; a self-nodal rational zero component with no attachments is exactly that excluded genus-one case. A genus-one component with an attachment still has positive degree after forgetting.

On the geometric fibre form $Z=C\mathrm{/}E$ by removing the tail and transferring its retained mark to the attachment point, or by removing the bridge and identifying its two attachment points. The latter is the finite affine pinching construction, valid also when the attachment points lie on one component. Its local ring is $k[[x,y]]\mathrm{/}(xy)$. These constructions yield a proper connected nodal curve of the same arithmetic genus, with n disjoint smooth markings. Neighbouring special-point counts are unchanged: the tail node is replaced by its transferred mark, and a bridge node branch is replaced by the branch of the new node. Thus every remaining component is stable.

The residue description of the dualizing line proves $L=\sigma \ast {\omega }_Z(\mathrm{\Sigma }{p}_i)$ for σ:C→Z. On a bridge the degree-zero bundle ${\omega }_{P^1}(\text{two attachments})$ has residues of opposite sign; they induce exactly the dualizing-fibre identification at the new node. On a tail the two simple poles are its node and its retained mark, so contraction replaces the node contribution on its neighbour by that marking. Away from E the identification is unchanged. This argument determines the gluing, rather than just comparing component degrees.

The normalization exact sequence at the attachments shows ${\sigma }_{\ast }{O}_C=O_Z$ and no higher H¹ contribution from E: a rational bridge or tail has $H^1(O)=0$, and its constant sections impose exactly the matching value at the identified branches or the one attachment. The same calculation after tensoring by a line bundle on Z gives $H^j(C,L^{4m})=H^j(Z,{\omega }_Z(\mathrm{\Sigma }{p}_i{)}^{4m})$, $j=0,1$.

For m≥1 the right-hand H¹ vanishes and the bundle is very ample by the cycle-2 normalized-component interpolation. Explicitly on a normalized stable component of genus h with v node branches and s total specials, 4(2h−2+s)≥2h+v+1. Subtracting its node-branch divisor and up to two further values/jets leaves degree≥2h−1, and the existing pinned Serre-duality theorem gives $H^1=0$. Branch values and at most two additional jets therefore interpolate; the normalization sequence gives $H^1=0$ and separation of every length-two subscheme, including tangent directions at a node. The same bound holds for every higher m. Hence $M^m$ is globally generated on the original fibre, $H^1(C_t,M_t^m)=0$, and $h^0(C_t,M_t^m)=4m(2g-2+n)+1-g$. If there is no zero component, use $Z=C_t$ directly.

For each fixed m≥1 take a local nonnegative finite-free complex K computing $R{\pi }_{\ast }{M}^m$ and every derived base change, as supplied by the actual proper-flat-finite-presentation cohomology theorem. On every geometric fibre the degree-one cohomology is zero by step 50.1. The finite-free base-change criterion at degree one therefore gives base change near the point and $H^1=0$ by Nakayama. The criterion's adjacent-degree assertion then makes the degree-zero comparison surjective; applying it at degree zero gives $E_m={\pi }_{\ast }{M}^m$ finite locally free, with every base change and with the rank in step 50.1. These constructions apply separately to every m, so there is no claim that infinitely many open shrinkings have a common open neighbourhood. The product maps $E_a\otimes E_b\to E_{a+b}$ are the canonical multiplication maps of sections; their base-change comparisons commute with multiplication by naturality of restriction. Thus the entire graded algebra, not just its ranks, commutes with arbitrary base change. The universal structure-sheaf calculation of step 13.1 gives $E_0=O_S$ with the same compatibility. Fibrewise generation and Nakayama make π*E₁→M surjective, defining the proper map v:C→P(E₁), with quotient-line convention.

Work on a Noetherian affine open of S trivializing E₁. The pinned proper-coherent-pushforward theorem makes $F=v_{\ast }{O}_C$ coherent on the relative projective space. Projection formula for the invertible O(m), checked locally by trivializing it, identifies $R_m=H^0(P(E_1),F(m))$

for every m≥0, compatibly with its polynomial coordinate action. Pinned lem-graded-section-module-finite-projective says the high-degree truncation of this section module is finitely generated over Sym(E₁). The finitely many lower terms are finite modules over the Noetherian base. Therefore the whole R is a finite graded module over Sym(E₁), and in particular a finitely presented graded algebra over the Noetherian base. This is the needed finite-generation argument; fibrewise embeddings alone do not supply it.

Consequently $C^{\mathrm{\prime }}=Proj$ R is finite over P(E₁). On a standard coordinate chart, finite homogeneous module generators divided by the corresponding coordinate powers give finite degree-zero module generators. These charts cover Proj R because integrality of R over Sym(E₁) puts every positive homogeneous element in the radical of its coordinate ideal. Thus C' is projective and finitely presented over S, using the existing finite-over-projective theorem.

Each $R_m$ is base-flat. For a homogeneous s of positive degree, the affine ring (R[s⁻¹])₀ is a filtered colimit of base-flat modules $R_{j\cdot \mathrm{deg}s}$, with transitions multiplication by s. It is therefore base-flat. These standard affine charts prove C' flat over S. Existing relative-Proj base change, together with $E_m$ base change in every degree, identifies its geometric fibre with Proj of the full section algebra of $M_t$. The latter is Z above: embed Z by its very ample ${\omega }_Z(\mathrm{\Sigma }{p}_i{)}^4$; its homogeneous coordinate ring agrees in high degrees with the full section ring by Serre vanishing for the embedding ideal. Changing finitely many low degrees does not change Proj. Thus C' has exactly the stable nodal geometric fibres claimed, with the fixed genus.

The canonical map q:C→C' is defined by ratios of the full sections, since the degree-one coordinate sections generate M. The same ratios identify its composition with C'→P(E₁) with v. On geometric fibres it is precisely σ. The images of the retained sections are disjoint and smooth: the bridge has no retained mark, and the tail has only the one mark transferred to its attachment, which was a node and so was not an old smooth mark of its neighbour. Thus C'/S with these sections is a stable n-pointed family. Its relative marked dualizing bundle has positive component degrees; applying the same interpolation and complete-embedding proof to C' proves relative ampleness, as required by the original definition.

Part 10: Universal direct image and base-change compatibility. For a coordinate section s of M, let $C_s$ be its nonvanishing locus. Since C is quasi-compact and separated, $\mathrm{\Gamma }(C_s,O_C)={\mathrm{colim}}_m$ $\mathrm{\Gamma }(C,M^m)$

where the transition is multiplication by s. To verify this identity without an extra contraction theorem, trivialize M on a finite affine cover of C: on each affine chart it is ordinary ring localization. A finite affine Čech equalizer computes H⁰, and filtered colimits commute with its finite products and kernels. The resulting ring is exactly (R[s⁻¹])₀. The corresponding D₊(s) cover C' by finite generation as above. Therefore $q_{\ast }{O}_C=O_C^{\mathrm{\prime }}$. The same calculation applies after every base change, because all graded pieces already commute with that base change. This also proves q is the categorical descent morphism used when a target map is constant on the contracted fibres, as follows: for a separated target and a map constant on each geometric q-fibre, choose an affine neighbourhood of that fibre image; the proper image of its inverse-image complement avoids the chosen base point. On the resulting neighbourhood downstairs $q_{\ast }O=O$ gives the unique ring-map factorization. Uniqueness glues these factors on overlaps.

The construction is canonical under isomorphisms and arbitrary pullback: the dualizing/marking line comparisons induce the graded algebra comparisons and hence the Proj comparisons. For an arbitrary test base T, the stable (n+1)-pointed family is, fppf locally, the pullback of the actual finite-type framed Hilbert family H constructed in parts 1–5 with target a point. This H is locally Noetherian over k, so steps 42–58 apply to its universal family. On the overlap of two such lifts, the isomorphism of the original pointed curves identifies the dualizing lines, every graded piece and its multiplication, and therefore the contracted Proj and q. These identifications satisfy the cocycle identity because they come from functorial pullback of the original line. The resulting contracted families have their canonical relatively very ample fourth marked-dualizing line. Their section bundles and embedded homogeneous ideals descend by the explicit faithfully flat descent argument of step 30.1, so the descent is a projective scheme family, not merely an algebraic-space quotient. The old sections and q descend as maps. After any further base change the graded comparisons identify this descended construction with the pullback. The equality $q_{\ast }O=O$ is universal by the coordinate-chart computation in steps 57–58. This proves the all-test forgetful morphism, not only a fibrewise stabilization rule.

Part 18: Regular surface factorization over the testing traits. Let f:X→Y be a proper birational morphism of integral regular surfaces projective over a complete equicharacteristic DVR with algebraically closed residue field. The complete base and every finite-type model and localization satisfy the normalization, normal-completion and domination hypotheses of [F4]. The morphism is an isomorphism outside finitely many closed points of Y: over a height-one point properness extends the generic inverse from its DVR, and the normal codimension-one modification argument supplies an open isomorphism locus there. Its contracted curves form a finite set. If there are none, it is proper quasi-finite and hence finite; finite birationality over normal Y makes it an isomorphism. Induct on their number N.

Choose a closed point y where f is not an isomorphism, put $A=O_{Y,y}$, and $B=B{l}_m$ Spec A. The regular-point chart calculation in the internal construction beginning at part 25 proves that B is regular, with exceptional $E^{\mathrm{\prime }}=P_k^1$ and conormal O(1). Normalize the graph of the rational lift $X_A⇢B$. The local domination supplier in [F4] applies to A, since A is essentially finite type over the complete equicharacteristic R. Its finite normalized point sequence dominates this graph; the normalization finiteness used in it has the same scope. Since the starting source is regular its point blowups stay regular, so their normalizations are identities. Since $X_A$ is regular, domination gives a finite roof $Z=X_n\to \dots \to X_0=X_A$ consisting of ordinary regular point blowups and a morphism g:Z→B. Every centre is closed over y and has residue k. Choose such a roof with minimal length.

Suppose its length is positive and let E be its last exceptional P¹. If g maps E to a point, it descends through the last blowup: off E the blowup is an isomorphism; near E choose an affine neighbourhood of its one image and shrink downstairs by properness so the entire inverse image maps into that affine. The identity $q_{\ast }O=O$ for the regular-point blowup factors its ring map. The descended maps glue uniquely, contradicting minimality. Thus g maps E onto E'. The map g is proper and birational. It is an isomorphism at the generic DVR of E', so E→E' is finite birational, hence an isomorphism. Their conormals are both $O_P^1(1)$. The induced conormal homomorphism is generically nonzero, hence multiplication by a nonzero k-scalar and invertible everywhere. Consequently the local pullback of E' has precisely E as its Cartier divisor near E. A positive-dimensional g-fibre through a point of E would lie there in E, impossible since g|E is an isomorphism. Thus g is quasi-finite at every point of E.

Apply the actual characteristic-free lem-proper-birational-normal-target-isomorphism-at-quasi-finite-point: g is an isomorphism over neighbourhoods covering E', so $g^{-1}(E^{\mathrm{\prime }})=E$. This is the full fibre of Z→Spec A over y. But $Z\to X_A$ maps E to a single point, and is surjective. Hence $f^{-1}(y)$ is a singleton and f is quasi-finite there. The same actual lemma makes f an isomorphism near y, a contradiction. Therefore the minimal roof had length zero: $X_A\to B$ exists.

The ideal $f\ast m_y$ is consequently invertible over the whole fibre over y. Elsewhere it is the unit ideal. Its noninvertibility locus is closed (the rank-one locally free locus of a finitely presented module is open), so it is empty. The blowup universal property factors f through $Y_1=B{l}_{y}Y$. This new target is regular by the same characteristic-free point charts; the new morphism f₁:X→Y₁ is proper birational. Exactly one contracted curve of f dominates the exceptional curve of Y₁, since f₁ is an isomorphism at its generic DVR. Every other contracted curve remains contracted. Thus $N(f_1)=N(f)-1$. Induction terminates and proves that f is a finite composition of ordinary closed-point blowups and an isomorphism.

This minimal-roof argument uses the exact complete-equicharacteristic suppliers and the explicit regular-point charts, without extending a finite-type-field Statement to an arithmetic base. No Jacobian-divisor or relative log-MMP assertion is substituted. It applies to the regular roofs used for specified-isomorphism extension over complete traits, and hence to the dual-graph stabilization argument.

Part 19: The explicit nodal-chain resolution used in uniqueness. A nonpersistent split node over the complete trait has étale-local equation $xy={\pi }^a$ after absorbing a base unit, a≥1. The $a=1$ total space is already regular. For a≥2 blow up its closed singular point, an operation intrinsically defined before choosing the étale chart. On the x-chart write $y=xv$ and $\pi =xw$. Cancellation in the common function field gives $v=x^{a-2}{w}^a$, so this chart is R[x,w]/(π−xw), regular with reduced nodal special fibre. The y-chart is the identical reversed calculation. On the π-chart write $x=\pi u$, $y=\pi v$; its equation is $uv={\pi }^{a-2}$. For $a=2$ this is $uv=1$ and is smooth, for $a=3$ the total space is regular, and otherwise it is the same problem with exponent smaller by two. The chart identifications have no hidden torsion: for example R[x,w]/(π−xw) has no x-torsion by reduction modulo x and cancellation, and after inverting x it embeds into the common function field. The analogous argument applies to the π-chart. The $uv-{\pi }^b$ hypersurface is normal: it is CM and its only possible nonregular point is the two-dimensional closed point $u=v=\pi =0$, so its height-one local rings are regular.

These calculations commute with étale base change, and regularity, normality and the asserted node charts descend étale-locally. Thus the actual sequence of ordinary blowups at its finitely many singular closed points is algebraic on the original curve model. The exponent drops by two along every possible singular continuation, so it terminates. For $a=2$ the exceptional tangent conic $XY=T^2$ is a smooth P¹, also in characteristic two. For $a>2$ the tangent conic $XY=0$ consists of two P¹'s meeting at the one π-chart point; resolving that point inserts the recursively obtained chain. Induction gives a chain of a−1 smooth rational curves, with its two ends attached to the two original local branches. Each special multiplicity is one, as seen from $\pi =xw$ on the end charts and from the recursive π-chart. The reduced divisor remains normal crossings. There are no other attachments. The original markings avoid this chain, since they lie in the relative smooth locus, and a given target map is constant on it because it maps to the original node.

This supplies the explicit normal nodal resolution used before applying part 18 factorization. It handles self-nodes by keeping their two branch flags separate. Persistent generic nodes are first normalized and later pinched along their paired smooth sections; no normal-total-space argument is applied across $xy=0$.

Part 20: Exact complete-trait inputs. Let R be a complete equicharacteristic DVR with algebraically closed residue field k, and C/K a smooth projective geometrically connected curve over its fraction field. Every testing DVR of the fixed-field stable-map problem has this characteristic after the dominating-trait construction. The completed local rings of finite-type R-models remain equicharacteristic.

The normalization supplier in [F4] makes every finite-type integral R-model and its localizations have finite normalization. The open-regular-locus supplier makes its singular locus closed. The regular-fibre supplier makes the completion of each normal local ring normal, hence a domain. Finally, the complete-equicharacteristic resolution supplier gives a finite normalized closed-point blowup sequence resolving each complete normal local ring of dimension two. These are exact existing local mathematical inputs, each at its own hypotheses, rather than an appeal to stable reduction.

The following reconstruction turns these complete local sequences into a regular projective model of C. At no stage is an arbitrary formal modification declared algebraic. Only closed-fibre point ideals are transported, and normalization is compared on the actual affine chart algebras.

Part 21: Normalization commutes with the relevant completion. Let (A,m) be one of these normal closed local surface rings and let T→Spec A be an integral modification. On an affine chart write B⊂Frac A, and let $B^{\nu }$ be its finite normalization in Frac A. Finiteness is one of the exact inputs [F4], also after localization. Set $D=B{\otimes }_A$ Â, $D^{\mathrm{\prime }}=B^{\nu }{\otimes }_A$ Â.

Faithful flatness injects both algebras into (Frac A)⊗_A Â, which is the localization of the domain  at the nonzero elements of A. Thus D and D' are domains. They contain  and lie in Frac Â, so both have fraction field Frac Â.

The map $B^{\nu }\to D^{\mathrm{\prime }}$ is flat. Its fibre at a prime q of $B^{\nu }$ is the formal fibre of A→Â at the contracted prime, extended to κ(q). The formal fibres in the normalization-completion supplier [F4] are geometrically regular, so these fibres are regular. Consequently the unrestricted flat-regular-fibres normality-ascent theorem makes D' normal. It is a finite integral D-algebra in their common fraction field. Every element integral over D is also integral over D', hence belongs to D' by normality. Therefore D' is exactly the normalization of D.

These identifications agree on chart overlaps inside Frac  and glue. Hence $(T^{\nu }){\times }_A$  ≅ (T×_A Â)^ν.

This proves the needed normalization-completion comparison within the exact equicharacteristic scope of [F4]. In particular no arbitrary finite algebra with formal coefficients has been declared algebraic.

Part 22: Descend the finite normalized point sequence, one actual centre at a time. Apply the complete local resolution input to Â. Write its finite sequence as

${\widehat{S}}_N\to {\widehat{S}}_{N-1}\to \dots \to {\widehat{S}}_0=Spec$ Â, where each arrow is the normalization of the blowup at a closed point over mÂ, and Ŝ_N is regular. We construct normal projective A-models $S_i$ with ${\widehat{S}}_i\cong S_i{\times }_A$ Â.

Start with $S_0=Spec$ A. Suppose $S_i$ is constructed. The closed fibres of $S_i$ and Ŝ_i are canonically identical, since A/m≅Â/mÂ. The next completed centre therefore corresponds to a unique closed point $x_i$ on the closed fibre of $S_i$. This is a closed point of $S_i$, because the closed fibre is closed and proper over k. Its point ideal pulls back exactly, not just up to radical, to the ideal of that completed centre. On a chart B, for its prime p containing mB, (B⊗_A Â)/(p(B⊗_A Â)) ≅ (B/p)⊗_A Â ≅ B/p.

The last ring is the residue field of $x_i$, so the extended ideal is precisely the corresponding maximal ideal. This is also the scheme-theoretic closed-fibre identification.

Blowups commute with the flat base change A→Â. Normalize the algebraic blowup to obtain $S_{i+1}$; its normalization is finite by [F4]. part 21 identifies its completed base change with the normalization of the completed blowup, which is the prescribed Ŝ_{i+1}. This completes the induction. Projectivity is preserved at each step by the blowup and by finite normalization over the projective A-model.

The terminal $S_N$ is regular. Indeed $S_N{\times }_A$ Â=Ŝ_N is regular, and this base change is faithfully flat and surjective. The actual regularity-descent lemma for a Noetherian local base and its completion gives regularity at every point of $S_N$. Equivalently choose a point upstairs over any given point and apply the local flat descent of regularity. No open formal regularity condition is merely assumed to algebraize.

We have thus produced a projective regular resolution of Spec A by a finite sequence of algebraic normalized point blowups. The finite completed sequence was used only to choose successive closed-fibre centres. All intervening schemes were reconstructed by blowup and finite normalization over A, and exact base-change comparison verified each reconstruction.

Part 23: A projective normal model with finitely many singular points. Choose a projective embedding $C\subset P_K^n$ and take its scheme-theoretic closure X₀ inside $P_R^n$ with its integral structure: on homogeneous affine charts use the kernel of the map to the generic coordinate domain. Then X₀ is integral, projective and finitely presented over R, with generic fibre C. Multiplication by every nonzero element of the DVR is injective on its coordinate rings, so X₀ is R-flat. Flat projective Hilbert-polynomial constancy gives one-dimensional fibres.

Normalize X₀. The finite-normalization input [F4] gives a finite normalization X₁→X₀. It is projective over R and integral, hence R-flat by torsion-freeness. Its generic fibre is still C, because C is already smooth and normal. It is a normal surface. Its regular locus is open by the open-regular-locus input [F4]. Normal surface local rings of height at most one are regular; consequently its singular locus is a closed subset of codimension at least two. It is disjoint from the generic fibre, and therefore consists of finitely many closed points $x_1,\dots ,x_t$ of the special fibre. Their residue fields are k because the fibre is finite type over the algebraically closed field k.

This step uses normality of the actual algebraic model and the proved open-locus assertion. It does not infer that a neighbourhood is normal just because a spread multiplication table is finite.

Part 24: Realize the local finite point sequences on the global model. Fix one singular point x of the current normal projective R-model X₁, and put $A=O_{X_1,x}$. part 22 supplies a finite normalized point sequence over Spec A terminating regularly. Realize the same sequence globally.

At an intermediate global model $T_i$, its base change to Spec A is the corresponding local $S_i$, by induction. A closed centre of $S_i$ lies over the original closed point x. Its residue field is a finite extension of $\kappa (x)=k$, hence equals k. It corresponds to a point of the special fibre of $T_i$ with that residue field, and is therefore a global closed point. Blow up that actual global point and normalize. The normalization is finite by the finite-normalization supplier [F4], so the result remains projective and normal over R.

The restriction of this operation to Spec A is exactly the desired local normalized blowup. Blowup commutes with flat localization. Normalization commutes with localization: an element integral after localization can be multiplied by one denominator to become integral before localization, and the reverse inclusion is immediate. Thus there is no approximation of a completed centre or finite algebra in this global realization.

All these centres lie over the one original point x, so the global modification is an isomorphism over X₁{x}. Its terminal scheme is regular at every point over x, because those local rings are the same as the local rings on the regular local-base terminal resolution. This equality is ordinary localization at base elements already units at those points. Hence the modification resolves x and leaves all other original singular points unchanged; it creates no singular points elsewhere.

Repeat for the finitely many original singular points. The resulting model X is regular everywhere. It is projective and integral over R and has generic fibre C. It is flat because its coordinate modules are torsion-free over the DVR. Its special fibre is a Cartier curve: on a regular surface the nonzero uniformizer cuts a pure one-dimensional fibre. No zero-dimensional fibre component appears; such a component would give a nonempty open quasi-finite locus over R, whose intersection with the dense generic curve would contradict its pure dimension one. Thus X is the required regular projective flat curve model.

This construction is finite: there are finitely many original singular points and the complete-equicharacteristic resolution sequence [F4] at each point is finite. It does not run an unproved global termination algorithm on an unbounded list of newly appearing singularities.

The special fibre is connected, if this is included in the curve-model convention. First $\mathrm{\Gamma }(X,O_X)=R$: proper coherent pushforward makes it finite over R; flat localization embeds it in $H^0(C,O_C)=K$, and integrally closedness of R forces equality. We can prove the needed closed-fibre connectedness without assuming degree-zero cohomology base change. The existing nonnegative perfect-cohomology complex for $O_X$ is a finite-free R-complex K. Put its first differential into Smith normal form over the DVR. For a nonzero diagonal entry $t^a$, the inverse limit of its kernels on $R\mathrm{/}{t}^n$ is zero: a compatible element at level n is the reduction of an element divisible by $t^{N-a}$ at every larger level N. Zero diagonal entries contribute lim $R\mathrm{/}{t}^n=R$. Consequently the canonical map $\mathrm{\Gamma }(X,O_X)$ → ${\lim }_n$ Γ(X×_R $R\mathrm{/}{t}^n,O$)

is an isomorphism, by this explicit degree-zero calculation and the natural cohomology comparisons. If the special fibre were disconnected, its nontrivial idempotent would lift uniquely through every nilpotent thickening, yielding a nontrivial idempotent in that limit and hence in R, a contradiction. Idempotent lifting uses only that 2e−1 is a unit, since its square is 1 modulo the nilpotent ideal; it does not divide by 2. As k is algebraically closed, connectedness here is geometric connectedness. This argument allows nonreduced special fibres and does not assert that $\mathrm{\Gamma }(X,O_X)\otimes k$ equals $\mathrm{\Gamma }(X_k,O)$.

Part 25: Setup. Let R be a complete Noetherian DVR, k its algebraically closed residue field, and π:X→Spec R an integral regular projective flat surface with one-dimensional fibres. Let E⊂X be a vertical exceptional curve of the first kind: E is an effective Cartier divisor, E≅P¹_k and $O_X(E)\mathrm{\mid }E\cong O_P^1(-1)$.

Then there is a projective flat regular integral R-surface Y and a proper R-morphism q:X→Y such that q is an isomorphism away from E, q(E) is a k-rational closed point y, $q_{\ast }{O}_X=O_Y$, and q is the ordinary blowup of the regular two-dimensional point y. The generic curve is unchanged. Starting with any such regular projective model, successively contracting vertical exceptional curves of the first kind terminates and yields a relatively minimal regular projective model. This conclusion assumes an initial regular model; it does not construct one or prove stable reduction.

Part 26: The elementary twist-vanishing argument. Choose an ample line bundle H on X, replacing it by a sufficiently high power so that it defines a closed embedding $X\to P_R^n$, its degree h on E is at least two, and $H^i(X,H^m)=0$ for every $i>0$ and every m≥1.

This is the actual Noetherian projective Serre-vanishing theorem, applied after one fixed sufficiently large power. Since R is affine, the chosen very ample bundle is ample on X itself. Put $L=H(hE)$. Its restriction to E has degree zero, hence is trivial. For every other special component C, O(E)|C has a nonzero canonical section and therefore nonnegative degree; consequently L has positive degree on C. The generic restriction of L is H. Here is the needed vanishing, without a formal-functions theorem or a filtration of a nonreduced special fibre. If $B\mathrm{\mid }E\cong O_E(d)$, d≥0, then $0\to B((j-1)E)\to B(jE)\to O_E(d-j)\to 0$.

For 1≤j≤d+1 the last sheaf has zero H¹, and all its higher cohomology vanishes. Thus, if B has no positive-degree cohomology, the same holds for B(jE) throughout that range, by induction and the long exact sequence. Apply this with $B=H^m$ and $d=mh$. Since h≥2, it gives $H^i(X,L^m(-aE))=0$ for all $i>0$, m≥1, and $a=0,1,2$.

In particular $H^1(X,L(-E))=0$. A nowhere-zero section of L|E lifts to a section s∞ of L. The embedding sections $t_0,\dots ,t_n$ of H map to sections $s_0,\dots ,s_n$ of L by H⊂H(hE). They vanish along E and still generate away from E. Together with s∞ they generate L everywhere. They define a morphism

$\phi :X\to P_R^{n+1}$.

It maps E to $z=(0:\dots :0:1)$. No point outside E maps to z. On the complement of z in the target, projection to the first n+1 coordinates composes with φ|X\E to the original closed embedding defined by H. Hence φ is quasi-finite on X\E, and indeed its restriction there is a locally closed immersion.

For reference, positive-degree cohomology/base change is also available: the exact sequence $0\to L^m$ --t→ $L^m\to L^m\mathrm{\mid }F\to 0$ and the vanishing of H¹ and H² above give $H^1(F,L^m\mathrm{\mid }F)=0$ for the possibly nonreduced special fibre F. Localization gives generic $H^1=0$. The already proved finite-free cohomology criterion therefore gives universal base change for ${\pi }_{\ast }{L}^m$, m≥1. No degree-zero structure-sheaf base-change assertion for the nonreduced F is being assumed.

Part 27: A projective contraction from the full section algebra. Let $T={\oplus }_{m\ge 0}$ $\mathrm{\Gamma }(X,L^m)$, $Y=Pro{j}_R$ T.

Each $T_m$ is a finite R-module by proper coherent pushforward and is torsion-free, since X is R-flat and L is invertible. Thus each is R-flat. In degree zero this does not assert that T₀ commutes with residue-field base change.

Let W be the finite free R-module with the selected generators $s_0,\dots ,s_n,s\mathrm{\infty }$ as its labelled basis; evaluation $W\otimes O_X\to L$ is surjective. For $F={\phi }_{\ast }{O}_X$, proper coherent pushforward makes F coherent on P(W). The projection formula identifies $T_m=\mathrm{\Gamma }(P(W),F(m))$

compatibly with its Sym(W)-module structure. The actual pinned high-degree section-module finiteness theorem applies over the Noetherian ring R. Its high-degree tail is finite over Sym(W); the finitely many remaining $T_m$ are finite R-modules. Hence T is a finite graded module algebra over Sym(W).

Every positive homogeneous element of T is integral over this coordinate algebra. Its homogeneous monic equation shows that it lies in the radical of the ideal generated by the coordinate sections. Thus the degree-one coordinate charts cover Proj T. Localizing finite homogeneous module generators by the corresponding coordinate powers shows that Y→P(W) is finite. Therefore Y is projective and finitely presented over R, and $A=O_Y(1)$, pulled back from $O_P(W)(1)$, is invertible. The degree-one chart description gives a canonical morphism q:X→Y with $q^{\ast }A=L$ and with composite φ.

On an affine chart D₊(s), its inverse image is $X_s$. Since X is quasi-compact and separated, a finite affine trivializing cover of L and its finite Čech equalizer give $\mathrm{\Gamma }(X_s,O_X)={\mathrm{colim}}_m$ $\mathrm{\Gamma }(X,L^m)=(T[s^{-1}]{)}_0$,

where the transition maps multiply by s and the equality is the usual ratio identification. On each affine of that cover it is ordinary ring localization; filtered colimits commute with the finite products and kernels. These charts prove $q_{\ast }{O}_X=O_Y$.

The map q is proper: X is proper over R and Y is separated over R. The pushforward identity implies surjectivity, because a nonempty open disjoint from its proper image would have zero structure sheaf. The section ring embeds, after a rational trivialization of L, into K(X)[u]; thus Y is integral. Its standard chart rings are filtered colimits of the R-flat modules $T_{jm}$, so Y is R-flat. Equivalently their torsion-freeness over a DVR suffices.

The image q(E) is a single point y: it lies in the finite fibre over z of Y→P(W), and E is connected. Since y is a closed special point and k is algebraically closed, $\kappa (y)=k$. No point outside E maps to y, because φ separates E from its complement. Hence $q^{-1}(y)=E$ set-theoretically. Off y, q is proper and quasi-finite; it is finite there, and $q_{\ast }{O}_X=O_Y$ makes that finite map an isomorphism. In particular the generic curve is unchanged and q is birational. All these statements precede, and do not assume, regularity of Y at y.

Part 28: Two descended functions generate the exceptional ideal. The first-step vanishings give a surjection $\mathrm{\Gamma }(X,L(-E))\to \mathrm{\Gamma }(E,L(-E)\mathrm{\mid }E)=H^0(P_k^1,O(1))$, because $H^1(X,L(-2E))=0$. Choose h₁,h₂ mapping to a basis of that two-dimensional space, and set $h_0=s\mathrm{\infty }$, which is nonzero everywhere on E. On D₊(h₀) the ratios $f=h_1\mathrm{/}{h}_0$, $g=h_2\mathrm{/}{h}_0$

are functions on Y, whose pullbacks vanish on E. Their images generate $I_E\mathrm{/}{I}_E^2=O_E(1)$ at every point. Nakayama makes them generate $I_E$ on a neighbourhood of E. Properness of q allows an affine neighbourhood V of y inside D₊(h₀) to be shrunk so that on all $U=q^{-1}(V)$, $I_E=(f,g)$. Indeed the support of $I_E\mathrm{/}(f,g)$ is closed and disjoint from E, so its proper image avoids y. The bundle A is trivial on V, and $q^{\ast }A=L$. Therefore on U, $O_U(-hE)\cong H\mathrm{\mid }U$.

The right side is ample; restriction of an ample bundle to a quasi-compact open remains ample. The positive-power ampleness criterion gives that $O_U(-E)$ is ample. The morphism U→V is projective: the graph of q is closed in X×_R Y, and projection to Y is projective. Since V is affine Noetherian, the actual projective Serre-vanishing theorem gives $H^1(U,O_U(-nE))=0$ for n≫0.

The sequences $0\to O_U(-(n+1)E)\to O_U(-nE)\to O_E(n)\to 0$

and $H^1(P_k^1,O(n))=0$ for n≥−1 then descend this vanishing to every n≥−1. In particular $H^1(U,O_U(E))=0$. This argument does not use formal functions, completion of the target, or an already constructed contraction theorem.

Because $I_E$ is invertible and is generated by f,g, its ordinary two-generator syzygy sequence is $0\to O_U(E)\to O_U^2$ $--(f,g)\to O_U(-E)\to 0$.

Write $B=\mathrm{\Gamma }(V,O_Y)=\mathrm{\Gamma }(U,O_X)$, using $q_{\ast }{O}_X=O_Y$. Restriction to E identifies $\mathrm{\Gamma }(U,I_E)$ with the kernel of B→k, which is the maximal ideal $m_y$ of the point y in B. Taking global sections of the syzygy and using $H^1(U,O(E))=0$ yields a surjection $B^2\to m_y$. Hence $m_y=(f,g)$.

We verify the local dimension rather than infer it from the number of generators. Write the whole special divisor as $F=aE+\mathrm{\Sigma }{b}_j{C}_j$. Regular local rings are factorial, so these component divisors are Cartier. Restricting the principal line bundle $O_X(F)$ to E gives $0=-a+\mathrm{\Sigma }{b}_j$ ${\mathrm{deg}}_E$ $O_X(C_j)$.

For $C_j\mathrm{\ne }E$ each degree is nonnegative and is positive exactly when $C_j$ meets E. Since $a>0$, at least one other component $C_j$ meets E. Its image under q is a curve through y, because q is an isomorphism off E. In the local domain $O_Y,y$ the prime of this curve contains the nonzero uniformizer and is strictly smaller than $m_y$. Thus there is a chain $(0)⊊p⊊m_y$, and the dimension is at least two. The two generators of $m_y$ give dimension at most two. Consequently $O_Y,y$ is regular of dimension two. Away from y, Y is isomorphic to the regular X\E. Thus Y is regular everywhere. Its special fibre remains a pure one-dimensional Cartier fibre, since the uniformizer is a nonzerodivisor in these regular surface local rings.

Part 29: Identification with an ordinary point blowup. The equality $m_y$ $O_X=I_E$ holds scheme-theoretically. Its invertibility gives a factorization X --b→ $B{l}_y$ Y --β→Y

by the actual blowup universal property. For clarity the regular-point blowup calculation needed here is valid for the regular local ring at a closed point of this R-model, without requiring finite type over a field. If $Q=O_Y,y$ and $m=(f,g)$ is its regular parameter ideal, f,g is a regular sequence: neither lies in m²; regular-local factoriality makes the irreducible f prime, and g is not in (f). The two charts are

Q[v]/(fv−g), Q[w]/(gw−f).

For example the first quotient has no f-torsion: reduce $fP=(fv-g)T$ modulo f, use regularity of g modulo f to conclude T is divisible by f, then cancel f. It embeds into Q[g/f]. The exceptional chart is k[v], with conormal generator f; the analogous second chart glues by $v=w^{-1}$, giving P¹_k and conormal O(1). At a closed point of the first exceptional chart the local maximal ideal is generated by f and v−c for a lift c of its k-coordinate. Its dimension is two because (f) is prime with curve quotient k[v]. At the generic exceptional point its local ring is one-dimensional with principal maximal ideal. These charts, together with the unchanged complement, prove that $B{l}_{y}Y$ is regular and normal, by direct calculation in its regular parameters.

On E the morphism b is the map to this P¹ defined by the basis f,g of its conormal $O_E(1)$, so it is an isomorphism. Off E it is already an isomorphism. It is proper, and all its geometric fibres are finite; hence it is finite. It is birational with normal target, so the finite-birational-normal isomorphism theorem makes b an isomorphism. Therefore q is precisely the ordinary blowup of y, and its scheme-theoretic exceptional fibre is E. This proves the contraction assertion.

Part 30: Termination from one given regular model. The special fibre has finitely many irreducible components. Each contraction above removes exactly E and is an isomorphism on the complement of its image point. Every other special component has a distinct one-dimensional image, with unchanged generic point. Thus the number of special components decreases by one. Flatness and projectivity of the new regular model have been proved, so the construction can be repeated whenever another vertical exceptional curve of the first kind remains. A strictly decreasing positive integer cannot continue indefinitely. The output is a regular projective flat model with no vertical exceptional curve of the first kind, that is, a relatively minimal regular model.

This proves existence of a relatively minimal model given a regular projective model. It proves neither existence of that initial model, uniqueness/universal minimality, vertical adjunction, the Picard torsion specialization theorem, nor stable reduction. Those are distinct author components. In genus at least two, the usual deduction of vertical canonical nefness from absence of (-1)-curves additionally uses the actual adjunction and connected-fibre data; no such data are silently manufactured here.

Part 31: Numerical input and geometric scope. Let $A=(a_{ij})$ be a symmetric integral n×n matrix, with $a_{ij}\ge 0$ for i≠j. Let $m_i$ be positive integers with $Am=0$, and assume the graph with one edge for each pair $i<j$ with $a_{ij}>0$ is connected. Let $p_i$ be nonnegative integers and put $k_i=2p_i-2-a_{ii}\ge 0$, ${\mathrm{\Sigma }}_i$ $m_i$ $k_i=M:=2g-2>0$.

For a proper regular minimal model over a DVR with algebraically closed residue field, these are exactly the required inputs: $F=\mathrm{\Sigma }{m}_i$ $C_i$ is the whole special fibre, $a_{ij}=C_i\cdot C_j$, $p_i=p_a(C_i)$, $k_i=K\cdot C_i$ and $K\cdot F=2g-2$. Here the $p_i$ are arithmetic genera; no component or intersection is assumed nodal. Integral proper components over an algebraically closed field have $H^0(O_{Ci})=k$, so the source numerical-type factors $w_i$ are all1. Existence of the model, connectedness, minimality/vertical K-nefness, adjunction and Euler-characteristic specialization are geometric inputs owned separately. This document does not derive model existence from the numerical theorem.

Part 32: Weighted negative form and proper subgraphs. For $x\in R^n$, the identity $Am=0$ gives $x^{t}Ax=-{\mathrm{\Sigma }}_{i<j}{a}_{ij}$ $m_i$ $m_j(x_i\mathrm{/}{m}_i-x_j\mathrm{/}{m}_j{)}^2$.

Indeed $a_{ii}$ $m_i=-{\mathrm{\Sigma }}_{j\mathrm{\ne }i}{a}_{ij}$ $m_j$; substitute this into the diagonal terms and group the two terms for each unordered pair. The expression is nonpositive, and vanishes precisely when $x_i\mathrm{/}{m}_i$ is constant along every edge, hence everywhere by connectivity. Thus ${\ker }_R$ $A=Rm$ and $ran{k}_Q$ $A=n-1$. Every proper principal submatrix is negative definite: extend its vector by zero outside the proper vertex set; equality in the displayed formula would make that extension a multiple of the everywhere-positive m, hence zero.

If $n>1$, connectivity and $Am=0$ give $a_{ii}<0$ for every i. Let $J=i:k_i>0$. It is nonempty because $\mathrm{\Sigma }{m}_i$ $k_i=M$. For i∉J, adjunction gives $a_{ii}=2p_i-2<0$, whence $p_i=0$ and $a_{ii}=-2$. In the geometric setting arithmetic genus0 implies that such a component is smooth rational: its normalization genus plus its nonnegative normalization defect is0. This does not make the other components smooth.

Part 33: Complete elementary ADE classification needed here. Let I be a connected component of the subgraph on Jᶜ. It is a proper vertex subset. The positive-definite matrix $-A_I$ has diagonal2. First every off-diagonal entry $a_{ij}$ within I is0 or1: if $a_{ij}\ge 2$, its two-vertex principal determinant $4-a_{ij}^2$ is nonpositive. Thus I is a simple graph. It contains no cycle: on the vertices of any cycle the all-ones vector has quadratic value at most0 for 2I−adjacency. It has no degree≥4 vertex: the principal four-leaf star has quadratic value0 on the vector taking2 at its centre and1 at its leaves. It has at most one degree3 vertex. Otherwise take a shortest path between two degree3 vertices and one edge in each of the other two directions at both ends. This two-fork principal tree has quadratic value0 on the vector2 along the joining path and1 at the four leaves. Each contradicts positive definiteness.

A graph with no degree3 vertex is a path $A_r$. If there is one degree3 vertex, let its arm lengths be 1≤a≤b≤c. The path Cartan matrix of length a is positive definite: its quadratic form is the sum of squared adjacent differences plus the two endpoint squares. Solving its linear recurrence with a unit source at the attaching endpoint gives inverse endpoint entry a/(a+1). Eliminating the three arms by a Schur complement therefore shows that positive definiteness is equivalent to

$2-a\mathrm{/}(a+1)-b\mathrm{/}(b+1)-c\mathrm{/}(c+1)>0$,

or $1\mathrm{/}(a+1)+1\mathrm{/}(b+1)+1\mathrm{/}(c+1)>1$. If a≥2 the sum is≤1, so $a=1$. If $b=1$, c is arbitrary, giving $D_{c+3}$ with arms(1,1,c). If b≥3 the sum is≤1, so the remaining case has $b=2$. Then $c=2,3,4$, giving E6,E7,E8. These exhaust the possibilities. In particular the exceptional E-type graph has at most8 vertices. No ADE-classification theorem is being imported as a premise.

Part 34: Uniform multiplicity and intersection bounds. For j∈J, integrality $k_j\ge 1$ gives $m_j\le M$. Put $q_j=-m_j$ $a_{jj}$. If $p_j\ge 1$ then $q_j=m_j(k_j+2-2p_j)\le m_j$ $k_j\le M$. If $p_j=0$ then $q_j=m_j(k_j+2)\le M+2M=3M$. Thus $q_j\le 3M$ for every positive-K vertex. Also, for every distinct i,j, the fibre equation at j gives $m_i$ $a_{ij}\le {\mathrm{\Sigma }}_{t\mathrm{\ne }j}{m}_t$ $a_{tj}=-m_j$ $a_{jj}=q_j$.

Consequently any zero-K vertex i adjacent to J has $m_i\le 3M$. Call such vertices boundary vertices of their zero-K connected component I. Every I has at least one: the whole graph is connected and J is nonempty. Set $P=\max \{m_i:i\in I\text{ meets }J\}\le 3M$. On I the equations are $2m_i={\sum }_{j\in I\text{ adjacent to }i}{m}_j+b_i$, $b_i={\mathrm{\Sigma }}_{j\in J}{a}_{ij}$ $m_j\ge 0$,

and $b_i>0$ exactly at those boundary vertices.

For $A_r$, ${\max }_I$ $m_i\le P$. If a maximum were larger than P, it would be at a vertex with $b_i=0$. An endpoint cannot be a maximum since its sole neighbour would have twice its value. At an interior maximum both neighbours must have that same value. Propagate to an endpoint or a boundary vertex, contradicting the preceding statements.

For $D_r$, let v be its branch vertex and u,w the two short leaves. If v is boundary, $m_v\le P$. If u or w is boundary, $2m_u=m_v+b_u$ or $2m_w=m_v+b_w$ gives $m_v\le 2P$. If none of v,u,w is boundary, the two leaf values are $m_v\mathrm{/}2$ and the branch equation makes the first long-arm value equal $m_v$. Along that arm the consecutive differences are nonincreasing, since $2m_i=m_{i-1}+m_{i+1}+b_i$. Starting with difference0, the first vertex with $b_i>0$ has value $m_v$ and is boundary, so $m_v\le P$. Such a first vertex must exist, since otherwise the values stay constant through the endpoint, contradicting its equation $2m_{\mathrm{e}\mathrm{n}\mathrm{d}}=m_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{v}\mathrm{i}\mathrm{o}\mathrm{u}\mathrm{s}}$. Finally the maximum on the long arm is at its branch end or a boundary vertex by the same maximum argument as for $A_r$. Thus ${\max }_I$ $m_i\le 2P$ in all D cases.

For E6,E7,E8, choose any boundary vertex. Across an internal edge, the equation at one endpoint gives $m_{\mathrm{n}\mathrm{e}\mathrm{x}\mathrm{t}}\le 2m_{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}}$. A path to any vertex has at most7 edges, so ${\max }_I$ $m_i\le 2^7$ P. Combining all cases gives $m_i\le 384M$ for every i, $q_i=-m_i$ $a_{ii}\le 768M$ for every i. Positive-K vertices and the $n=1$ case satisfy these bounds as well: for $n=1$, $A=0$ and $m_1{k}_1=M$ with $k_1>0$, so $m_1\le M$. The previous fibre inequality now gives $m_i\mathrm{\mid }{a}_{ij}\mathrm{\mid }\le 768M=1536(g-1)$ for every i,j.