Proof. Choose coordinates in which \(Q(x,y)=xy\), so that \((x,y)\mathbin{\cdot}(x',y')=(xy'+yx')/2\). Set \[h=\sqrt{90}>0,\quad R=\sqrt{1+h^2/4}>0,\quad b=\frac{2h}{R},\quad d(z)=\left(Rz-\frac h2,\frac Rz-\frac h2\right).\] Direct expansion gives \[d(z)^2+d(z)\mathbin{\cdot}(h,h)=R^2-h^2/4=1.\] For a sufficiently small general nonzero complex number \(\lambda\), let \(z_1,z_2,z_3\) be the roots of \[X^3-bX^2+b\lambda X-\lambda=0.\] Vieta’s formulas give \[\sum_{i=1}^3z_i=b,\qquad \sum_{i=1}^3z_i^{-1}=b,\qquad 2\sum_{i=1}^3d(z_i)=(h,h).\] Thus taking each \(d(z_i)\) twice gives (3).

For completeness, these vectors have the required distinct directions. As \(\lambda\to0\), two roots approach zero and the third approaches \(b\). The roots are distinct for general \(\lambda\): the cubic discriminant has nonzero leading term \(-4b^3\lambda\) at zero. Near zero the projective ratio of the two entries of \(d(z)\) is \[\frac{z(Rz-h/2)}{R-hz/2},\] whose derivative at zero is \(-h/(2R)\ne0\). The two small roots therefore give distinct nonisotropic directions. At \(z=b\) the entries are \(3h/2\) and \((R^2-h^2)/(2h)\), both nonzero, so the third limiting direction is different from their common limit. These properties persist for sufficiently small general \(\lambda\). Finally apply simultaneously to all six vectors an isometry \((x,y)\mapsto(\nu x,\nu^{-1}y)\). On nonisotropic projective directions this multiplies the ratio by \(\nu^2\). Only finitely many values of \(\nu\in\mathbb C^*\) are excluded by the prescribed finite set, and all equations in (3) are preserved. ◻

Proof. The restriction kernels on \(\Sigma\) are respectively \[\mathcal A_j-C=(-2-j,-2),\qquad \mathcal A_j-\Gamma=(4-j,0).\] Product cohomology on \(\mathbb P^1\times\mathbb P^1\) gives \(H^0=H^1=0\) for the first kernel and \(H^1=0\) for the second. Adjunction therefore gives the two assertions about restriction.

By (5), the ideals of the two length-\(\ell\) subschemes in the local surface are \[(y-\varphi(z),z^\ell)=(y,z^\ell)\qquad(\ell\le4).\] Thus these are the same embedded subscheme, and restrictions of the surface line bundle and its sections to them agree. The two surjectivity statements just proved imply equality of evaluation ranks.

On the elliptic curve, \(\deg F=2\), \(Z=3F+\Delta\), and \(2Z=6F\). These identities give every residual bundle in (9). A positive-degree line bundle on an elliptic curve has as many sections as its degree; a negative-degree bundle has no section. The degree-zero bundle \(\Delta\) is nontrivial, so both its zeroth and first cohomology vanish. Subtracting the residual section dimensions from \(h^0(\Gamma,N_j)\) gives the stated ranks. In particular, the twelve-dimensional surface system for \(j=3\) has a two-dimensional kernel on restriction to \(\Gamma\), and the resulting ten-dimensional system injects into its length-twelve evaluation space. ◻

Proof. The existence and contact assertions were established above. From \(C=(8,4)\) and \(K_C=(6,2)|_C\), intersection on the quadric gives \[\deg H=4,\qquad \deg K_C=6\cdot4+2\cdot8=40, \qquad g(C)=21.\] Since \(\deg Z=6\), one has \(\deg L=10\). The morphism given by \(H\) is the first projection, restricted to the irreducible curve \(C\), hence is finite of degree four. Tangency at the marked points identifies its differential there with that of \(F|_\Gamma\), which is nonzero. Its marked values are the corresponding \(F|_\Gamma\)-values, already seen not to be all equal. Choose a regular fiber away from these finitely many values. A determinant-one change of the basis \(u\) makes this fiber the zero divisor of \(u_1\). Such a change preserves \(u_1\,du_2-u_2\,du_1\) on both curves, and hence preserves the quadratic form and every normalization already chosen.

For \(0\le j\le3\), the sequence restricting \(\mathcal O_\Sigma(j,0)\) to \(C\) has kernel \(\mathcal O_\Sigma(j-8,-4)\), with \(H^0=H^1=0\) by product cohomology. This proves \(h^0(C,jH)=j+1\). For any line bundle \(B\) on \(C\) and effective divisor \(D\), Riemann–Roch and Serre duality give \[ h^0(C,B+D)-h^0(C,B) =\deg D-\operatorname{rank}\left( H^0(C,K_C-B)\longrightarrow H^0(D,(K_C-B)|_D)\right). \tag{13}\] Apply this with \(B=jH\) and use the ranks in Lemma 3. Its first three rows give \[h^0(C,\mathcal O_C(Z))=1, \quad h^0(C,H+2Z)=2, \quad h^0(C,2H+2Z)=3.\] The inclusions of \(H^0(C,H)\) into the spaces for \(H+Z\) and \(H+2Z\) then give \(h^0(C,L)=2\). The fourth row gives \[h^0(C,3H+2Z)=h^0(C,3H)+12-10=6.\] This proves (10), and equality of the dimensions in the natural multiplication inclusions proves (11). Because the pencil \(u\) has no base point and \(Z\) is reduced, the assertion about the common zeros of \(r\) follows as well.

Finally, the divisor sequence for \(N=(k+1)L\), followed by Serre duality, shows that (12) is surjective if and only if the natural inclusion \[H^0(C,(3-k)L)\longrightarrow H^0(C,(3-k)L+Z)\] is an isomorphism. For \(k=2\) this is the inclusion from \(L\) to \(L+Z=H+2Z\), between two-dimensional spaces. For \(k=3\) it is the inclusion from \(\mathcal O_C\) to \(\mathcal O_C(Z)\), between one-dimensional spaces. For \(k\ge4\) both spaces vanish, since the target bundle has degree \(10(3-k)+6\le-4\). The final assertion follows from Lemma 2 and (4). ◻

Proof. The morphism \(f\) is finite and flat of degree four. The identity \(2\Lambda=K_C+2H\) identifies \(\Lambda^2\) with the relative dualizing bundle. We verify directly the normalization and perfection of its trace pairing. Locally at a ramification point one can choose parameters with \(x=w^e\) on a single component over a target disk. A frame of \(H\) pulled back from that disk can be chosen so that \(\eta_C=dx\,\mathfrak f^2\). If \(\tau^2=a(w)\,dw\,\mathfrak f^2\), with \(a\) a unit, the local pairing is \[(b,c)\longmapsto \mathop{\mathrm{Tr}}\!\left(\frac{a(w)bc}{e w^{e-1}}\right).\] On the basis \(1,w,\ldots,w^{e-1}\), the pairing without the unit \(a\) has matrix \[\mathop{\mathrm{Tr}}\!\left(\frac{w^{i+j}}{e w^{e-1}}\right) =\begin{cases}1&i+j=e-1,\\0&i+j\ne e-1, \end{cases} \qquad 0\le i,j<e.\] It is therefore holomorphic and perfect. Multiplication by \(a\) is an automorphism, and taking the direct sum over the points above the target disk proves the assertion. On an unramified fiber the trace is the sum of its four contributions; there is no division by the degree.

By the splitting theorem for vector bundles on \(\mathbb P^1\) [17], \(E\) is a direct sum of line bundles. Self-duality implies that its splitting degrees occur in opposite pairs. Proposition 4 gives \[h^0(E(-1))=h^0(\theta)=3,\qquad h^0(E(-2))=h^0(H+2Z)=2.\] A positive splitting degree \(a\) contributes \(a\) and \(a-1\), respectively, to these two numbers, and all nonpositive degrees contribute zero. There is consequently exactly one positive degree, equal to three. Thus \[E\simeq\mathcal O(3)\oplus\mathcal O^{\oplus2}\oplus\mathcal O(-3).\] The nonzero map \(\mathcal O(3)\to E\) obtained from multiplication by \(\xi^2\) has zero components in the three lower-degree summands. Its component in \(\mathcal O(3)\) is a nonzero constant. Hence its image \(\mathcal P\) is the entire positive summand, and is a subbundle, including over the images of the points of \(Z\). The injective pullback \(H^0(\mathbb P^1,\mathcal O(3))\to H^0(C,3H)\) is an isomorphism, since both spaces have dimension four. Thus \(H^0(\mathcal P)=\xi^2H^0(C,3H)\).

The line \(\mathcal P\) is isotropic, since \(\mathop{\mathrm{Hom}}(\mathcal O(6),\mathcal O)=0\). Likewise it pairs trivially with the two degree-zero summands. Perfection gives \[\mathcal P^\perp\simeq\mathcal O(3)\oplus\mathcal O^{\oplus2},\qquad \mathcal P^\perp/\mathcal P\simeq\mathcal O^{\oplus2},\] with a nondegenerate constant form on the quotient. All global sections of \(E\) lie in \(\mathcal P^\perp\), and \(H^1(\mathcal P)=0\). Taking sections proves the assertions about \(W\) and its fibers.

At \(q\in Z\), evaluation annihilates \(\mathcal P_{f(q)}\), because the map defining that line is multiplication by \(\xi^2\). The finite fiber algebra is a product of its Artin-local factors, and the trace pairing is the orthogonal sum of their pairings. Since \(f\) is unramified at \(q\), the factor at \(q\) is one-dimensional, with form coefficient \(s(q)/\mathcal E(q)\). Evaluation at \(q\) vanishes on the other factors, whether or not they are ramified. The metric dual of coefficient evaluation thus has square \(\mathcal E(q)/s(q)\) in \(E_{f(q)}\). It lies in \(\mathcal P_{f(q)}^\perp\), because the evaluation kills \(\mathcal P_{f(q)}\). Passing to the orthogonal quotient preserves this square and gives the functional on \(W\). Finally \(\mathcal E(q)\ne0\) by the unramifiedness in Proposition 4. ◻

Proof. Set \(N_\Sigma=\Gamma+F\), \(N_\Gamma=N_\Sigma|_\Gamma\), and \(D=2Z\). Fourth-order contact identifies \(D\) on \(C\) and on \(\Gamma\) as the same embedded length-twelve subscheme of \(\Sigma\). The isomorphism \(N_\Sigma|_C\simeq K_C-3H\) is the one supplied by adjunction with the fixed volume. The differential \(\omega\) trivializes \(K_\Gamma\). For a line bundle \(B\) on a smooth curve, the local pairing between principal parts of \(B(D)\) and jets of \(K-B\) is \[(a,s)\longmapsto\sum_{q\in Z}\mathop{\mathrm{Res}}_q(as).\] It is perfect: in a parameter and mutually dual frames the two coefficients of \(a_{-2}z^{-2}+a_{-1}z^{-1}\) pair with a jet \(s_0+s_1z\) as \(a_{-2}s_1+a_{-1}s_0\). Consequently the two principal part spaces \[H^0\!\left(C,\frac{\mathcal O_C(3H+2Z)}{\mathcal O_C(3H)}\right), \qquad H^0\!\left(\Gamma, \frac{\mathcal O_\Gamma(-N_\Gamma+2Z)}{\mathcal O_\Gamma(-N_\Gamma)}\right)\] are both identified by residues with \(H^0(D,N_\Sigma|_D)^*\). This defines their comparison without any choice of coordinates or frames. It compares the full local principal-part spaces before any condition of global realizability is imposed.

Both complete adjoint systems are restrictions of the same ambient system \(H^0(\Sigma,N_\Sigma)\). Indeed, the kernels in the restriction sequences to \(C\) and to \(\Gamma\) are respectively \[\mathcal O_\Sigma(-\Gamma-3F)=\mathcal O_\Sigma(-5,-2), \qquad \mathcal O_\Sigma(F)=\mathcal O_\Sigma(1,0),\] and both have vanishing first cohomology. Their images on the common scheme \(D\) therefore coincide. Equivalently, an ambient section vanishing on \(\Gamma\) vanishes to order at least four on \(C\) at \(Z\) and has no effect on these residue tests.

The principal part exact sequence and Serre duality now say that the image of \(H^0(C,\Lambda)\), by \(T\mapsto\operatorname{pp}(T/\xi^2)\), is precisely the annihilator of this common adjoint jet space; its kernel is \(\xi^2H^0(C,3H)\). On \(\Gamma\) the same annihilator is the image of \[H^0(\Gamma,\Phi)\longrightarrow H^0\!\left(\Gamma, \frac{\mathcal O_\Gamma(-N_\Gamma+2Z)}{\mathcal O_\Gamma(-N_\Gamma)}\right), \qquad \widetilde T\longmapsto \operatorname{pp}(\widetilde T/\beta^2),\] because \(\Phi=2M-N_\Gamma\) and \((\beta)=Z\). This map is injective: its kernel would give a section of \(-N_\Gamma\), which has degree \(-10\). These facts establish (18). In particular, this is an identification of full principal parts. Its dimension can also be read directly on \(\Gamma\): \(h^0(N_\Gamma)=10\), and restriction to \(D\) is injective since \(\deg(N_\Gamma-2Z)=-2\); its annihilator has dimension \(12-10=2\).

We compute the local comparison, including its first derivative and the adjunction scalar. Near a point of \(Z\) choose coordinates \((y,z)\) and frames \(\mathfrak g\) of \(\Gamma\) and \(\mathfrak f\) of \(F\) such that \[\gamma=y\mathfrak g,\qquad \Omega=e(y,z)\frac{dy}{y}\wedge dz,\qquad e(0,z)=1,\] so \(\omega=dz\) on \(\Gamma\). Write \[\kappa=k(y,z)\mathfrak g^2\mathfrak f^4, \qquad \beta=b(z)\mathfrak m, \qquad \mathfrak m^4=\mathfrak g^2\mathfrak f^4\big|_\Gamma.\] Here \(k(0,z)=b(z)^4\) and \(b'(0)k_y(0,0)\ne0\). With \(K(z)=k_y(0,z)\), the equation of \(C\) gives \[ y|_C=-\frac{b(z)^4}{K(z)}+O(z^8),\qquad e|_C=1+O(z^4),\qquad k_y|_C=K(z)+O(z^4). \tag{19}\] The normal-first adjunction convention sends the frame \(\mathfrak g\mathfrak f^4\) to \[ \mathop{\mathrm{Res}}_C\!\left(\frac{e(y,z)\,dy\wedge dz}{k(y,z)}\right) =\frac{e|_C}{k_y|_C}\,dz. \tag{20}\] The sign is positive here, since \(dy\wedge dz=(dk\wedge dz)/k_y\).

Let \(a=\xi/z\) be the local frame of \(\mathcal O_C(Z)\) and \(\tau=\mathfrak f^3a^2\) the resulting frame of \(\Lambda\). The fixed fourth-power identification sends \(a^4\) to \((y|_C/z^4)\mathfrak g\). Using (20), its square has the form \[ \tau^2=s(z)\,dz\,\mathfrak f^2, \qquad s(z)=\frac{e|_C\,y|_C}{z^4 k_y|_C},\qquad s(0)=-\frac{b'(0)^4}{k_y(0,0)^2}. \tag{21}\] Thus the negative sign comes from solving the equation of \(C\).

Write \(T=\pi(z)\tau\), and write \(\widetilde T=\iota([T])\) in the frame \(\widetilde\tau=\mathfrak m^2/(\mathfrak g\mathfrak f)\) of \(\Phi\) as \(\widetilde T=\widetilde\pi(z)\widetilde\tau\). For an ambient test section \(A(y,z)\mathfrak g\mathfrak f\), the two residue tests are \[ \mathop{\mathrm{Res}}_0\!\left( \frac{A|_C\,\pi\,e|_C}{k_y|_C\,z^2}\,dz\right), \qquad \mathop{\mathrm{Res}}_0\!\left( \frac{A(0,z)\,\widetilde\pi}{b(z)^2}\,dz\right). \tag{22}\] The restrictions of \(A\) to the two curves agree modulo \(z^4\). It follows from (19) that equality of their residue functionals on all local test jets is exactly \[ \operatorname{pp}\!\left(\frac{\pi}{K(z)z^2}\right) =\operatorname{pp}\!\left(\frac{\widetilde\pi}{b(z)^2}\right), \qquad \widetilde\pi\equiv\frac{(b(z)/z)^2}{K(z)}\pi\pmod{z^2}. \tag{23}\] To display both coefficients, put \[b=b_1z+b_2z^2+O(z^3),\quad K=K_0+K_1z+O(z^2),\quad \pi=\pi_0+\pi_1z+O(z^2).\] The respective principal parts are \[\frac{\pi_0}{K_0}z^{-2} +\left(\frac{\pi_1}{K_0}-\frac{K_1\pi_0}{K_0^2}\right)z^{-1}, \qquad \frac{\widetilde\pi_0}{b_1^2}z^{-2} +\left(\frac{\widetilde\pi_1}{b_1^2} -\frac{2b_2\widetilde\pi_0}{b_1^3}\right)z^{-1}.\] Hence \[ \begin{split} \widetilde\pi_0&=\frac{b_1^2}{K_0}\pi_0,\\ \widetilde\pi_1&=\frac{b_1^2}{K_0}\pi_1+ \left(\frac{2b_1b_2}{K_0} -\frac{b_1^2K_1}{K_0^2}\right)\pi_0. \end{split} \tag{24}\] This invertible triangular transformation is the local expression of the intrinsic residue-dual identification above; therefore its expressions in different coordinates and frames agree. In particular, no choice of independent pointwise scalars is being substituted for a global map.

It remains to identify its metric. In these frames write \(\eta_C=\mathcal E_C\,dz\,\mathfrak f^2\) and \(\eta_\Gamma=\mathcal E_\Gamma\,dz\,\mathfrak f^2\). Tangency gives \(\mathcal E_C(0)=\mathcal E_\Gamma(0)\). Let \(\ell\) and \(\widetilde\ell\) be coefficient evaluations in \(\tau\) and \(\widetilde\tau\), respectively. By (24), \[\iota^*\widetilde\ell=\alpha\ell, \qquad \alpha=\frac{b_1^2}{K_0}.\] Equations (17) and (21) therefore give \[ (\iota^*\widetilde\ell)^2 =\alpha^2\frac{\mathcal E_C(0)}{s(0)} =-\mathcal E_\Gamma(0). \tag{25}\] The compatible bundle identifications give \(\widetilde\tau^2=\mathfrak f^2\). Thus the final quantity in (25) is exactly \(Q(\widetilde\ell)\), since the section defining \(Q\) multiplies to \(-\eta_\Gamma/\omega\). The evaluations at \(Z\) have three distinct projective directions. The difference of the two binary quadratic forms vanishes on those three directions, and hence vanishes identically. This proves the isometry, with its sign and scalar fixed. ◻

Proof. A determinant-one change of basis preserves \(\eta_C\) and \(\eta_\Gamma\), and hence all the forms already fixed. Avoiding the finitely many branch values and the image of \(Z\), we may arrange that the zero divisor of \(u_1\) is the required fiber. The isometry of Lemma 6 transports the equations for \(w_j\) to (27). It also transports their evaluation directions. Since each \(l_j\) is nonzero, there is a unique nonzero \(c_j\) with \(c_jl_j=v_j^*\). Locally \(R_j=c_ju\), so it is holomorphic and nonzero at the center, and (26) gives \[R_j(0)\wedge R_j'(0)=c_j^2l_j^2=(v_j^*)^2.\]

Every prescribed set of simple principal parts of \(\theta\) at \(Z\) is realizable by a section of \(\theta+Z\). Indeed, the obstruction in \(H^1(C,\theta)\) pairs, by Serre duality, with \(H^0(C,K_C-\theta)=H^0(C,\theta)\) by the sum of residues. All these test sections vanish at \(Z\), by Proposition 4. Their residue against any simple principal part is therefore zero. Apply this separately to the two entries to obtain \(P\) with the stated pole parts. Since \(r=z_ju\), the row \(m_jr'\) has exactly the same simple pole part as \(P\); thus \(p_j\) is holomorphic.

The simple pole of \(P\) at \(q_j\) is proportional to \(u(q_j)\). Its wedge with \(u\) has zero residue coefficient, so \(J=u\wedge P\) is holomorphic there; away from \(Z\) it is already holomorphic. Hence it is a global section of \(\Lambda\). To compute its class, fix \(T\in H^0(C,\Lambda)\) and consider the global meromorphic canonical differential \[\nu=\frac{P_1T}{u_1}, \qquad \theta+(\theta+H)-H=2\theta=K_C.\] Its only possible poles lie in \(Z\) and in the simple zero fiber of \(u_1\). At \(q_j\), the compatible frames make the canonical product exactly multiplication of coefficients followed by \(dz_j\), and therefore \[\mathop{\mathrm{Res}}_{q_j}\nu=c_jl_j(T).\] At a zero \(q\) of \(u_1\), the section \(u_2\) is nonzero and \[J(q)=-u_2(q)P_1(q),\qquad \eta_C(q)=-u_2(q)\,du_1(q).\] Both expressions have the same minus sign. In a local parameter, the residue is consequently \[\mathop{\mathrm{Res}}_q\nu=\left(\frac{JT}{\eta_C}\right)(q).\] The zero fiber is unramified, so the sum of these four residues is the value of \(\mathop{\mathrm{Tr}}(JT/\eta_C)\) there. This trace is a global holomorphic function on \(\mathbb P^1\) by Lemma 5, hence is constant. The residue theorem gives \[\langle[J],[T]\rangle =-\sum_jc_jl_j(T)=-V([T]),\] which proves (30) with the stated sign.

On a fixed disk suppress the subscript \(j\). Since \(r=zu\), \[P=p+\frac{c}{z}u+cu',\qquad cJ=R\wedge p+R\wedge R'.\] In particular, \[c_jl_j(J)=R_j(0)\wedge p_j(0)+W_j.\] On the other hand, the metric identification in (30) gives \[c_jl_j(J)=-v_j^*\cdot V=W_j-1.\] Thus \(R_j(0)\wedge p_j(0)=-1\), or equivalently \(p_j(0)\wedge R_j(0)=1\). Summing \(W_j-1=-v_j^*\cdot V\) proves the last identity in (29). Notice also the useful local expansion \[ P=\frac{R_j(0)}{z_j} +p_j(0)+2R_j'(0)+O(z_j). \tag{31}\] If \(P\) is changed by a holomorphic pair \(S\), the equality \(H^0(C,\theta)=\xi^2H^0(C,2H)\) shows that \(u\wedge S\in\xi^2H^0(C,3H)\); thus \([J]\) is unchanged.

For completeness, let \(\zeta=\psi(z)\) be another centered parameter, and let the corresponding compatible frame of \(L\) be \(g(z)\ell^{\mathrm{fr}}\). Compatibility says \(g^4=\psi'\). The associated changes of the other frames are \[a_{\rm new}=\frac{z}{\psi(z)}a,\qquad \mathfrak f_{\rm new}=g\frac{\psi(z)}{z}\mathfrak f, \qquad \tau_{\rm new}=g^3\frac{\psi(z)}{z}\tau.\] Writing \(g_0=g(0)\), it follows that \(l_{\rm new}=g_0^{-7}l\), so the same vector \(v^*\) prescribes \(c_{\rm new}=g_0^7c\). Also \[r_{\rm new}=g^{-1}r,\qquad R_{\rm new}(0)=g_0^2R(0).\] The coefficient row of the global \(\theta\)-section transforms as \(P_{\rm new}=g^{-2}P\). Its coefficient of \(\zeta^{-1}\) is \(\psi'(0)g_0^{-2}R(0)=g_0^2R(0)\), exactly the newly prescribed \(R_{\rm new}(0)\). Thus the same global \(P\) realizes the new prescription. The row \(R_{\rm new}\) is a scalar multiple of \(R\), with scalar \(g_0^2\) at the center; differentiating with respect to \(\zeta\) shows \(W_{\rm new}=g_0^4W/\psi'(0)=W\). Finally, subtracting \(m_{\rm new}\,d r_{\rm new}/d\zeta\) from \(P_{\rm new}\) shows that \(p_{\rm new}(0)-g_0^{-2}p(0)\) is proportional to \(R(0)\). Hence \(p_{\rm new}(0)\wedge R_{\rm new}(0)=p(0)\wedge R(0)\). All the local conclusions are therefore compatible with the fixed fourth-root structure. ◻

Proof. Use \(E_t(t^a)=at^a\) and \([E_t,\partial_z]=0\) in the two components of the bracket. For the divergence, the coefficient of \(dt\wedge dz\) in \(\sigma\) is \(t^{-5}\), so \[\operatorname{div}_{\sigma}D =t^5\left(\partial_t(t^{k-4}b) +\partial_z(t^{k-5}h)\right) =t^k((k-4)b+h').\] Setting \(\ell=0\), \(A=r\), \(B=0\) gives the last formula. ◻

Proof. Write \(D^{(2)}=A_2\); this notation denotes the vertical row, not a transition generator. For \(i,j\in\{1,2\}\) set \[ \Psi_{ij}=\left(r_iD_j^{(2)}+\frac14r_i'U_j +\frac34r_j'U_i\right)dz. \tag{44}\] Once the horizontal rows have matched, the difference of these expressions for the transported inside lift and the outside lift is exactly \(r_i\) times the \(j\)th vertical difference, multiplied by \(dz\). The derivative terms therefore change neither the obstruction nor its four tests.

First compute the total for the outside lift. Put \(\epsilon_{12}=1\), \(\epsilon_{21}=-1\), and \(\epsilon_{11}=\epsilon_{22}=0\). Under a split change with \(a=(\log g)'\), formula (36) gives \[\widetilde r_i=g^{-1}r_i,\qquad \partial_{\widetilde z}\widetilde r_i=g^{-5}(r_i'-ar_i),\qquad \widetilde U_i=gU_i,\qquad \widetilde D_i^{(2)}=g^{-3}(D_i^{(2)}+aU_i).\] Substituting in (44), and expressing both differentials in the old coordinate, yields \[ \widetilde\Psi_{ij}-\Psi_{ij} =\frac34(r_iU_j-r_jU_i)d\log g =\frac34\epsilon_{ij}\,d\log g \quad\hbox{on }U_0. \tag{45}\] The last equality uses \(r\wedge U=1\) on the outside. Use the section \(r_1=\xi u_1\) of \(L\). Its coefficient transforms by \(\widetilde r_1=g^{-1}r_1\), so \(\Psi_{ij}+\frac34\epsilon_{ij}d\log r_1\) is an intrinsic differential on \(U_0\). There it has precisely the four possible simple poles at the reduced zero fiber of \(u_1\), each of residue \(\frac34\epsilon_{ij}\). The residue theorem on the complement of small disks therefore gives total contour residue \(-3\epsilon_{ij}\) around \(Z\) for this intrinsic differential. At each of the six points of \(Z\), the section \(r_1\) has a simple zero. Subtracting these six logarithmic residues yields \[ \sum_{q\in Z}\mathop{\mathrm{Res}}_q\Psi_{ij}^{\mathrm{out}} =-\frac34\epsilon_{ij}\deg L. \tag{46}\] Only contour integrals at \(Z\) were used, so this argument does not assume meromorphic extension of the outside lifts.

Next compute the transported inside total. Before transport, \(\Psi_{ij}\) is holomorphic on the disk. The order-two transition changes its rows by \[\delta D^{(2)}=hr'-\frac12h'r,\qquad \delta U=-2hr.\] In (44) this is the exact differential \[ \delta\Psi_{ij}=-\frac12d(hr_ir_j), \tag{47}\] so it has zero contour residue. It suffices to calculate the effect of the order-one transition, including its second exponential term.

The bracket of \(t(m'E_t/3+m\partial_z)\) with the inside order-one term \(t^2(pE_t+mr\partial_z)\) contributes \[\left(\frac13m'p+mp'-\frac13mm''r, \frac23mm'r+m^2r'-mp\right)\] to \((A_2,U)\). Half of its second bracket with the leading term \(trE_t\) contributes \[\left(\frac23mm'r'+\frac12m^2r''+\frac16mm''r, -\frac13mm'r-m^2r'\right).\] Thus the full effect is \[\begin{align*} \Delta A_2&=\frac13m'p+mp'+\frac23mm'r' +\frac12m^2r''-\frac16mm''r, \\ \Delta U&=-mp+\frac13mm'r. \tag{48}\end{align*}\] The \(m^2r'\) horizontal terms cancel in this sum.

Fix one disk and abbreviate \(R=mr\), \(m=c/z\), so \(r=zR/c\). Substitution in (48) gives \[\begin{aligned} \Delta A_2&=-\frac{cR}{z^3}+\frac{cR'}{3z^2} +\frac{cR''}{2z}-\frac{cp}{3z^2}+\frac{cp'}z, \\ \Delta U&=-\frac{cp}z-\frac{cR}{3z^2}. \end{aligned}\] It follows, with every coefficient on the right evaluated at \(0\), that \[\begin{align*} F_{ij}:=\mathop{\mathrm{Res}}_0(r_i\Delta A_{2,j}\,dz) &=-\frac13R_ip_j-\frac23R_iR_j'-R_i'R_j,\\ H_{ij}:=\mathop{\mathrm{Res}}_0(r_i'\Delta U_j\,dz) &=-R_ip_j-\frac13R_iR_j'-\frac23R_i'R_j. \tag{49}\end{align*}\] For example, the double pole \(-R_iR_j/z^2\) in the first expression contributes \(-R_i'R_j-R_iR_j'\) to its residue; this accounts for both derivative terms. The complete residue of the change in (44) is consequently \[ \begin{aligned} T_{ij}&=F_{ij}+\frac14H_{ij}+\frac34H_{ji}\\ &=-\frac7{12}R_ip_j-\frac9{12}R_jp_i -\frac54R_iR_j'-\frac{17}{12}R_i'R_j. \end{aligned} \tag{50}\]

Decompose this matrix into half its sum and half its difference with its transpose. Directly from (50), \[\begin{align*} T_{ij}^{\mathrm{sym}} &=-\frac23\bigl(R_ip_j+R_jp_i +2R_iR_j'+2R_i'R_j\bigr),\\ T_{ij}^{\mathrm{alt}} &=\frac1{12}\epsilon_{ij} \bigl((R\wedge p)(0)+(R\wedge R')(0)\bigr). \tag{51}\end{align*}\] To identify the symmetric term intrinsically, observe that \[P=p+mr'=\frac{R(0)}z+p(0)+2R'(0)+O(z).\] Since \(P_iP_j\) is a meromorphic section of \(L^4=K_C\), its local expression is \(P_iP_j\,dz\), and \[\mathop{\mathrm{Res}}_0(P_iP_j\,dz) =R_ip_j+R_jp_i+2R_iR_j'+2R_i'R_j.\] Thus the sum of every symmetric entry in (51) is zero, by the residue theorem on \(C\). For the alternating entry, (32) gives \(R(0)\wedge p(0)=-1\) and therefore \[\sum_{q\in Z}T_{ij}^{\mathrm{alt}} =\frac1{12}\epsilon_{ij}\sum_{\nu=1}^6(W_\nu-1) =-\frac34\epsilon_{ij}\deg L.\] In particular, the sums of the four entries \((11),(12),(21),(22)\) of the transported inside matrix are respectively \[0,\qquad -\frac34\deg L,\qquad \frac34\deg L,\qquad 0.\] They coincide with the four outside totals in (46). Hence every Serre-duality pairing of the vertical difference vanishes. The corresponding class in \(H^1(C,L^3)^2\) is zero, so holomorphic vertical corrections on the charts complete the order-two matching. ◻

Proof. Let \(N=L^{-1}\) be the split normal bundle. All changes to its disk-to-outside identifications are finite compositions of flows of holomorphic vector fields \[D=t^k\bigl(b(z)t\partial_t+h(z)\partial_z\bigr),\qquad k\geq1,\] on punctured disk neighborhoods. On a compact annulus in any such overlap, these vector fields vanish along the zero section. The local existence theorem for holomorphic differential equations, applied uniformly on that compact annulus, therefore gives the required time-one maps and their inverses after shrinking the fiber radius. There are only finitely many flows. Shrinking successively gives their compositions on neighborhoods of the same compact annular bands. Denote the resulting inside-to-outside maps by \(\Phi_j\), \(1\leq j\leq6\).

Here is a quotient construction that also verifies the separation property. Choose mutually disjoint coordinate disks \(D_j\) about the six points, and in each choose radii \[0<a_j<a'_j<b'_j<b_j\] so that \(\Phi_j\) and its inverse are defined on neighborhoods of the closed annulus \(a_j\leq |z_j|\leq b_j\) in their respective split bundles. Their domains may be taken somewhat wider than these closed annuli. Retain a small open neighborhood \(V_j\) of the zero section over \(|z_j|<b'_j\), and a small open neighborhood \(V_0\) in \(N\) over \[C\setminus\bigcup_{j=1}^6\{|z_j|\leq a'_j\}.\] Choose the transverse thicknesses small enough that the maps are defined throughout the retained parts of the closed annuli and \[ \bigl|z_j(\Phi_j(p))-z_j(p)\bigr| <\frac{a'_j-a_j}{2} \tag{60}\] there. This is possible because each map restricts to the identity on the zero section. We also make the images of the retained annuli lie over their respective disjoint disks \(D_j\).

Set \[A_j=\{p\in V_j:a_j<|z_j(p)|<b_j,\ \Phi_j(p)\in V_0\},\] and identify \(p\in A_j\) with \(\Phi_j(p)\in V_0\). These are biholomorphic identifications between open subsets. A point of \(V_0\) is in at most one of their images, since those images lie over disjoint disks. Each equivalence class in \(V_0\sqcup V_1\sqcup\cdots\sqcup V_6\) thus has at most two elements. In particular the equivalence relation consists exactly of the diagonal, these six graphs, and their inverses; no further identifications or triple-overlap equations occur.

Each graph is closed relative to \(V_j\times V_0\). Indeed, suppose \(p_\nu\in A_j\), \(p_\nu\to p\in V_j\), and \(\Phi_j(p_\nu)\to q\in V_0\). The source base radii have limit at least \(a_j\) and less than \(b'_j<b_j\). The flow is defined across the closed annulus, so continuity gives \(q=\Phi_j(p)\). If the limiting source radius were \(a_j\), inequality (60) would give \(|z_j(q)|<a'_j\), contrary to \(q\in V_0\). Thus \(a_j<|z_j(p)|<b_j\) and \(p\in A_j\), as required. Possible fiber boundaries cause no difficulty because both limiting points are assumed to belong to the retained open pieces.

The equivalence relation is therefore closed. The quotient map is open, since it glues open sets by biholomorphisms. An open quotient map with closed equivalence relation has Hausdorff target: two inequivalent representatives have product neighborhoods disjoint from the relation, and the images of these neighborhoods are disjoint open sets. Moreover, the quotient map restricts on every \(V_i\) to an injective open map. The complex structures on these pieces consequently give the quotient a smooth complex surface structure. The images of their zero sections give the original compact curve \(C\) with its original complex structure.

Every modifying flow fixes \(C\) and induces the identity on its conormal bundle. In the split identification its coordinate expansion has \(t'=t+O(t^2)\), although its base coordinate may change by \(O(t)\). Composing with the original split changes therefore leaves \(\mathcal I/\mathcal I^2\simeq L\). This proves \(C^2=-\deg L=-10\).

In the logarithmic basis \((t\partial_t,\partial_z)\) the prescribed fields have coefficients \(tA_i,tB_i\). Rows with indices \(0\) through \(7\) prescribe these coefficients through degree \(8\) in \(t\). The matching in Proposition 10 is therefore precisely matching modulo \(\mathcal I^9\mathcal F\) on the actual overlap maps. It gives the claimed quotient sections. Their logarithmic determinant is \[t^2(A\wedge B)=t^4(1+O(t)).\] Replacing \(t\partial_t\) by \(\partial_t\) multiplies this determinant by \(t\), giving the stated ordinary determinant. ◻

Proof. Since \(K_C\simeq L^4\), tensoring (61) by \(L^n\) gives \[0\longrightarrow L^n\longrightarrow\mathcal F|_C\otimes L^n \longrightarrow L^{n-4}\longrightarrow0.\] Serre duality identifies the duals of the first cohomology groups of the outer terms with \[H^0(C,L^{4-n})\quad\hbox{and}\quad H^0(C,L^{8-n}),\] respectively. Both vanish for \(n\geq9\), as \(\deg L>0\). The first assertion follows from the cohomology sequence.

Because \(C\) is a Cartier divisor and \(\mathcal F\) is locally free, \[\mathcal I^n\mathcal F/\mathcal I^{n+1}\mathcal F\simeq\mathcal F|_C\otimes L^n.\] The quotient \(\mathcal E/\mathcal I^r\mathcal E=\mathcal I^9\mathcal F/\mathcal I^{9+r}\mathcal F\) has a finite filtration with precisely these graded terms for \(9\leq n<9+r\). Their first cohomology vanishes by the first assertion. Successive cohomology sequences give the second assertion; the quotient is supported on the compact curve \(C\). ◻

Proof. The intersection matrix of \(C\) is the negative definite matrix \((-10)\). Grauert’s contraction criterion therefore contracts \(C\) to a point \(x\) of a normal complex surface: \[\pi:Y\longrightarrow X,\qquad \pi^{-1}(x)=C, \qquad Y\setminus C\simeq X\setminus\{x\},\] where the fiber equality is set-theoretic, and \(\pi_*\mathcal O_Y=\mathcal O_X\). We use the criterion in [14]. The contraction is proper and is a biholomorphism off the exceptional curve. It is the holomorphic reduction in that theorem, whose target is normal when the source is normal; these properties are part of the reduction and contraction statements in [14]. We choose a sufficiently small Stein representative \(X\) about \(x\) and replace \(Y\) by its inverse image. In particular \(X\setminus\{x\}\) is smooth.

We prove \(R^1\pi_*\mathcal E=0\) for \(\mathcal E=\mathcal I^9\mathcal F\), keeping track of the two ideal filtrations. Let \(\mathfrak m_x\) be the ideal sheaf of \(x\) in \(X\) and put \(\mathcal J=\mathfrak m_x\mathcal O_Y\). Both \(\mathcal J\) and \(\mathcal I\) vanish exactly on \(C\), and \(\mathcal I\) is radical. The analytic Nullstellensatz [7] and Noetherianity give an inclusion \(\mathcal I^a\subseteq\mathcal J\) locally at each point of \(C\) for some positive integer \(a\). A finite cover of the compact curve gives a single exponent. After shrinking the representative, \[ \mathcal I^a\subseteq\mathcal J\subseteq\mathcal I, \qquad \mathcal I^{ar}\subseteq\mathcal J^r\subseteq\mathcal I^r \quad(r\geq1). \tag{62}\] Such a shrinking may be made on \(X\): for a proper map, every open neighborhood of its compact fiber contains the inverse image of a sufficiently small neighborhood of the image point.

The proper analytic direct-image theorem makes \(R^1\pi_*\mathcal E\) coherent. The analytic theorem on formal functions identifies its completed stalk with \[ \widehat{(R^1\pi_*\mathcal E)_x} \simeq \varprojlim_{r\geq1}H^1(Y,\mathcal E/\mathcal J^r\mathcal E). \tag{63}\] We use [13]: the hypotheses are a proper holomorphic map and a coherent analytic sheaf. In particular no flatness of \(\pi\) is required. The quotients in (63) are supported on the infinitesimal fibers over \(x\), so their cohomology is independent of a further shrinking of the representative.

For clarity, the right side vanishes by an explicit factorization, without identifying the individual cohomology groups for the two filtrations. The inclusions in (62) give \[\mathcal J^{ar}\mathcal E \subseteq\mathcal I^{ar}\mathcal E \subseteq\mathcal J^r\mathcal E.\] Hence the transition map from index \(ar\) to index \(r\) in (63) factors through \[H^1(Y,\mathcal E/\mathcal I^{ar}\mathcal E)=0\] by Lemma 12. The inverse system is therefore pro-zero, and its inverse limit is zero. Since the stalk \((R^1\pi_*\mathcal E)_x\) is a finite module over the Noetherian local ring \(\mathcal O_{X,x}\), its zero completion implies that the stalk itself is zero, by Nakayama’s lemma. Away from \(x\) the map \(\pi\) is an isomorphism, so the same higher direct image vanishes there. We obtain \(R^1\pi_*\mathcal E=0\).

The sheaf \(\pi_*\mathcal E\) is also coherent. Cartan’s Theorem B on the Stein space \(X\), together with the Leray spectral sequence for \(\pi\), now gives \[H^1(Y,\mathcal E)=0.\] The cohomology sequence of \[0\longrightarrow\mathcal E\longrightarrow\mathcal F \longrightarrow\mathcal F/\mathcal I^9\mathcal F\longrightarrow0\] therefore lifts both \(\bar v_i\) to actual holomorphic sections \(v_i\) of \(\mathcal F\).

It remains to check that arbitrary such lifts retain the required frame. In each divisor chart, let \(v_i^{(8)}\) be the polynomial representative of the prescribed jet. It belongs to \(\mathcal I\mathcal F\), and \[v_i-v_i^{(8)}\in\mathcal I^9\mathcal F.\] Thus the change in their wedge belongs to \(\mathcal I^{10}\det\mathcal F\). By Lemma 11, the logarithmic determinant of the lifted pair is still \(t^4(1+O(t))\); its leading coefficient is \(1\) even at the common zeros of the leading row \(r\). Its ordinary determinant is \[ v_1\wedge v_2=t^5 u(t,z)\,\partial_t\wedge\partial_z, \qquad u(0,z)=1. \tag{64}\] The unit condition holds on a neighborhood of each point of \(C\). Compactness gives one neighborhood on which all these local factors are nonvanishing. Properness allows a further shrinking of \(X\) whose inverse image lies in this neighborhood. Equation (64) then proves that \(v_1,v_2\) frame \(\mathcal T_{Y\setminus C}\). ◻

Proof. Take the contraction constructed in Lemma 13. Suppose first that \(x\) were a smooth point. A sufficiently small neighborhood of \(x\) would admit a nowhere-vanishing holomorphic two-form \(\omega\). Its pullback would be holomorphic on \(Y\) and nonvanishing off \(C\). Thus \[\operatorname{div}(\pi^*\omega)=aC\qquad(a\geq0), \quad\hbox{and hence}\quad K_Y\cdot C=aC^2\leq0.\] Adjunction, however, gives \[K_Y\cdot C=2g(C)-2-C^2=40+10=50,\] a contradiction. The point \(x\) is singular, and it is the only singular point after the chosen shrinking.

Write \(X^\circ=X\setminus\{x\}\) and \(j:X^\circ\hookrightarrow X\). The two fields of Lemma 13 descend to a holomorphic frame \(w_1,w_2\) on \(X^\circ\). Normality in dimension two gives the Hartogs extension property [7] \[ \mathcal O_X\simeq j_*\mathcal O_{X^\circ}. \tag{65}\] This property extends the fields as derivations, as follows. Embed a representative of \((X,x)\) in a smooth analytic germ with coordinate functions \(x_1,\ldots,x_e\). For each \(i\) and \(\alpha\), the holomorphic function \(w_i(x_\alpha)\) on \(X^\circ\) extends uniquely to a function \(a_{i\alpha}\) on \(X\). If \(f\) is any local defining equation of the embedding, then \[\sum_{\alpha=1}^e a_{i\alpha}\frac{\partial f}{\partial x_\alpha}=0\] on \(X^\circ\) and hence on \(X\), by density and reducedness. The tuple \((a_{i\alpha})_\alpha\) consequently defines a section of \(\mathcal{H}om_{\mathcal O_X}(\Omega_X^1,\mathcal O_X)\) extending \(w_i\). We continue to denote that extension by \(w_i\).

For any local analytic derivation \(w\) on \(X\), the frame on \(X^\circ\) gives unique holomorphic coefficients \(h_1,h_2\) there such that \(w=h_1w_1+h_2w_2\). Both coefficients extend by (65). The extended equality holds on \(X\), since applying its difference to the embedding coordinates gives holomorphic functions vanishing on the dense open set \(X^\circ\). This proves surjectivity of the map \[\mathcal O_X^{\oplus2}\longrightarrow\mathcal T_X, \qquad(h_1,h_2)\longmapsto h_1w_1+h_2w_2.\] If a pair lies in its kernel, both coefficients vanish on \(X^\circ\) because \(w_1,w_2\) are a frame there, and therefore vanish on \(X\). The map is also injective, proving the assertion. ◻

Proof. We improve the equations one order at a time. The exact relation \(gB=0\) makes the equation error, reduced modulo \(I\), nearly belong to \(K\). Artin–Rees lets us correct it into \(K\); the finite length of \(K/J\) then expresses the sufficiently high-order corrected class by coordinate derivatives. A change of generators removes the remaining \(I\)-valued error. We arrange these corrections so that the equation precision increases at every step while the relation precision stays fixed. The resulting coordinate and generator changes converge in \(P\).

We first choose the fixed bounds. For a map \(u:M\to M'\) of finite modules over a Noetherian local ring with maximal ideal \(\mathfrak a\), the Artin–Rees lemma [28], applied to \(\operatorname{im}u\subset M'\) gives a constant \(a\ge0\) such that \[\operatorname{im}u\cap\mathfrak a^rM' \subset u(\mathfrak a^{r-a}M)\qquad(r\ge a).\] Choose nonnegative constants \(\alpha,\beta,d\) so that \[\begin{align*} \operatorname{im}\psi\cap\mathfrak m_R^rR^t &\subset\psi(\mathfrak m_R^{r-\alpha}R^s), \tag{68}\\ K\cap\mathfrak m_R^rR^s &\subset\mathfrak m_R^{r-\beta}K, \tag{69}\\ I^{\oplus s}\cap\mathfrak n^rP^s &\subset\{fC:C\in\mathfrak n^{r-d}P^{s\times s}\} \tag{70}\end{align*}\] whenever the corresponding exponents are nonnegative. In (70), Artin–Rees is applied to the map \(P^{s\times s}\to P^s\), \(C\mapsto fC\), with image \(I^{\oplus s}\); it is the filtration induced from \(P^s\) that is relevant.

Choose \(\ell\ge0\) with \(\mathfrak m_R^\ell K\subset J\) and set \[ c_0=\beta+\ell,\qquad c=c_0+d,\qquad N\ge\alpha+1,\qquad n_0\ge\max\{c+N,c+2,2c+1\}. \tag{71}\] We first describe one improvement step. Suppose \(n\ge n_0\) and \[h=g-f\in\mathfrak n^nP^s, \qquad B-D\in\mathfrak n^NP^{s\times t}, \qquad gB=0.\] The exact relation, reduced modulo the fixed ideal \(I\), gives \[ \bar h\bar D =\bar g\bar D =-\bar g(\bar B-\bar D) =-\bar h(\bar B-\bar D) \in\mathfrak m_R^{n+N}R^t. \tag{72}\] The factor \(\bar h\) on the right is what provides order \(n+N\). By (68), there is \[\bar\epsilon\in\mathfrak m_R^{n+N-\alpha}R^s \subset\mathfrak m_R^{n+1}R^s, \qquad \psi(\bar\epsilon)=\psi(\bar h).\] Hence \[\kappa=\bar h-\bar\epsilon \in K\cap\mathfrak m_R^nR^s \subset\mathfrak m_R^{n-\beta}K \subset\mathfrak m_R^{n-c_0}J.\] Here the last inclusion uses \(\mathfrak m_R^\ell K\subset J\), with the remaining factor of order \(n-\beta-\ell=n-c_0\). Lift coefficients expressing \(\kappa\) in the displayed generators of \(J\) to elements \(a_j\in\mathfrak n^{n-c_0}\), and lift \(\bar\epsilon\) to \(\epsilon\in\mathfrak n^{n+1}P^s\). Then \[w=h-\sum_{j=1}^e a_j\partial_jf-\epsilon \in I^{\oplus s}\cap\mathfrak n^{n-c_0}P^s.\] The order of this ideal-valued remainder need only be \(n-c_0\), since the lifted derivative combination may have lower order than \(h\): a derivative of a defining equation can be a unit. Using (70), write \(w=fC\) with \(C\in\mathfrak n^{n-c}P^{s\times s}\). We have proved \[ h=\sum_{j=1}^e a_j\partial_jf+fC+\epsilon, \qquad a_j,C\in\mathfrak n^{n-c}, \qquad \epsilon\in\mathfrak n^{n+1}P^s. \tag{73}\] Thus the loss \(d\) for lifting the ideal-valued remainder has been included in the fixed constant \(c\).

Put \(q=n-c\), and define \[ \phi(x_j)=x_j-a_j,\qquad M=(1+C)^{-1},\qquad g_{\mathrm{new}}=\phi(g)M,\qquad B_{\mathrm{new}}=(1+C)\phi(B). \tag{74}\] Our bounds ensure \[q\ge\max\{N,2\},\qquad 2q\ge n+1, \qquad n+q-1\ge n+1.\] In particular, \(\phi\) is a formal automorphism with identity linear part, and \(M\) is invertible. Formal Taylor expansion gives \[\phi(f)-f+\sum_j a_j\partial_jf\in\mathfrak n^{2q}P^s, \qquad \phi(h)-h\in\mathfrak n^{n+q-1}P^s.\] Combining these estimates with (73) gives \[\phi(g)\equiv f(1+C)\pmod{\mathfrak n^{n+1}P^s}, \qquad g_{\mathrm{new}}-f\in\mathfrak n^{n+1}P^s.\] The relations in (74) transform exactly: \[g_{\mathrm{new}}B_{\mathrm{new}} =\phi(g)(1+C)^{-1}(1+C)\phi(B)=0.\] Moreover, substitution by \(\phi\) changes any power series by an element of \(\mathfrak n^q\), and \(C\) has order at least \(q\). Therefore \[B_{\mathrm{new}}-D =C\phi(B)+(\phi(B)-B)+(B-D) \in\mathfrak n^NP^{s\times t}.\] The fixed relation precision \(N\) is preserved, while the equation precision increases from \(n\) to \(n+1\).

Starting with \((g_{n_0},B_{n_0})=(g,B)\), repeat this step for \(n=n_0,n_0+1,\ldots\). Write \(\phi_n,M_n\) for its corrections. For completeness, the cumulative coordinate and generator changes obey \[\Phi_{n_0}=\mathop{\mathrm{id}},\quad U_{n_0}=1,\qquad \Phi_{n+1}=\phi_n\circ\Phi_n,\qquad U_{n+1}=\phi_n(U_n)M_n,\] so that \(g_n=\Phi_n(g)U_n\) at every stage. Since \[(\phi_n-\mathop{\mathrm{id}})(P)\subset\mathfrak n^{n-c}, \qquad M_n-1\in\mathfrak n^{n-c}P^{s\times s},\] the consecutive images \(\Phi_n(x_j)\) and the consecutive matrices \(U_n\) differ in orders tending to infinity. They converge in the complete ring \(P\) to \(\Phi(x_j)\) and \(U\). The resulting substitution \(\Phi\) has identity linear part and hence is an automorphism, while \(U\equiv1\pmod{\mathfrak n}\) is invertible. Taking limits in \(g_n=\Phi_n(g)U_n\) yields \(f=\Phi(g)U\), as required. ◻

Proof. The completed analytic presentation. Embed the germ in a smooth analytic germ and write \[S=\mathcal O_{X,x}=\mathbb C\{x_1,\ldots,x_e\}/I_{\mathrm{an}}.\] Choose a finite row \(f\) generating \(I_{\mathrm{an}}\) and a finite matrix \(D\) generating all its relations over the convergent power-series ring. Completion is exact on finite modules, so they give the presentation (66) over \(P=\mathbb C[[x_1,\ldots,x_e]]\), with \(R=\widehat S=P/(f)\). Complex analytic local rings are excellent [6]; thus the completion of the normal local ring \(S\) is a normal local domain. Also \[ \dim R=\dim S=d, \qquad R\text{ is regular if and only if }S\text{ is regular}. \tag{75}\] We use here the usual exactness and permanence properties of completion [28].

The convergent version of (67) is the conormal dual of the analytic embedding. On the smooth locus the tangent-to-normal map is surjective: a smooth analytic subspace of a complex manifold is locally a regular embedding. Its cokernel is consequently a coherent module supported at the isolated point \(x\), and has finite length. Flat completion identifies this cokernel with the module \(T^1=K/J\) above. Lemma 16 therefore applies to \(f,D\).

We also record precisely which differentials are being completed. The module of holomorphic differentials of \(S\) has the finite Jacobian presentation with generators \(dx_j\) and relations \(df_i\). Its dual is the holomorphic tangent module. Flat completion of this kernel yields \[ \mathop{\mathrm{Der}}^{\mathrm{cont}}_{\mathbb C}(R,R) =\ker\left(R^e\longrightarrow R^s, (v_j)\longmapsto\sum_{j=1}^e v_j\overline{\partial_jf}\right) \simeq R^d. \tag{76}\] Here “continuous” refers to the maximal-ideal topology. A continuous derivation of a power-series quotient is determined by its values on the \(x_j\), and the displayed relations are precisely the conditions for it to descend from \(P\). All completed differential modules used below are these finite continuous differential modules.

Approximation of equations and relations. Let \[H=\bigl(\mathbb C[x_1,\ldots,x_e]_{(x_1,\ldots,x_e)}\bigr)^h\] be the henselization, whose completion is \(P\). This is precisely a henselization of a finite-type algebra over the field \(\mathbb C\) at a prime, as covered by Artin’s Theorem 1.10. Apply that approximation theorem over \(H\) to the polynomial system in the \(s+st\) entries of a row \(G\) and a matrix \(B\), \[ \sum_{i=1}^s G_i B_{i\nu}=0\qquad(1\le\nu\le t), \tag{77}\] with formal solution \((G,B)=(f,D)\). It gives a row \(g\) and a matrix \(B\) with entries in \(H\), satisfying \(gB=0\) exactly and agreeing with \(f,D\) to any prescribed finite order [1]; see also [29]. Choose that order at least \(\max\{n_0,N\}\), with \(n_0,N\) from Lemma 16. The lemma gives \[ P/(g)\simeq R. \tag{78}\]

The henselization is the filtered colimit of pointed étale neighborhoods of the origin, with the same residue field [30]. Thus the finitely many entries of \(g,B\) are represented on a single such neighborhood. After passing to a later neighborhood, the finitely many identities (77) hold there as well. If this stage is expressed over the local polynomial ring, finite presentation and clearing finitely many denominators spread it to a neighborhood of the origin in affine space; shrinking preserves the marked point and makes the morphism étale. After an affine shrinking the identities hold in its coordinate ring. We obtain an affine étale neighborhood \(\mathop{\mathrm{Spec}}E\to\mathbb A^e_{\mathbb C}\) with a marked point \(\mathfrak q\) over the origin, residue field \(\mathbb C\), and a row \(g\) of elements of \(E\) vanishing at \(\mathfrak q\). Its completed ambient local ring is \(P\). Put \[A'=E/(g),\qquad \mathfrak p'=\mathfrak q/(g),\qquad T=A'_{\mathfrak p'}.\] Then \(A'\) is of finite type over \(\mathbb C\), and exactness of completion together with (78) gives \[ \widehat T=P/(g)\simeq R. \tag{79}\] In particular, this construction preserves the complete local ring, including its dimension and its normality.

Descent of tangent freeness and local ring properties. Because the ambient neighborhood is étale over \(\mathbb A^e_{\mathbb C}\), its differentials have basis \(dx_1,\ldots,dx_e\) near the marked point. Completing the finite presentation of \(\Omega^1_{T/\mathbb C}\) therefore gives the continuous Jacobian presentation of \(P/(g)\). For a finitely presented module, flat base change commutes with its dual: this follows by expressing the dual as the kernel of a map between finite free modules. The continuous chain rule under (79) and (76) consequently give \[ \mathop{\mathrm{Der}}_{\mathbb C}(T)\otimes_T\widehat T \simeq\mathop{\mathrm{Der}}^{\mathrm{cont}}_{\mathbb C}(\widehat T,\widehat T) \simeq\widehat T^{d}. \tag{80}\] This is an equality obtained from finite presentations; it does not identify ordinary algebraic differentials of a power-series ring with its continuous differentials.

Let \(M=\mathop{\mathrm{Der}}_{\mathbb C}(T)\). By (80), its residue vector space has dimension \(d\). Lift a basis to \(d\) elements of \(M\). Nakayama’s lemma gives a surjection \(T^d\to M\), which after completion is a surjection between free modules of rank \(d\) over the local ring \(\widehat T\), and hence an isomorphism. Faithful flatness of completion annihilates its kernel, proving \(M\simeq T^d\).

The ring \(T\) injects into its domain completion, so it is a domain. Normality also descends. Indeed, if \(a/b\in\operatorname{Frac}(T)\) is integral over \(T\), it is integral over the normal domain \(\widehat T\) and hence belongs to \(\widehat T\). Thus \(a\in b\widehat T\cap T=bT\), where the last equality follows from faithful flatness, and \(a/b\in T\). Finally, completion preserves both Krull dimension and the dimension of the maximal ideal modulo its square. Equations (75) and (79) show that \(T\) has dimension \(d\) and is nonregular whenever \(S\) is nonregular.

The isolated-singularity property can also be retained. Let \(\mathfrak j_{\mathrm{an}}\) be the \(d\)th Fitting ideal of the holomorphic differential module of \(S\). The analytic Jacobian criterion identifies its zero set with the nonsmooth locus, so some power of the maximal ideal of \(S\) is contained in \(\mathfrak j_{\mathrm{an}}\); in the smooth case this ideal is the unit ideal. Fitting ideals commute with base change. The completed Jacobian presentations and (79) thus imply, by contraction from the faithfully flat completion, that \[ \mathfrak m_T^b\subset \operatorname{Fitt}_d(\Omega^1_{T/\mathbb C}) \quad\text{for some }b\ge0. \tag{81}\] On the reduced pure-\(d\)-dimensional neighborhood constructed below, the algebraic Jacobian criterion interprets this containment as nonsmoothness supported at the marked point.

An integral normal affine neighborhood with a free tangent module. The finite module \(\mathop{\mathrm{Der}}_{\mathbb C}(A')\) localizes at \(\mathfrak p'\) to \(\mathop{\mathrm{Der}}_{\mathbb C}(T)\), because \(\Omega^1_{A'/\mathbb C}\) is finitely presented. Represent the basis just obtained after inverting an element outside \(\mathfrak p'\). It defines a map from a free module of rank \(d\) to the derivation module whose kernel and cokernel vanish at \(\mathfrak p'\). Their supports are closed, so a further principal shrinking makes this map an isomorphism.

Since \(T\) is a domain, exactly one irreducible component of \(\mathop{\mathrm{Spec}}A'\) passes through \(\mathfrak p'\), and the support of the nilradical avoids \(\mathfrak p'\). Remove the other components and that support. The remaining neighborhood is reduced and irreducible. Its normal locus is open, since schemes of finite type over \(\mathbb C\) are excellent, and contains \(\mathfrak p'\) by the normality of \(T\); restrict to this normal locus as well. This irreducible neighborhood has dimension \(d\), since its marked point is closed and its local ring has dimension \(d\). Its nonsmooth locus is closed and, by (81), has no positive-dimensional component through \(\mathfrak p'\). Remove any components of that locus that do not contain \(\mathfrak p'\).

All the restrictions just made are open neighborhoods of \(\mathfrak p'\). Choose a principal affine neighborhood contained in their intersection, and write it as \(\mathop{\mathrm{Spec}}A\), with marked maximal ideal \(\mathfrak p\). The algebra \(A\) is finitely generated, normal and integral, its dimension is \(d\), and \[A_{\mathfrak p}=T,\qquad \mathop{\mathrm{Der}}_{\mathbb C}(A)\simeq A^d.\] The marked residue field and completed local ring are unchanged, and the variety is smooth away from that point. This proves all the assertions. ◻

Proof of Theorem 1. Proposition 14 supplies a singular normal complex analytic surface germ, smooth away from its marked point, whose holomorphic tangent module is free of rank two. Apply Proposition 17 and fix any of the approximations and affine neighborhoods furnished by its proof. Its coordinate ring \(A\) is a finitely generated normal integral \(\mathbb C\)-algebra of dimension two, with \[\mathop{\mathrm{Der}}_{\mathbb C}(A)\simeq A^2, \qquad A_{\mathfrak p}\text{ nonregular}.\] Thus the derivation module is free, and in particular projective, whereas \(A\) is not regular. This is the required counterexample over \(\mathbb C\). ◻

Proof. The free tangent sheaf on \(X\) restricts to a trivial tangent bundle on \(U\). If \(\overline\kappa(U)\le1\), the theorem of Biswas, Gurjar and Kolte [4] would make the normal affine surface \(X\) smooth, contrary to Theorem 1. The only remaining value of logarithmic Kodaira dimension for a surface is two. ◻

Proof. Since \(\mathop{\mathrm{Der}}_{\mathbb C}(A)\simeq A^2\), the identity of \(X\) already has the universal property just stated. Its blowup at the derivation module is therefore \(X\) itself, and the operation remains unchanged at every iteration. Normalization also leaves \(X\) unchanged because \(A\) is normal. The relation with the Lipman–Zariski conjecture was noted in [2]; the surface above gives a negative answer to that question in characteristic zero. ◻