Proof. Consider the scheme defined by the three spatial partial derivatives. Its central fiber is supported at \(0\) and has length \(\mu\). The local finiteness theorem for analytic maps gives a finite representative over the disk. In the regular local ring \(\mathbb C\{x_1,x_2,x_3,t\}\), the three partials together with \(t\) form a system of parameters. They therefore form a regular sequence. The quotient by the three partials is Cohen–Macaulay of dimension one, and \(t\) is a nonzerodivisor in it. It is consequently finite flat over the disk, of degree \(\mu\). In every nearby fiber its local length at \(0\) is already \(\mu\); there can be no further points. This is conservation of the spatial-gradient number; compare (Denkowski 2018, Proposition 3.1).

Each surface fiber is a hypersurface, hence Gorenstein and Cohen–Macaulay. Its isolated singularity has codimension two, so Serre’s criterion gives normality. Similarly, the total hypersurface is Gorenstein and Cohen–Macaulay; it is smooth off \(\Gamma\), which has codimension two in \(X\), and is therefore normal. Since no fiber of the ambient projection is contained in \(X\), its parameter is a nonzerodivisor on \(X\). Torsion-freeness over the regular one-dimensional base gives flatness. After a disk base change the equation is again a relative hypersurface, with the same individual surface germs and the same spatial critical locus. These arguments apply again. ◻

Proof. Choose \(B\) inside the representative of Lemma 2, with radius a Milnor radius for \(f_0\). Transversality at the compact set \(f_0^{-1}(0)\cap\partial B\) persists for small \(t\) and small \(c\). Points elsewhere on the sphere do not enter these small levels, after shrinking, by compactness. This proves the first two assertions.

Fix \(c\) with \(0<|c|<\eta\). The spaces \(f_t^{-1}(c)\cap B\) form a proper smooth family with boundary over the smaller parameter disk. Smooth lifts tangent to the boundary give the proper-submersion theorem in this setting. Their Euler characteristic is therefore that of the central Milnor fiber, namely \(1+\mu\); see (Milnor 1968, Theorems 6.5 and 7.2) and, for the analytic Jacobian-algebra formulation, (Lê and Ramanujam 1976, sec. 1, Theorem 1.5).

For a fixed \(t\), choose an inner Milnor ball for \(f_t\), with closure in the interior of \(B\). On the closed region between the two spheres, \(f_t\) has no critical point, and both boundary restrictions are submersive over a sufficiently small disk of values. Thus the shell has the same Euler contribution at value \(0\) and at sufficiently small nonzero values. In the inner ball, passing from value \(0\) to a small nonzero value replaces a contractible singular fiber by a Milnor fiber of Euler characteristic \(1+\mu\). Additivity of \(\chi_c\), or a decomposition using the inner boundary and the two complementary pieces, therefore gives \[\chi(f_t^{-1}(c)\cap B)-\chi(X_t\cap B)=\mu.\] The inner ball and the admissible nonzero value may depend on \(t\); the preceding computation of the outer smooth level applies to every such value. It follows that \(\chi(X_t\cap B)=1\). Removing its one origin point gives the second equality. Finally, transversality persists for radii in a small interval around the chosen radius, and the identical argument applies there. ◻

Proof. Let \(\mathfrak m=(x_1,x_2,x_3)\) and \(J(f)=(f_{x_1},f_{x_2},f_{x_3})\). The local algebra \(\mathbb C\{x\}/J(f)\) has length \(\mu\), so its maximal ideal has \(\mu\)th power zero, by the descending filtration and Nakayama’s lemma. Hence \(\mathfrak m^\mu\subset J(f)\). For the given \(b\in\mathfrak m^{\mu+2}\) we have \[b\in\mathfrak m^{\mu+2}\subset\mathfrak m^2J(f), \qquad b_{x_i}\in\mathfrak m^{\mu+1}\subset\mathfrak m J(f).\] Write \(b=a^{\mathsf T}g\), where \(g=(f_{x_i})_i\) is a column vector and the entries of \(a\) lie in \(\mathfrak m^2\). There is a matrix \(C\) with entries in \(\mathfrak m\) such that the gradient column of \(f+sb\) is \((I+sC)g\). On a sufficiently small neighborhood this matrix is invertible simultaneously for \(0\leq s\leq1\). The holomorphic spatial vector field \[w_s=-(I+sC)^{-\mathsf T}a\] vanishes to second order and satisfies \(d(f+sb)(w_s)=-b\). Its time-dependent flow \(h_s\), initially the identity, exists as a holomorphic germ throughout \(0\leq s\leq1\), after shrinking the spatial neighborhood. This follows from a uniform quadratic bound on \(w_s\); the reverse equation supplies the inverse germ. The chain rule gives \(\frac{d}{ds}((f+sb)\circ h_s)=0\), and therefore \(h=h_1\) has the required property. Taking \(b=j^Nf-f\) proves the assertion about Taylor polynomials. ◻

Proof. Choose a log resolution \(\pi:Z\to S\), isomorphic outside \(B\), and put \(\rho=b\pi\). Let \(E\) be the full reduced exceptional divisor of \(\rho\). With compatible canonical divisors, log canonicity gives \[K_Z+E=\pi^*(K_S+B)+N,\] where \(N\) is effective and \(\pi\)-exceptional. The pullback on the right is \(\rho\)-nef and has intersection zero with every component of \(N\). It is therefore the positive part \(P\) of the relative decomposition.

The divisor \(Z_P=P-\rho^*K_V\) is exceptional: the two summands agree outside \(\rho^{-1}(p)\). Moreover, \(\rho^*K_V\) has intersection zero with every exceptional curve, so \(Z_P=\nu(P)\). The projection formula gives \[P^2=K_V^2+Z_P^2=(K_S+B)^2.\] Together with (1), this proves the formula. ◻

Proof. By Varchenko’s theorem (Varchenko 1982), as recovered by Steenbrink, the spectrum of an isolated hypersurface singularity is constant in a \(\mu\)-constant deformation (Steenbrink 1985, Theorem 2.8). Its smallest number plus one is the complex singularity index (Steenbrink 1985, Theorem 2.11), and the log canonical threshold is the minimum of this index and one (Kollár 1997, Theorem 9.5). For a normal hypersurface in a smooth ambient space, adjunction and inversion of adjunction identify log canonicity with threshold one (Kawakita 2007). To apply the algebraic inversion theorem to an individual analytic germ, replace its defining function by a sufficiently high Taylor polynomial using Lemma 4; analytic right equivalence preserves both the threshold and log canonicity. Thus log canonicity, and its failure, is constant in the family. This use of the spectrum concerns the individual isolated function germs; it does not assume topological triviality of the family. ◻

Proof. Work first in the finite-dimensional algebra \(A_\mu=\mathbb C[x_1,x_2,x_3]/\mathfrak m^{\mu+1}\). The ideal generated by the partials of \(g\) is the image of a linear map \(A_\mu^{\oplus3}\to A_\mu\) whose matrix entries depend polynomially on the coefficients of \(g\). The condition that its cokernel have dimension \(\mu\) is a locally closed rank condition.

This condition tests the full local length, including exclusion of infinite length. Indeed, for the local algebra \(A=\mathbb C\{x\}/J(g)\) with maximal ideal \(\mathfrak n\), suppose \(\dim_{\mathbb C}A/\mathfrak n^{\mu+1}=\mu\). The \(\mu+1\) successive layers \(\mathfrak n^j/\mathfrak n^{j+1}\), \(0\leq j\leq\mu\), cannot all be nonzero. If a layer vanishes, Nakayama’s lemma applied to the finitely generated ideal \(\mathfrak n^j\) shows that this ideal is zero. Consequently \(A\) already has length \(\mu\). The converse follows from the same filtration. Polynomial, localized algebraic, and convergent power-series computations give the same truncated quotient. This proves the first assertion.

For constructibility, let \(B\) be any irreducible locally closed subvariety of \(U_\mu\); replace it first by a smooth dense open. Consider its universal hypersurface with the origin section. After another dense open restriction on \(B\), there is a Zariski neighborhood of this section on which the relative singular locus is exactly the section. To see this, isolatedness in each fiber implies that the section is an irreducible component of the relative critical locus: a larger component containing it would have a positive-dimensional fiber at a general section point, by the dimension inequality. The other components meet the section over proper closed subsets of \(B\). Remove those subsets from \(B\), then remove those components from the total space. The resulting total hypersurface is Cohen–Macaulay and regular off a subset of codimension two, hence normal. Its fibers have dimension two, so miracle flatness over the smooth base gives flatness; it is smooth over \(B\) off the section.

Take a projective resolution, isomorphic off the section, which principalizes its ideal and has simple normal crossing reduced exceptional divisor \(E\). Characteristic-zero resolution of singularities and principalization provide this construction; see (Włodarczyk 2005, Theorems 1.0.1–1.0.3). Generic smoothness, applied to the resolution space and every intersection of components of \(E\), permits a further dense open restriction on \(B\) such that the resolution is smooth over \(B\) and \(E\) is a relative simple normal crossing divisor. Nondominating exceptional strata are discarded in making this restriction. Since the resolution is projective and all of \(E\) lies over the section, \(E\) and all intersections of its components are proper over \(B\).

The fibers are now good resolutions of the surface germs at the origins. Their weighted exceptional dual graphs are locally constant in the complex topology, up to permutation of the vertices. In fact smooth proper variation preserves the connected components and genera of the exceptional curves, as well as their intersection points and incidence. Their self-intersections are degrees of the normal line bundles obtained by restricting the divisor line bundles on the resolution space. These degrees are locally constant in smooth proper curve families. If a total divisor has several components in a fiber, they are disjoint because that relative divisor is smooth, so the same degree calculation applies to each component.

The weighted graph determines \(\beta\). It determines the exceptional intersection matrix, and adjunction gives \(K\cdot E_i=2g(E_i)-2-E_i^2\) for each exceptional curve. Thus it determines all intersections of \(K+E\) with the exceptional curves, and hence its relative Zariski decomposition and the square of its numerical exceptional representative. An irreducible smooth complex base is connected, so \(\beta\) is constant on this dense open of \(B\). Noetherian induction on the remaining closed subsets yields a finite stratification of \(U_\mu\) on which \(\beta\) is constant. ◻

Proof. Take \(N\geq\mu+1\) and consider the holomorphic coefficient map \(a:\Delta\to V_N\), \(a(t)=j^Nf_t\). By Lemma 4, it takes values in \(U_\mu\) and preserves the analytic type of every individual surface germ, hence its Wahl invariant. Write \(a_0=a(0)\) and \(\beta_0=\beta(V(a_0),0)\).

If constancy fails on every smaller disk, \(a_0\) lies in the closure of the constructible set \[Z_{\ne}=\{g\in U_\mu:\beta(V(g),0)\ne\beta_0\}.\] The closure here may be taken in the Zariski topology: an accumulating sequence of coefficient vectors already forces this inclusion. Choose an irreducible locally closed subset of \(Z_{\ne}\) whose closure in \(U_\mu\) contains \(a_0\). After restricting to an affine neighborhood of \(a_0\), its closure is an irreducible affine variety \(Z\) of positive dimension, containing a dense open \(Z^\circ\subset Z_{\ne}\).

There is an algebraic curve in \(Z\) through \(a_0\) whose general point lies in \(Z^\circ\). For completeness, choose a finite Noether normalization \(\pi:Z\to\mathbb A^d\) and translate so that \(\pi(a_0)=0\). The image of \(Z\setminus Z^\circ\) is a proper closed subset. Choose a line through \(0\) not contained in that image. Its inverse image is cut out in \(Z\) by \(d-1\) equations, so every irreducible component through \(a_0\) has dimension at least one. Finiteness of \(\pi\) gives the opposite inequality and prevents a one-dimensional component from mapping to a point. Such a component therefore dominates the line and has general point in \(Z^\circ\). Normalize it, choose a point over \(a_0\), and remove the finitely many other points needed to obtain the desired marked smooth curve \(C\). The inclusion into coefficient space supplies regular polynomial coefficients on \(C\).

Every member of this curve lies in \(U_\mu\), its marked fiber is the polynomial \(a_0\) with non-log-canonical surface germ, and its general Wahl invariant differs from \(\beta_0\). This contradicts the assumed algebraic assertion. Therefore the invariant is constant in the original analytic family after shrinking. ◻

Proof. First choose a projective log resolution \(g:Z\to X\) which principalizes the ideal of \(\Gamma\) and resolves the union with \(X_0\). We choose it to be an isomorphism over \(X\setminus\Gamma\): that open set is smooth and its special fibre is already smooth. Thus \(Z\) is smooth, \(g^{-1}(\Gamma)\) has divisorial support, and the union of this support with the support of \(Z_0\) has simple normal crossings. All exceptional divisors of \(g\) lie over \(\Gamma\). By generic smoothness for \(Z\to C\) and for the finitely many strata of its exceptional divisor, we may delete finitely many base points other than \(0\) so that \(Z\to C\) is smooth away from \(0\) and every fibre support together with the exceptional divisor has simple normal crossings. After the same shrinking, every vertical exceptional divisor lies over \(0\).

Let \(H_Z\) be the reduced horizontal exceptional divisor of \(g\), and write \(Z_0=\sum a_jZ_j\). The preceding shrinking makes \((Z_0)_{\mathrm{red}}+H_Z\) simple normal crossing, with \(H_Z\) relatively simple normal crossing over \(C\setminus\{0\}\). Choose a positive integer \(b\) divisible by every \(a_j\), and a parameter \(\tau\) at \(0\) regular with no other zero on a Zariski neighbourhood. The cover \(s^b=\tau\), with its smooth normalization, is finite, ramified to order \(b\) over \(0\), and étale elsewhere. We again write \(X\), \(C\) and \(\Gamma\) for the pulled-back objects.

The pulled-back target \(X\) is still normal and Gorenstein. Indeed the chosen curve cover is locally \(s^b=\tau\), so the target is locally the hypersurface \(A[s]/(s^b-\tau)\) over a Gorenstein ring \(A\). It is therefore Gorenstein and satisfies Serre’s condition \(S_2\). Off the section it is smooth over the new smooth curve, while the section has codimension two. Thus it satisfies \(R_1\) and is normal. Its fibres remain integral, so the flat total space over the connected covering curve is integral.

Apply Proposition 31 to \(Z\) with marking \(H_Z\) and the chosen cover. It gives a smooth variety \(V\), projective over the normalized pullback of \(Z\), whose special fibre is reduced and simple normal crossing together with the transformed horizontal marking. The construction is an isomorphism on the normalized pullback away from \(0\), where the family and marking already have these properties. Over \(X\setminus\Gamma\), the original morphism was smooth and the marking was empty, so the same assertion shows that \(V\to X\) is an isomorphism there. Thus all exceptional divisors still lie over \(\Gamma\), and every vertical exceptional divisor lies over \(0\).

Let \(H_V\) be the reduced sum of the horizontal exceptional prime divisors on \(V\). Since the modification is an isomorphism away from \(0\), \(H_V\) is precisely the transformed reduced marking \(H_Z\). For every \(c\), the pair \((V,H_V+V_c)\) is simple normal crossing. Consequently \((V,H_V)\to C\) is a dlt morphism. The relative minimal model program of (Kollár and Mori 1998, Theorem 7.9) applies to the threefold pair \((V,H_V)\) over the flat target \(X\to C\). All its contractions and flips are projective over \(X\), it terminates, and the resulting morphism \((Y,H)\to C\) is again dlt. Its alternative of a Mori fibre space cannot occur: such a space would factor the dominant morphism from the threefold to \(X\) through a variety of dimension less than three. We therefore obtain \(K_Y+H\) nef over \(X\).

The program consists of divisorial contractions and flips, so it extracts no divisors. Every surviving horizontal exceptional prime consequently retains coefficient one in \(H\), and every vertical exceptional prime still lies over \(0\). Each step is an isomorphism over \(X\setminus\Gamma\), since there were no relative curves to contract there. The usual preservation of \(\mathbb Q\)-factoriality in this program gives the asserted \(\mathbb Q\)-factorial model.

For completeness, the scheme-theoretic fibre is precisely the boundary fibre used above. The pullback of a local parameter at \(c\) is a nonzerodivisor on the integral normal variety \(Y\), so \(Y_c\) is Cartier. Its order at every prime component is one: the corresponding divisorial valuation occurred on \(V\), where the fibre was reduced, and the program did not extract divisors. A principal divisor with these orders on a normal variety defines a reduced subscheme. Locally, if \(u^n\in(t)\), the height-one valuations show that \(u/t\) is regular at every height-one prime; normality then gives \(u/t\in \mathcal O_Y\). Thus \((t)\) is radical.

For \(c\ne0\) there are no vertical exceptional divisors, and \(X_c\) is integral. Its strict transform is therefore the only component of \(Y_c\). This component is normal by the dlt property. At the generic point of each curve in \(H\cap Y_c\), the pair is simple normal crossing; hence the Cartier fibre cuts \(H\) with multiplicity one, giving the reduced Weil divisor \(H|_{Y_c}\). Over \(0\) there is again just one strict transform of \(X_0\); all other components are exceptional. They map into \(\Gamma\cap X_0\) and are projective, because \(h\) is projective. This proves all the assertions. ◻

Proof. Shrink \(U\) so that the only boundary components present are those through \(z\), and choose regular functions \(g_j\) with \(\operatorname{div}(g_j)=m_jB_j\), where \(m_j\) is a positive integer. Normalize \(U\) in a field extension obtained by adjoining roots \(a_j^{m_j}=g_j\) for all \(j\). Normalization is finite here, and taking one field extension gives a finite surjective map of some degree \(d\). This is the usual local boundary-Cartier cover; see also (Kollár 2023, Proposition 11.25).

At a codimension-one point, write each \(g_j\) as a unit times the \(m_j\)-th power of a local equation, or as a unit if \(B_j\) is absent. Adjoining its root is therefore unramified there. The same argument shows that the cover is étale wherever all the \(B_j\) are Cartier, in particular over \(U\setminus B\) and over the simple normal crossing locus of \((U,B)\). Consequently \(\operatorname{div}(a_j)=\rho^*B_j\) has coefficient one on every prime in its support. Thus the total pullback is reduced and Cartier.

Put \(\widetilde B=\rho^*B\). The codimension-one ramification formula gives \[K_{\widetilde U}+\widetilde B =\rho^*(K_U+B)\] with compatible canonical divisors. More generally, if a divisorial valuation \(F\) upstairs restricts to \(e\) times a normalized divisorial valuation \(E\) downstairs, then \[ A(F,\widetilde U,\widetilde B)=eA(E,U,B), \tag{5}\] where \(A\) denotes log discrepancy. To see the formula, choose a birational model on which \(E\) is a divisor and normalize it in the upper function field. At the generic point of \(F\), the ramification divisor has coefficient \(e-1\). Hence the discrepancy coefficients satisfy \(a_F=e a_E+(e-1)\), giving (5). Restriction of a divisorial valuation through a finite extension is divisorial, since its value group has finite index and its residue field has the same transcendence degree. This is the finite-cover discrepancy formula of (Kollár and Mori 1998, Proposition 5.20); see also (Kollár 2023, 11.22–23).

It follows that the upper pair is log canonical and that every log canonical center upstairs maps onto a log canonical center downstairs. Each lower center meets the simple normal crossing locus because the lower pair is dlt. The cover is étale there, so each upper center also meets the simple normal crossing locus. This is the dlt criterion. We use the standard facts that the log canonical centers of a dlt pair are the normal strata of its coefficient-one boundary, with the expected codimensions, and that intersections of log canonical centers are unions of such centers; see (Fujino 2007, Proposition 9.2 in the author draft) and (Kollár 2023, Definition 11.5 and 11.10).

Fix \(y\in\rho^{-1}(z)\). There is at least one upper boundary prime through \(y\) over each lower boundary prime through \(z\). Indeed each root \(a_j\) vanishes at \(y\), so its Cartier divisor contains \(y\). Primes over distinct lower primes are distinct, since a finite map preserves their dimensions. Thus at least two upper primes meet at \(y\). If at least three meet, the dlt stratum theorem makes \(y\) a zero-dimensional log canonical center. The dlt criterion then says that the upper pair is simple normal crossing at \(y\).

Suppose instead that precisely two upper boundary primes \(T_1,T_2\) meet at \(y\). They lie over distinct lower primes; hence each agrees near \(y\) with its entire reduced Cartier pullback and is itself Cartier there. The surface \(T_1\) is normal by dlt adjunction, and a normal surface is Cohen–Macaulay. Therefore the Cartier curve \(C=T_2|_{T_1}\) has no embedded points. It is generically reduced, because the pair is simple normal crossing at the generic points of its curve strata, so it is reduced. Each irreducible component of \(C\) is a normal curve, hence smooth. Two such components cannot meet at \(y\): their intersection would make \(y\) a log canonical center, forcing simple normal crossings there, whereas a simple normal crossing threefold pair with only two boundary branches has no zero-dimensional log canonical center. Consequently \(C\) is a smooth curve near \(y\).

Let \(R=\mathcal O_{\widetilde U,y}\) and let \(a,b\) define \(T_1,T_2\). Since \(R/(a,b)\) is a regular local ring of dimension one, \[\operatorname{edim}R \le \operatorname{edim}(R/(a,b))+2=3=\dim R.\] Thus \(R\) is regular. Moreover \(a,b\) have independent images in its cotangent space, so they are part of a regular system of parameters. The two boundary components are transverse. This proves the claimed simple normal crossing assertion in every case. ◻

Proof. Use the finite cover of Lemma 14, and let \(d\) denote its degree. At each point \(y\) over \(z\), choose analytic simple normal crossing coordinates. As the cover is unramified in codimension one, the vertical multiplicities remain one. After absorbing a unit into one coordinate, the pulled-back function and horizontal divisor are \[f\circ\rho=z_1\cdots z_r, \qquad \rho^{-1}H=\{z_{r+1}\cdots z_{r+h}=0\}, \qquad r\ge1,\quad r+h\ge2.\] When \(h=0\), the horizontal divisor is empty. For \(r\ge2\), the local Milnor fiber of the coordinate product has the homotopy type of \((S^1)^{r-1}\) and therefore Euler characteristic zero. The same is true after restricting to any intersection of horizontal coordinate hyperplanes. Inclusion–exclusion therefore gives zero for the complement of the horizontal boundary. For \(r=1\), each such restricted Milnor fiber is a polydisc, with Euler characteristic one. Here \(h\ge1\), so inclusion–exclusion gives \(\sum_{J\subseteq\{1,\ldots,h\}}(-1)^{|J|}=(1-1)^h=0\). These computations may be made on small adapted polydiscs, or on Milnor balls, with the same local nearby Euler values.

Let \(\widetilde\alpha=\mathbf1_{\widetilde U\setminus\rho^{-1}H}\). Over \(U\setminus V\), its finite pushforward is \(d\mathbf1_{U\setminus H}\): away from \(H\) the cover is étale of degree \(d\), and on \(H\) all the terms are zero. Since nearby cycles only use the restriction off \(V\), proper compatibility gives \[d\,\psi_f(\mathbf1_{U\setminus H})(z) =\sum_{y\in\rho^{-1}(z)} \psi_{f\circ\rho}(\widetilde\alpha)(y)=0.\] This is an identity of constructible Euler functions; no triviality of the covering local system is needed. Since \(d>0\), the result follows. ◻

Proof. Write \(\lambda=\psi_q(\mathbf1_{Y\setminus H})\) using a local coordinate on \(C\) at \(0\). Proper compatibility gives \[\chi_c(Y_t\setminus H)=\int_{Y_0}\lambda\,d\chi_c.\] At every point lying on at least two components of \(Y_0+H\), Lemma 15 gives \(\lambda=0\). Every remaining point lies in exactly one \(S^\circ\). The surface \(S\) is normal by dlt adjunction, so its singular points form a finite set. At a smooth point of \(S^\circ\), the local reduced Cartier fiber is precisely \(S\). Its regular local ring is a quotient of the ambient local ring by the base parameter, so the ambient ring is regular and that parameter is part of a regular system of parameters. Thus \(q\) is smooth there and \(\lambda=1\).

At a singular point \(p\in S^\circ\), a neighborhood misses \(H\) and all other special-fiber components. Flatness and the hypothesis on the nearby open fibers therefore give a smoothing of the normal isolated surface singularity \((S,p)\). If \(M_p\) is its Milnor fiber, then \(\lambda(p)=\chi(M_p)\ge1\) by the assumed interior smoothing statement. Integrating the disjoint open surfaces and these finitely many corrections yields (6), with \[\delta=\sum_{S\subset Y_0} \sum_{p\in\operatorname{Sing}(S)\cap S^\circ} \bigl(\chi(M_p)-1\bigr)\ge0.\] ◻

Proof. Choose \(r>0\) such that \(rL\) is Cartier. Flatness and properness imply that \(\chi(Y_c,\mathcal O_Y(nrL)|_{Y_c})\) is constant in \(c\) for every integer \(n\) (The Stacks Project Authors 2026, Lemma 36.32.2). This is a polynomial in \(n\) of degree at most two, whose quadratic coefficient is \(r^2(L|_{Y_c})^2/2\). Its leading term is the sum of the leading terms on the two-dimensional components, weighted by their generic lengths (The Stacks Project Authors 2026, Lemmas 33.45.1, 33.45.2 and 33.45.6). Comparing these coefficients and dividing by \(r^2\) proves the assertion. ◻

Proof. The integral threefold \(Y\) dominates the smooth curve, so it is flat: its local rings are torsion-free modules over the corresponding discrete valuation rings. Its fibres have pure dimension two, and it is projective over \(C\). Lemma 19 and the assumed vanishing give, for \(c\ne0\), \[(K_{Y_c}+B_c)^2 = (L|_{Y_c})^2 = (L|_{S_0})^2+\sum_a(L|_{S_a})^2 = (K_{S_0}+B_{S_0})^2.\] The restrictions of \(L\) are nef over the respective \(X_c\), since \(L\) is \(h\)-nef. The full-boundary hypotheses therefore allow Lemma 5 on both the central and noncentral birational models. The displayed equality and \(K_{X_c}^2=K_{X_0}^2\) yield the claimed equality of local invariants. ◻

Proof. The zero-fiber family is a flat deformation of a normal Gorenstein surface singularity. Each fiber has just the one singular point in the chosen representative. Thus constancy of the invariant at the section is also constancy of the sum over the singularities of the fiber, as used in Okuma’s notation.

We first check the small-deformation setup in (Okuma 2004, secs. 3–4). Resolve the analytic total space, and then principalize the pulled-back ideal of the section on the smooth resolution (Włodarczyk 2009, Theorems 2.0.1 and 2.0.3, Lemma 4.0.3). The composite is proper and isomorphic off the section: the total space is smooth there and the pulled-back ideal is the unit ideal there. Its full reduced inverse image of the section is a simple normal crossing divisor. The exceptional support is proper over the disk. The critical loci for the parameter map on the resolution and on all closed intersections of exceptional components lie in this proper support. These intersections are smooth. On each of them a critical component has constant parameter value; an intersection contained in a fiber also has a single value. Properness makes their images closed analytic subsets of the disk. After shrinking, their only possible value is zero. Hence the resolution over the punctured disk is smooth, with relative normal crossing exceptional divisor. These relative normal-crossing charts give a locally trivial deformation of the exceptional divisor there. This supplies the punctured-disk weak simultaneous resolution required by Okuma’s setup.

Okuma’s Theorem 1.2 (Okuma 2004) now gives a simultaneous resolution after a finite base change, with minimal semigood fiber resolutions and a reduced divisor \(E\) whose every restriction is the full reduced exceptional divisor over the distinguished non-log-canonical section. That section is our origin section, because there are no other singularities. Take the normalization of a branch of the finite base change. A local parameter then puts its map to the original disk in the form \(t=u^e\). Flatness and smooth fibers make the resulting resolved family smooth over this disk. Formation of the Cartier divisor restrictions commutes with this further base change. In particular their reducedness also holds at \(u=0\). The divisor \(E\) contains no fiber, and its restrictions are effective Cartier divisors; hence it is a relative effective Cartier divisor. Its support is exactly \(b^{-1}(\{0\}\times\Delta_u)\).

It remains to prove relative normal crossings, since fiberwise normal crossings alone do not give that conclusion. Choose the fixed closed Milnor ball \(B\) from the preliminary Euler calculation and set \(N_B=b^{-1}(X'\cap(B\times\Delta_u))\). Its projection to the disk is a proper smooth submersion with boundary. Near the sphere boundary the modification is an isomorphism, and the required submersion follows from the uniform boundary transversality of the original family. Consequently \(\chi((N_B)_u)\) is constant. The complement of \(E_u\) in this fiber is isomorphic to \((X_{u^e}\cap B)\setminus\{0\}\). The preliminary identity \(\chi(X_t\cap B)=1\) and additivity, using compact supports on the complements, give \[\chi(E_u)=\chi((N_B)_u),\] so the compact reduced nodal curves \(E_u\) have constant Euler characteristic.

Here is the local consequence of that constancy. The divisor \(E\) is Cartier in the smooth total space \(N\). At a smooth point of \(E_0\) it is already smooth over the parameter. At each node of \(E_0\), the holomorphic Morse lemma with parameter gives relative coordinates in which its equation is \[z_1z_2=a(u),\qquad a(0)=0.\] There are finitely many nodes. Choose disjoint small neighborhoods of them with transverse boundaries. Smoothness over the parameter near the remaining points of \(E_0\), together with properness of \(E\to\Delta_u\), excludes new nodes outside these neighborhoods after shrinking the disk. The remainder is a proper smooth family with boundary, so its topology is unchanged. Inside a node neighborhood a crossing has Euler characteristic one, whereas a smoothing is an annulus and has Euler characteristic zero. Thus for general small \(u\), \[\chi(E_u)=\chi(E_0)- \#\{\text{nodes whose smoothing function }a\text{ is not identically zero}\}.\] No global labeling of components is needed; the calculation includes self-intersection nodes. Since the Euler characteristic is constant, every smoothing function vanishes identically. The resulting charts, together with the charts at the smooth points, prove relative normal crossings near \(E_0\). Properness of \(E\to\Delta_u\) extends this conclusion over a smaller disk. ◻

Proof. On the ambient tube with the origin section removed, \((G,u)\) is a smooth submersion, including on the sphere boundary. Extend each \(v_a\), \(a\in\{1,i\}\), to a smooth ambient field \(V_a\) satisfying \[du(V_a)=a,\qquad dG(V_a)=0, \qquad V_a\text{ tangent to the sphere boundary}.\] Here the prescribed extension along \(Z^*\) can be made in submersion coordinates, with boundary coordinates where necessary. Equivalently, subtract a preliminary lift and extend the remaining section of the kernel bundle of \(d(G,u)\) from the closed submanifold \(Z^*\) of the axis complement. A partition of unity patches these extensions: the specified projections, boundary tangency, and prescribed values on \(Z^*\) are affine linear conditions. There is no assertion that \(V_a\) extends to the axis.

For \(u\in D\), integrate \(\operatorname{Re}(u)V_1+\operatorname{Im}(u)V_i\) for time one, starting at \((x,0)\). For \(x\ne0\) on the zero level, existence for the whole interval is the hypothesis. For \(G(x,0)=c\ne0\), the trajectory stays in the compact set \[\{(y,s):y\in B,\ s\in[0,u],\ G(y,s)=c\},\] which misses the axis. Smooth continuation and boundary tangency give existence for the whole interval here as well. Reverse transport gives the inverse on \(T_u\setminus\{0\}\). Both maps depend smoothly on their arguments off the axis, also in charts at the sphere boundary, and preserve the value of \(G\).

Set \(H_u(0)=0\). To prove joint continuity there, let \(x_j\to0\) and \(u_j\to u\). A subsequential limit \(y\) of \(H_{u_j}(x_j)\) exists in \(B\), and function preservation gives \(G(y,u)=0\). If \(y\ne0\), smooth dependence of reverse transport near \((y,u)\) would imply that \(x_j\) tends to its nonzero endpoint over \(0\), a contradiction. Thus every such limit is \(0\). The same argument proves inverse continuity: if \(y_j\in T_{u_j}\) tends to \(0\), any nonzero subsequential limit of \(H_{u_j}^{-1}(y_j)\) contradicts smooth forward transport. Consequently the maps and inverses are jointly continuous everywhere. Reverse transport also proves that each \(H_u\) is onto exactly \(T_u\). Finally, the tube interiors form open neighborhoods of the respective origins and their union is an open neighborhood of the origin section. This proves the last assertion. ◻

Proof. Write \(\rho(x,u)=\|x\|^2\). We claim that \((u,\rho)\) is a real submersion on the punctured zero locus sufficiently near \((0,0)\). Otherwise choose points \((x_j,u_j)\to(0,0)\) witnessing failure. After taking a subsequence, the unit spatial vectors \(x_j/\|x_j\|\) converge to \(v\), the real tangent spaces of the total zero locus converge to \(L\), and the real tangent spaces of its fibers converge to \(Q\). Whitney (a) implies that \(L\) contains the two real base directions. The fiber tangent spaces have real dimension \(2n-2\) and lie in \(\ker du\), so \[Q=L\cap\ker du:\] the inclusion holds by passage to limits, and both sides have dimension \(2n-2\). Whitney (b), applied to the secants from \((0,u_j)\), gives \((v,0)\in L\), hence \((v,0)\in Q\). On the other hand, failure of submersivity says that \(d\rho\) vanishes on each fiber tangent space. It follows that \((v,0)\) is orthogonal to \(Q\), a contradiction to \(\|v\|=1\).

Take a sufficiently small closed ball and parameter disk inside this submersion neighborhood. Smooth local right inverses to \(d(u,\rho)\), patched by a partition of unity, give lifts \(v_1,v_i\) of the parameter directions with \(d\rho(v_a)=0\). Their trajectories preserve the positive spatial radius. Along any fixed parameter segment they therefore remain in a compact set disjoint from the axis, so transport is complete in both directions. These fields are tangent to the sphere boundary. The zero levels are transverse to that sphere; by compactness the same holds for all sufficiently small nonzero levels and parameters, after shrinking. Lemma 23 applies. ◻

Proof. Choose the ball and a smaller parameter disk within representatives of \(b\). Above the closed ball, \(q\) is a smooth submersion with boundary: near the sphere, \(b\) is an isomorphism and the fibers meet the sphere transversely. Relative normal-crossing coordinates give local lifts of the two real parameter directions tangent to every branch of \(E\). Away from \(E\) use submersion coordinates, and near the sphere use boundary submersion coordinates. A partition of unity gives smooth lifts on \(M\) tangent both to \(E\) and to the sphere boundary.

Properness makes the inverse image of the closed ball over each compact parameter segment compact. The smooth lifts are therefore complete along that segment in both directions. Their flows preserve \(E\) and its complement: tangency preserves each local branch, and uniqueness applied in both time directions prevents a trajectory in the complement from reaching \(E\). Push the fields down through the isomorphism off \(E\). These are precisely the complete fields on the punctured zero family required by Lemma 23. ◻

Proof. Put \(\zeta=\exp(2\pi i/e)\). For real \(s\ge0\) sufficiently small, \[J_s=H_{\zeta s}^{-1}\circ H_s:T_0\longrightarrow T_0\] is a homeomorphism preserving \(f_0=F(\cdot,0)\), and \(J_0=\mathrm{id}\). These compositions have exact common domains: the two target tubes \(T_s\) and \(T_{\zeta s}\) coincide since \(s^e=(\zeta s)^e\). The families \(J_s\) and \(J_s^{-1}=H_s^{-1}\circ H_{\zeta s}\) are jointly continuous.

For \(t=r^e\exp(i\theta)\), \(r>0\) and \(0\le\theta\le2\pi\), define \[\phi_t =H_{r\exp(i\theta/e)}\circ J_{r\theta/(2\pi)}.\] The two values at the angular seam agree, because \[H_{\zeta r}\circ J_r=H_r.\] Figure 1 displays this endpoint identity.

The inverse is \[\phi_t^{-1} =J_{r\theta/(2\pi)}^{-1}\circ H_{r\exp(i\theta/e)}^{-1},\] and its seam identity is \(J_r^{-1}\circ H_{\zeta r}^{-1}=H_r^{-1}\). Thus both maps glue. At \(r=0\) every constituent map is the identity, independently of \(\theta\); joint continuity of the constituent families and their inverses therefore gives joint continuity on the polar disk with its central circle collapsed. Set \(\phi_0=\mathrm{id}\).

For every \(t\), the source is the same tube \(T_0\) and the target is \(\{x\in B:|f_t(x)|<\eta\}\). Moreover, \[f_t\circ\phi_t=f_0\circ J_{r\theta/(2\pi)}=f_0, \qquad \phi_t(0)=0.\] Restricting to the interiors gives the asserted homeomorphism germ \((x,t)\mapsto(\phi_t(x),t)\) over the original disk. ◻

Proof. Suppose first that \(\operatorname{mult}_0(f_0)=2\). Choose a spatial direction in which the second derivative of \(f_0\) is nonzero. The implicit function theorem solves the corresponding first-derivative equation as \(z=a(y,t)\), where \(y\in\mathbb C^2\) and \(a(0,t)=0\). Translating \(z\) by \(a(y,t)\) and taking a holomorphic square root of the resulting nonvanishing quadratic coefficient gives biholomorphic coordinates, depending holomorphically on \(t\) and preserving the origin, in which \[f_t=z^2+g_t(y).\] The Jacobian quotient is then the Jacobian quotient of \(g_t\), so \(g_t\) has an isolated critical point and \(\mu(g_t)=\mu(f_t)\). Equimultiplicity for this plane family gives constant \(\mu^*(g_t)\), whose two entries are \(\operatorname{mult}_0(g_t)-1\) and \(\mu(g_t)\). Teissier’s criterion and Corollary 24 give a joint right-trivialization of \(g_t\). Extend its maps by the identity in \(z\) and conjugate by the coordinate changes at \(0\) and at \(t\). This gives the required right-trivialization of \(f_t\), with central map the identity.

Now suppose \(X_0\) is log canonical. Adjunction and inversion of adjunction identify this condition with log canonicity of \((\mathbb C^3,X_0)\) (Kawakita 2007); the different on the normal Cartier hypersurface in a smooth ambient space is zero. These single-germ algebraic statements also apply to the analytic germ by its analytic right-equivalence to a sufficiently high polynomial jet. Put \(m=\operatorname{mult}_0(f_0)\). On the blowup of the origin the exceptional plane has discrepancy \(2-m\), hence \(m\leq3\). Since the origin is a critical point, \(m\geq2\), and the case \(m=2\) is already settled.

For \(m=3\), let \(\widetilde X_0\) be the strict transform and \(P\simeq \mathbb P^2\) the exceptional plane. The crepant boundary on the blowup is \(\widetilde X_0+P\). Adjunction to \(P\) shows that \((P,C)\) is log canonical, where \(C=\widetilde X_0|_P\) is the tangent cubic with its scheme-theoretic multiplicities. A boundary component of a log canonical pair has coefficient at most one. Thus \(C\) is reduced.

Equimultiplicity keeps the order of \(f_t\) equal to three. Choose a projective line transverse to \(C\) at three distinct smooth points. This condition persists for the cubic initial forms of \(f_t\) when \(t\) is small. The corresponding plane sections, and hence general plane sections, are ordinary triple points. Their Milnor number is four: the two quadratic initial forms of their partial derivatives have no common projective zero, so their local intersection multiplicity is \(2\cdot2\). Consequently \[\mu^*(f_t)=(2,4,\mu(f_0))\] throughout a sufficiently small disk. Teissier’s criterion gives Whitney regularity, and Corollary 24 supplies the right-trivialization. ◻

Proof. On the ambient complement of \(A\), put \[k_a=-aG_u\, \frac{(\overline{G_{x_1}},\ldots,\overline{G_{x_n}})} {\|d_xG\|^2},\qquad K_a=(k_a,a).\] These smooth real fields satisfy \(dG(K_a)=0\). Equation (8) says precisely that \(k_a\to0\) at \(A\). On \(X^*\), the difference \(s_a=w_a-k_a\) is a section of the complex vector bundle \(K=\ker d_xG\), and it too tends to zero at \(A\).

We extend this correction while preserving its vanishing at the section. The smooth submanifold \(X^*\) is closed in the ambient complement of \(A\). Choose a smooth tubular retraction \(r:U\to X^*\), with a variable tube radius small enough that \[\|z-r(z)\|<\tfrac12\operatorname{dist}(r(z),A) \qquad(z\in U).\] Choose a smooth function \(\chi\) on the axis complement, equal to one on \(X^*\), taking values in \([0,1]\), and with support in \(U\). If \(\pi_{K_z}\) denotes Hermitian orthogonal projection onto \(K_z\), define \[\widetilde s_a(z)=\chi(z)\pi_{K_z}(s_a(r(z)))\quad(z\in U), \qquad \widetilde s_a=0\quad\text{outside }U.\] This is a smooth section of \(K\) agreeing with \(s_a\) on \(X^*\). If \(z\) approaches a point of \(A\) within \(U\), the displayed tube bound implies that \(r(z)\) approaches the same point. Since the projection has norm at most one, \(\widetilde s_a(z)\to0\) as well. Thus \[W_a=(k_a+\widetilde s_a,a)\] extends continuously across \(A\) by \(W_a=(0,a)\). It is smooth off \(A\), agrees with \(v_a\) on \(X^*\), and annihilates \(dG\) everywhere. At points of \(A\) the last assertion follows from \(G(0,u)=0\).

Peano’s existence theorem gives local solutions of each real ODE \(\dot z=W_a(z)\). They are unique off \(A\) because the field is smooth there. A solution meeting \(A\) has \(G=0\) throughout, by the chain rule. If it left \(A\), a connected interval on which it lies off \(A\) would give a trajectory of \(v_a\) with a finite-time endpoint on \(A\), contrary to the hypothesis. The same argument in reverse time excludes arrival from outside \(A\). On \(A\) the equation is \(\dot x=0\), \(\dot u=a\), so uniqueness holds there too.

For clarity, continuity of these local flows also holds at \(A\). On a fixed compact neighborhood the continuous field is bounded. Initial points in a smaller neighborhood therefore have solutions on a common short interval, all contained in that compact neighborhood. For a convergent sequence of initial points, the solutions are uniformly bounded and equicontinuous. Every uniformly convergent subsequence satisfies the integral equation for the limiting initial point. Uniqueness identifies its limit, and hence the whole sequence converges. The same argument applies to negative times. Uniqueness and reverse flow give continuous local inverse maps. Consequently these are local flows by homeomorphisms, jointly continuous in the initial point and time.

Write \(u=s+it\). Starting at \((x,0)\), first follow \(W_1\) for time \(s\) and then \(W_i\) for time \(t\). Uniform short-time existence permits both steps for all \(x\) and \(u\) in common smaller neighborhoods. The endpoint has parameter \(u\) and defines \(H_u(x)\). Each step preserves \(G\) and the section. Reverse the two steps, in the opposite order, to obtain the joint inverse. The reverse operations are defined on an open neighborhood of the section. Within that neighborhood the image of our open source is its inverse image under the reverse map, so this image is open. At \(u=0\) both times are zero. This proves all the assertions. ◻

Proof. Use the Hermitian inner product linear in its first entry, and set \(q(x,u)=\pi_{K_{(x,u)}}x\). We first show that \[ \frac{\|q(x,u)\|}{\|x\|}\longrightarrow1 \qquad\text{on }X^*\text{ approaching }A. \tag{9}\] Along any sequence, pass to a subsequence on which \(x/\|x\|\) and \(K_{(x,u)}\) converge, to \(v\) and \(Q\) respectively. Equation (8) implies that the tangent spaces of the total zero set converge to \(Q\times\mathbb C\). Whitney condition (b), applied to the secants between \((x,u)\) and \((0,u)\), puts \((v,0)\) in this limiting space. Thus \(v\in Q\), which proves (9).

In particular \(q\ne0\) sufficiently near the section. With \(k_a\) as in the preceding proof, define \[w_a=k_a- \frac{\operatorname{Re}\langle k_a,x\rangle}{\|q\|^2}\,q, \qquad v_a=(w_a,a),\qquad a=1,i.\] The correction belongs to \(K\), so \(dG(v_a)=0\). Moreover \(\operatorname{Re}\langle q,x\rangle=\|q\|^2\), giving \(d\|x\|^2(v_a)=0\). Its norm is bounded by \(\|k_a\|\|x\|/\|q\|\); hence \(w_a\to0\) by (8) and (9). Every constant real combination preserves the positive radius of a trajectory in \(X^*\), so no such trajectory can reach \(A\) in finite time. Proposition 33 and Remark 34 apply. ◻

Proof. Relative normal-crossing coordinates give smooth lifts of the real parameter directions \(1,i\) tangent to every local branch of \(E\). Submersion coordinates supply lifts away from \(E\). A partition of unity patches these to smooth fields \(\widetilde v_1,\widetilde v_i\) on \(M\) with the same projections and tangencies. Through the isomorphism off \(E\), push them to fields \(v_a=(w_a,a)\) on \(X^*\).

These spatial parts tend to zero at \(A\). Indeed, properness supplies convergent subsequences of the unique preimages of any sequence in \(X^*\) tending to a point of \(A\). Every limit lies in \(E\). The spatial component of \(b\) vanishes on \(E\), so its differential on a field tangent to a branch of \(E\) is zero there. Continuity of \(db(\widetilde v_a)\) now proves \(w_a\to0\).

To verify the trajectory condition, suppose that an integral arc of a constant real combination of the \(v_a\) had a finite-time endpoint in \(A\) inside the representative. Its lift to \(M\setminus E\) solves the corresponding smooth combined field. A terminal segment of the downstairs arc has compact closure, whose inverse image is compact by properness. Smooth ODE continuation therefore extends the lifted arc to a point of \(E\). Tangency preserves \(E\), and uniqueness in both time directions prevents an arc outside \(E\) from reaching it. This contradiction proves the required condition. The continuous-lift criterion applies. ◻