Introduction

All varieties are defined over $\mathbb C$. A smooth projective variety is Fano if its anticanonical divisor is ample. The classification of smooth Fano threefolds by Iskovskikh, Mori and Mukai gives $105$ deformation families; we use the usual numbering, whose first integer is the Picard rank. Determining their K-polystable members is a finer problem, related to the existence of Kähler–Einstein metrics and not constant in every deformation family. We consider the following families; see [3]. Each family is treated in a separate part.

1. In Family 1.9, $X\subset\mathbb P^{11}$ is a codimension-two linear section of the five-dimensional adjoint $G_2$-variety in $\mathbb P^{13}$. It is a prime Fano threefold of genus ten, with $\operatorname{Pic}(X)=\mathbb Z[-K_X]$ and $(-K_X)^3=18$. We prove K-stability for every smooth member; the final assembly is in Section ??.

2. In Family 1.10, $X$ is the zero locus of three sections of $\bigwedge^2\mathcal U^\vee$ on $\operatorname{Gr}(3,7)$, where $\mathcal U$ is the tautological rank-three bundle. It is a prime Fano threefold of genus twelve with $(-K_X)^3=22$. Part ?? contains a symmetric counterexample to the unrestricted anticanonical-singularity formulation of Donaldson’s conjecture in [3, Conjecture 7.1], and partial classification results, including local estimates from ordinary point and line blow-ups, a nodal reduction for terminal Gorenstein degenerations, an exclusion of a nonterminal boundary family, strict monomial-valuation estimates, and an exact ribbon description of the section ring under the transverse width-two hypotheses.

3. In Family 2.5, $X$ is the blow-up of a smooth cubic threefold $V\subset\mathbb P^4$ along a smooth plane cubic $C=V\cap\mathbb P^2$. The hyperplanes containing this plane give a cubic-surface fibration $X\to\mathbb P^1$, and $(-K_X)^3=12$.

4. In Family 2.9, $X$ is the blow-up of $\mathbb P^3$ along a smooth curve of degree seven and genus five whose ideal sheaf is generated by three cubics. The cubic net gives a conic bundle $X\to\mathbb P^2$, and $(-K_X)^3=16$.

5. In Family 2.10, $X$ is the blow-up of a smooth complete intersection of two quadrics $V\subset\mathbb P^5$ along the smooth elliptic quartic $C=V\cap\mathbb P^3$. The hyperplanes containing this linear subspace give a quartic del Pezzo fibration $X\to\mathbb P^1$, and $(-K_X)^3=16$.

6. In Family 2.11, $X$ is the blow-up of a smooth cubic threefold $V\subset\mathbb P^4$ along a line $\ell$. Projection from the line gives a conic bundle $X\to\mathbb P^2$, and $(-K_X)^3=18$. Both possible normal bundles, $\mathcal O_\ell^{\oplus2}$ and $\mathcal O_\ell(1)\oplus\mathcal O_\ell(-1)$, are included.

7. In Family 2.12, $X$ is a smooth complete intersection of three divisors of bidegree $(1,1)$ in $\mathbb P^3\times\mathbb P^3$. Each projection blows up a smooth nonhyperelliptic sextic curve of genus three, and $(-K_X)^3=20$.

8. In Family 2.13, $X$ is the blow-up of a smooth quadric $Q\subset\mathbb P^4$ along a smooth sextic curve of genus two cut out on $Q$ by quadrics. The quadrics give a conic bundle $X\to\mathbb P^2$, and $(-K_X)^3=20$.

9. In Family 2.14, $X$ is the blow-up of the quintic del Pezzo threefold $V_5\subset\mathbb P^6$ along the smooth elliptic quintic cut out by two hyperplanes. Their pencil gives a quintic del Pezzo fibration $X\to\mathbb P^1$, and $(-K_X)^3=20$.

10. In Family 2.16, $X$ is the blow-up of a smooth intersection of two quadrics $V_4\subset\mathbb P^5$ along a smooth conic. Projection from the plane of the conic gives a conic bundle $X\to\mathbb P^2$, and $(-K_X)^3=22$.

11. In Family 2.17, $X$ is the blow-up of $\mathbb P^3$ along a smooth elliptic quintic. Its second contraction is the blow-up of a smooth quadric along a smooth elliptic normal quintic, and $(-K_X)^3=24$. Write $E,E'$ for the exceptional divisors over $\mathbb P^3$ and the quadric, respectively. For a point $p\in\mathop{\mathrm{Sing}}(E\cap E')$, let $\ell'_p$ be the fibre of $E'$ through $p$, and put $\Sigma_X=\{p\in\mathop{\mathrm{Sing}}(E\cap E'):E|_{\ell'_p}=3p\}$.

12. In Family 2.20, $X$ is the blowup of $V_5$ along a smooth twisted cubic. Projection from its span gives a conic bundle over $\mathbb P^2$, and $(-K_X)^3=26$. Its K-stable, K-polystable and strictly K-semistable members are classified in Part XI.

13. In Family 3.2, $X$ is a divisor of class $2\xi+\mathcal O(2,3)$ in $\mathbb P_{\mathbb P^1\times\mathbb P^1} (\mathcal O\oplus\mathcal O(-1,-1)^{\oplus2})$, where $\xi$ is the tautological class and its restriction to $X$ is irreducible. The projection is a conic bundle with cubic and sextic del Pezzo pencils, and $(-K_X)^3=14$.

14. In Family 3.5, $X$ is the blow-up of $\mathbb P^1\times\mathbb P^2$ along the graph of a degree-five morphism $\eta:\mathbb P^1\to\mathbb P^1$, with the source embedded as a smooth conic in $\mathbb P^2$. Here $(-K_X)^3=20$. The stability of its smooth members is classified below.

15. In Family 3.6, $X$ is the blow-up of $\mathbb P^3$ along a smooth elliptic quartic and a disjoint line. The plane and quadric pencils give del Pezzo fibrations of degrees five and six, and their product is a conic bundle over $\mathbb P^1\times\mathbb P^1$. Here $(-K_X)^3=22$.

16. In Family 3.7, $X$ is the blow-up of a smooth divisor $V\subset\mathbb P^2\times\mathbb P^2$ of bidegree $(1,1)$ along a smooth complete intersection of two members of $|\mathcal O_V(1,1)|$. The resulting pencil consists of sextic del Pezzo surfaces, and $(-K_X)^3=24$.

17. In Family 3.8, $X$ is the blow-up of $\mathbb P^1\times\mathbb P^2$ along the graph of a degree-four morphism $\mathbb P^1\to\mathbb P^1$, with its source embedded as a smooth conic in $\mathbb P^2$. Equivalently, the centre is the intersection of divisors of bidegrees $(0,2)$ and $(1,2)$. Here $(-K_X)^3=24$. Its K-polystable members are classified in Part XVI.

1.1 Description of the results

Every smooth Fano threefold in Family 1.9 is K-stable. We prove $\delta_p(X)\ge6/5$ away from all lines and $\delta_p(X)\ge72/71$ at every point on an ordinary line, and exclude all curve centres. The remaining points lie on special lines. At a marked point with nonzero quadratic coefficient $\mu$, an ordinary point extraction gives $\delta_p(X)\ge9/8$. When $\mu=0$, the weight-$(1,2,2)$ extraction has expected order $287/72$. A surface estimate treats one sector of valuations. For the other sector, we blow up the strict transform of the line and apply adjunction with both exceptional divisors. A normal-jet bound makes the boundary effective, and the resulting inequality $A_X(v)-(288/245)S_X(v)\ge11v(a)/35>0$ completes the proof. These point extractions require no threefold Zariski decomposition.

We prove K-stability of every smooth Fano threefold in Mori–Mukai Families 2.5, 2.9, 2.10, 2.11, 2.12, 2.13, 2.14, 2.16, 2.17, 3.2, 3.6 and 3.7. We obtain $\delta(X)\ge96/95$ in Family 2.9, $\delta(X)\ge320/319$ in Family 2.13, and $\delta(X)\ge176/171$ in Family 2.16. The proofs use Abban–Zhuang refinement on del Pezzo fibres, cubic surfaces, and exceptional surfaces of point or curve extractions. We compute expected vanishing orders from filtered section rings or bound them by explicit spaces containing the refined sections. Each family is treated in a separate part, after Part I, which gives the definitions, adjunction tools and worked examples. In Family 2.11 we treat both normal-bundle types of the blown-up line. Weak cubic sections handle the special reduced fibre. At the remaining smooth tangencies, a weight-$(1,1,2)$ point extraction and a valuation centred on the ramification ruling give the exact local threshold $216/167$. Double conic fibres have local threshold at least $144/127$. For Family 2.12, the published reduction to points of the intersection of the two exceptional divisors is completed using ordinary point blow-ups and a weight-$(2,1,1)$ extraction. Exact initial rings and monomial moment certificates control all tangent configurations. The weighted calculations identify the actual refined section spaces and do not require a Zariski decomposition on an extracted threefold. For Family 2.13, quartic del Pezzo sections treat reduced conic fibres. Blowing up the node on such a section gives the bound $160/159$. For double fibres, the actual weighted refinement consists of ideal-power series for a length-two subscheme of $\mathbb P^1\times\mathbb P^1$. A degeneration preserving the different, together with an explicit correction for the dimensions of these series, gives the bound $320/319$. For Family 2.14, the quintic del Pezzo pencil reduces the proof to four types of singular points and to lines on singular fibres. We treat both normal-bundle types of lines and both positions of their intersection with the elliptic quintic. The refined threshold can equal one in the $A_3$ case; a positive discrepancy term completes that case. For Family 2.16, the known reduced-fibre estimates leave double lines. Their weighted extraction has expected order $57/22$. The refined series consists of powers of the ideal of one point on the different, and a degeneration preserving that different gives the local bound $88/85$ on every double fibre. An independent GIT proof for Family 2.16 computes the CM class $6\zeta+20\xi+66\eta$, proves stability of every smooth pair by the Hilbert–Mumford criterion, and uses relative reconstruction and CM-degree minimization to obtain K-stability.

For Family 2.17, two line extractions reduce the problem to a finite triple-contact locus. A weight-$(1,1,2)$ point extraction at every remaining point has expected order $29/8$. Its actual refined section ring is identified by an integral initial divisor in a symplectic flag variety. A second degeneration gives complete binary spaces on $\mathbb P(1,1,2)$, whose moments yield $\delta_p(X)\ge64/59$. This completes the proof for all smooth members, without a threefold Zariski decomposition on the weighted point blow-up. For Families 3.6 and 3.2, weighted curve extractions above double conics have expected orders $243/88$ and $135/56$, respectively. Exact section spaces give refined thresholds at least one, with the different retained. The positive discrepancy remainders complete both proofs, including the special double-conic configurations. Every smooth member of Family 3.5 is K-semistable. It is K-stable exactly when its associated degree-five map has no totally ramified point; the K-polystable members are also classified. For Family 3.7, sextic fibres and quintic sections reduce the proof to finitely many point configurations. Exact section algebras on ordinary and weighted point blow-ups complete the proof for every smooth member.

For Family 2.20 we identify every nonclosed smooth GIT orbit by a pencil of binary quartics and its Wronskian sextic. These give a one-parameter collection of strictly K-semistable threefolds with a common K-polystable smooth limit. Abban–Zhuang estimates exclude every divisor centre and every point on a smooth conic-bundle fibre. Exact singular-fibre surface calculations isolate the remaining valuative problem. At every double fibre the weighted extraction has expected order $297/104$ and discrepancy three. We determine its complete refined series by multiplication and a dimension count. Ordinary and weighted surface blowups prove $\delta_p(X)\ge104/99$ along every double fibre of a GIT-stable member. For reduced fibres, two flops give the expected order $43/26$ for the extraction of a component. Its complete refinement yields the same local bound along ordinary components when the fibre node is outside the original exceptional divisor. Negative-section flags and a retained discrepancy treat special-line components. The possible nonpositive-beta centres on a GIT-stable member reduce to at most three fibre nodes on the original exceptional divisor. Their ordinary point extractions have $S_X(J)=11/4<3=A_X(J)$. We identify their complete initial section algebras in every degree. Surface flags, including an extra fixed line in the degenerate-conic case, give $\delta_p(X)\ge104/103$ at all remaining nodes. This proves that a smooth member of Family 2.20 is K-stable exactly when its quartic pencil is GIT-stable, and K-polystable exactly when that pencil is GIT-polystable.

For Family 3.8, a smooth member is determined by a basepoint-free pencil of binary quartics. It is K-stable exactly when the associated degree-four morphism has no totally ramified point. There is one K-polystable member which is not K-stable; all other smooth members are strictly K-semistable and are parametrized by $\mathbb P(2,3)$. The proof treats every singular del Pezzo fibre by explicit surface flags and point or curve extractions. The actual refined sections, or complete containing series with the same asymptotic dimension, replace Zariski decompositions on the extracted threefolds.

For Family 1.10, we construct a symmetric counterexample to the unrestricted anticanonical-singularity formulation of Donaldson’s conjecture, stated as Conjecture 7.1 in The Calabi Problem for Fano Threefolds. A small deformation of the Mukai–Umemura threefold in the stable binary-octic direction $uv(u^6+v^6)$ is K-stable with finite automorphism group and has two points on no line exchanged by an $\mathfrak S_3$-action. Each point has an anticanonical divisor with exact germ $y^2=x^3+t^4x$; the involution exchanges the two divisors. We compute a central pencil, impose $j=1728$ at a simple root, and prove equivariant persistence. The additional hypothesis that a bad point lies on a line still implies failure of K-stability. We identify the exact bad-point-on-line locus with the locus of members having a special line meeting no other line, and with the complement of affine GIT-stable Mukai nets. In particular this locus is closed, and every member on it has $\delta(X)=1$. The line-incidence and closed-net revisions are therefore equivalent. We prove the revised criterion near Mukai–Umemura; the global implication from GIT stability to K-stability remains open. We also obtain the following partial classification results. Nonclosed smooth Mukai-net orbits yield strictly K-semistable members with the prescribed anticanonical singularity. The ordinary point blow-up gives $\delta_p(X)\ge264/259$ at every point on no line, including all nonordinary conic flops. We give sharper bounds according to the singularities of the associated cubic surface, and exhibit a smooth pointed member with one double conic parameter. For ordinary lines we classify all resolving-surface configurations and compute their fibre flags. In particular, $\delta_p(X)\ge88/83$ if exactly one line passes through $p$ and it is ordinary. The same bound holds along lines with smooth or nodal associated quintic surface and in further configurations treated by a point extraction on the exceptional surface. By changing the chosen line, the surface estimates reduce the points away from special lines to two explicit incidence configurations. A new point-jet estimate settles the directed three-line cycle for all divisorial valuations, giving $\delta_p(X)\ge270336/269791$. A weighted extraction also gives a strict bound along the special lines satisfying the explicit hypotheses of Definition ??. This includes all ramification cases on their exceptional surfaces. The proof retains actual fixed parts, differents and total-pullback corrections throughout. We also prove that every special test configuration with factorial terminal central fibre has zero Donaldson–Futaki invariant. Every terminal Gorenstein limit is nodal, with divisor-class rank one plus its number of nodes. We exclude all four one-nodal families as smooth test-configuration limits. An explicit fixed-point obstruction excludes the nonterminal boundary family obtained from smooth bidegree-$(4,1)$ curves on a quadric in $\mathbb P^3$. For the two exceptional ordinary-line configurations we obtain $A_X/S_X\ge132/125$ and $A_X/S_X\ge264/229$, respectively, for valuations monomial in coordinates straightening the lines. Under the transverse width-two hypotheses, the full anticanonical ring is described by powers of the ideal of a rational ribbon on a smooth quadric. The special-line links have four conductor configurations. Double conductor contact on $\mathbb F_1$ now has $\delta_p(X)\ge88/83$ along the entire special line, with the infinitely near conditions retained. On the GIT-stable locus, the remaining local cases are the mutually width-three ordinary pair and the distinguished point in the $\mathbb F_3$ width-two case. These results do not complete the global classification for Family 1.10, which is not included in the Main Theorem.

The three classifications confirm the conjectures in [3, Sections 7.2, 7.5 and 7.6].

We also construct a counterexample to Donaldson’s conjecture in The Calabi Problem for Fano Threefolds [3, Conjecture 7.1]. It concerns smooth prime Fano threefolds of genus twelve and predicts K-polystability from the absence of certain singular anticanonical divisors, apart from the members admitting a torus action. Our example contradicts the necessary direction of that prediction.

The construction has an $\mathfrak S_3$-action exchanging two points on no line and two anticanonical divisors with the prescribed germs. A single such divisor already contradicts the conjecture. At these points the space of sections of weighted order at least six has dimension two, whereas the torus degeneration giving the known obstruction has dimension eight. Thus the analytic equation does not determine the global anticanonical filtration. Subsection ?? discusses possible revisions. Theorem ?? identifies the exact bad-point-on-line locus with the complement of affine GIT-stable Mukai nets and proves that it is closed. Every member of this locus has $\delta(X)=1$. The line-incidence and closed-net conjectures are equivalent, and both hold near Mukai–Umemura. The global implication from a smooth GIT-stable net to a K-stable threefold remains unproved.

The valuative approach of Fujita and Li replaces test configurations by inequalities between log discrepancies and expected vanishing orders [18, 22]. The stability threshold, introduced through basis divisors and developed systematically by Blum–Jonsson [6], packages these inequalities in the invariant $\delta$. Its relation to K-stability and the existence of Kähler–Einstein metrics is described in [21]. Abban and Zhuang [4] developed refinement by admissible flags, allowing threefold estimates to be reduced to series on surfaces and curves. We use this method throughout.

Table 1 records the seventeen families for which a complete classification of K-polystable smooth members remained unresolved before the results of this paper, with the qualification concerning Family 3.5 below. Theorem 1.1 summarizes our results. The entry for Family 1.10 records partial results only; its global classification remains open. The geometric descriptions follow the classification in [3, Section 6]. All varieties and blow-up centres in the table are smooth, and the usual Fano conditions on the centres are understood.

In the table, $V_3\subset\mathbb P^4$ is a cubic, $V_4\subset\mathbb P^5$ is an intersection of two quadrics, $V_5\subset\mathbb P^6$ is a three-dimensional linear section of $\operatorname{Gr}(2,5)\subset\mathbb P^9$, and $Q\subset\mathbb P^4$ is a quadric. For Family 1.10, $\mathcal U$ is the tautological rank-three subbundle on $\operatorname{Gr}(3,7)$. For Family 3.2, put $$\pi:\mathcal B=\mathbb P_{\mathbb P^1\times\mathbb P^1} \bigl(\mathcal O\oplus\mathcal O(-1,-1)^{\oplus2}\bigr) \longrightarrow\mathbb P^1\times\mathbb P^1, \qquad \xi=\mathcal O_{\mathcal B}(1),$$ using the quotient convention, and let $D$ be the unique divisor in $|\xi|$.

$^{*}$Family 3.5 is retained in this historical list; its smooth classification is proved in Part XIII. Denisova–Papazachariou’s announced manuscript is listed under “In Preparation”, with a PDF available upon request, on Papazachariou’s research page, consulted on 30 September 2026 [15]. We have not used that manuscript as a theorem. Part XIII proves K-semistability for every smooth member and K-stability whenever there is no totally ramified point, including all mixed fibres.

Abban–Zhuang prove K-stability for smooth prime Fano threefolds of degree at most $16$ in [44]; their argument does not cover degree $18$. The published proof of [23, Lemma 4.2] treats Family 1.9 under a fixed-point-free abelian symmetry hypothesis. Its multiplier-ideal argument excludes curve centres of degree at least two without using that symmetry. Xu–Zhuang still list the universal K-semistability assertion for Family 1.9 in [51, Problem 10], in the revision dated 1 June 2026.

Part ?? proves K-stability for every smooth member of Family 1.9. The exceptional divisor of an ordinary line blow-up has expected order $101/72$; the weighted extraction over a special line has expected order $43/18$. Surface flags and changes of projection line treat every point on an ordinary line. The special-line analysis reduces to marked points on higher-width special lines. At those points, two-divisor adjunction retains the positive term $11v(G)/18$ for valuations whose centres on the smooth model are contained in $G$.

An ordinary point extraction gives $\delta_p(X)\ge9/8$ when the quadratic coefficient $\mu$ is nonzero. If $\mu=0$, the actual initial sections of the weight-$(1,2,2)$ extraction split into binary spaces. Their fixed orders and full Hilbert mass give $S_X(F)=287/72$ and a refined bound $144/119$ on $C\ne0$. The last valuations have centre $P=[1:0:0]$ on this extraction. The ordinary blowup of $P$ on $F$ has refined moment $115/72$, and its point moments on the required affine chart are at most $1/3$. To lift these surface estimates to the threefold, we blow up the strict transform of the original line. The normal bundle gives $$\operatorname{ord}_F(s)\le m+4\operatorname{ord}_L(s) \qquad (0\ne s\in H^0(X,mH)).$$ This inequality controls the extra boundary on the surface and allows both applications of inversion of adjunction to use effective boundaries. Retaining the threefold discrepancy gives $A_X(v)-(288/245)S_X(v)\ge11v(a)/35>0$. No Zariski decomposition on either extracted threefold enters this final calculation.

For Family 2.5, Cheltsov–Denisova–Fujita [8] proved K-stability under the assumption that the cubic-surface fibration has no singular point of type $D_5$ or $E_6$. Their local exclusions reduce the general case to these two singularities. Part II supplies the bounds $96/95$ and $16/15$ for the corresponding local thresholds and completes their reduction.

For Family 2.9, the argument in [3, Section 5.2] proves K-stability of a particular member using a cyclic group of order five and hence gives K-stability for a general member. Part III treats every smooth member. Flags on smooth cubic surfaces handle reduced conic fibres away from a distinguished exceptional fibre. At its remaining smooth-fibre points, blowing up the point on the cubic surface gives $\delta_p(X)\ge6912/5335$. We also give an independent proof using the exceptional plane of a threefold point blow-up. Double conic fibres require a weighted curve extraction; its exceptional surface is $\mathbb P^1\times\mathbb P^1$ with different one half of a smooth $(2,1)$-curve. The boundary is retained in the surface calculation.

For Family 2.10, Kaloghiros and Petracci proved K-stability of a general member [20, Theorem 1.3], and the equivariant results of [23] include special members. Part IV removes the generality and symmetry assumptions. Fibre estimates and nodal auxiliary sections treat the global reduction. Weighted point extractions treat $D_4$ and $D_5$ fibre singularities, while ordinary point blow-ups treat smooth points on the special lines of singular fibres. The latter computation requires an additional anticanonical section whose order exceeds the orders of its displayed summands.

The new exceptional-surface estimates do not require a complete Zariski decomposition on the extracted threefold. In Families 2.5 and 2.10, we identify the actual section filtrations, including cancellations, by regularity, localization and saturation. Weighted Hilbert functions and polynomial intervals then give the expected orders. In the exceptional-surface arguments for Family 2.9, explicit necessary vanishing conditions give spaces containing the refined sections. Their order sums provide upper bounds with the original normalization. We retain the volume calculations on the original threefolds, the Zariski decompositions on auxiliary surfaces, and the complete transverse-position calculation in Part III. Thus the absence of a full decomposition on an extracted threefold is justified by a description or comparison of section spaces, not assumed.

For Family 2.11, Part V treats both kinds of lines. The original exceptional divisor suffices for first-kind lines. For second-kind lines, its refined threshold is only $24/29$, so another argument is needed. A weak cubic section gives $\delta_p(X)\ge36/35$ on the special reduced fibre. After the curve centres and transverse points have been excluded, the remaining points lie on two ramification rulings over smooth conic fibres. A weight-$(1,1,2)$ blow-up at each such point has expected order $26/9$ and refined surface threshold at least $216/167$. We calculate the latter from complete binary spaces with explicit fixed factors. A valuation centred on the ramification ruling has discrepancy three and expected order $167/72$, giving the matching upper bound. Thus the local threshold at each of these tangency points is exactly $216/167$; see Corollary 29.9. For double fibres, the two possible expected orders are $91/36$ and $169/72$, according to whether the reduced fibre lies in the original exceptional divisor; both refined surface thresholds are at least $144/127$. None of these weighted extractions requires a threefold Zariski decomposition.

For Family 2.12, Cheltsov–Li–Ma’u–Pinardin [1] reduced every possible nonpositive-beta valuation to a closed point of $E\cap E'$ and proved stability for several special members. Part VI completes the point analysis. Smoothness of the determinant quartic forces the coprimality needed for an exact ordinary initial ring. Three cases on its exceptional plane are treated by explicit monomial moment certificates. In one of them the refined threshold equals one, but the extraction discrepancy retains a strictly positive term. Proportional residual directions require weights $(2,1,1)$; the expected order is $13/4$ and the refined threshold is at least $40/37$. We compute both ordinary and weighted expected orders in a separate Hilbert-weight lemma. These computations, too, require no Zariski decomposition on an extracted threefold.

For Family 2.13, the symmetric construction in [3, Section 5.5] proves K-stability of a particular member and hence of a general member. The smooth-conic estimate in the proof of [24, Lemma 5.6] gives $\delta_p(X)\ge80/77$. Part VII treats all remaining fibres. The ordinary blow-up of a node on a quartic del Pezzo section improves the polarized surface estimate to $\delta_p(S,-K_S+tF)\ge4/(3+2t)$. For a double conic, the blow-up of its scheme-theoretic fibre ideal is a weight-$(1,2)$ curve extraction. Its expected vanishing is $5/2$, and its exceptional pair is $(\mathbb P^1\times\mathbb P^1,\tfrac12D)$ with $D$ a smooth $(2,1)$-curve. We identify the actual weighted initial ring and its refined ideal-power spaces before estimating their thresholds. A degeneration keeps $D$ fixed. The limiting spaces can be smaller than the natural monomial spaces containing them; the resulting dimension correction is essential for the strict inequality. This weighted argument uses no Zariski decomposition on the extracted threefold.

For Family 2.14, the symmetric construction in [3, Section 4.3] proves K-stability of a particular member, and hence of a general member. Part VIII treats every smooth member. Every pencil fibre is a normal quintic del Pezzo surface with Du Val singularities. The exact fibre refinement reduces possible nonpositive-beta centres to certain lines and singular points. Ordinary line blow-ups and weight-$(1,2)$ line extractions treat the two normal-bundle types. The latter require separate calculations at the intersection with the original exceptional divisor. Point extractions treat $A_1,A_2,A_3,A_4$. Prime Pfaffian initial rings and full-volume section certificates account for all cancellations. A restriction lemma retains the zeros on a second divisor, giving the complementary point bounds. At an $A_3$ point, the first extraction has expected order $71/16$ and discrepancy five; its positive margin completes the argument even when the refined threshold equals one. None of these extractions requires a threefold Zariski decomposition.

For Family 2.16, Cheltsov–Hwang–Küronya–Laface–Mangolte–Massarenti–Park–Zhao [28] prove strict local estimates at every point on a reduced conic fibre. Their main theorem consequently proves K-stability when the discriminant is smooth, and more generally under an explicit equivariant condition. Kaloghiros–Liu–Petracci–Zhao [34, Appendix A, Theorem A.1] describe the geometry of all K-semistable degenerations of this family; that result does not by itself prove stability of every smooth member. Part IX treats the remaining double fibres. Blowing up their fibre ideals gives a plt exceptional surface $F\cong\mathbb P^1\times\mathbb P^1$ with different $D/2$ of bidegree $(2,1)$. A prime initial complete intersection identifies the actual refined spaces as $H^0(\mathcal I_P^k(m-\epsilon,r))$ for one point $P\in D$. Their dimensions give $S_X(F)=57/22$. Moving $P$ to a ramification point while preserving $D$ gives the exact monomial spaces used for the bound $88/85$. The reduced-fibre estimates then give $\delta(X)\ge176/171$. This proof also uses no Zariski decomposition on the weighted extracted threefold. A second proof in Sections 57–60 uses GIT. We compute the CM polarization on the incidence space of a conic and a pencil of quadrics and prove a strict Hilbert–Mumford inequality for every smooth pair. The relative reconstruction of [34, Proposition A.13], together with CM-degree minimization, then gives K-stability. This argument is independent of the local threshold estimates and does not assume the conjectural description of the entire K-moduli space as a GIT quotient.

For Family 2.17, Cheltsov–Pokora [40] reduce possible nonpositive-beta centres to points of $E\cap E'$. Part X uses the two contracted lines to reduce further to $\Sigma_X$. The remaining triple-contact points are treated by a weight-$(1,1,2)$ blow-up. The exact initial section ring gives $S_X(G)=29/8$, and a second degeneration gives refined threshold at least $64/59$. The exact Hilbert count ensures that no higher-order sections are lost. This completes K-stability for every smooth member. No Zariski decomposition on this weighted point blow-up is required.

For Family 3.6, Cheltsov [42] proves K-stability when the conic bundle has reduced fibres. Part XIV treats the omitted double conics. The blow-up of their fibre ideals gives a weighted curve extraction of discrepancy three and expected order $243/88$. We compute its actual refined spaces, retain the different $D/2$, and prove that the refined threshold is at least one. The positive remainder $21/88$ then proves strict K-stability. The argument includes the case where the line passes through a quadric-cone vertex and the corresponding sextic fibre is nonnormal. As in the preceding weighted arguments, no threefold Zariski decomposition is used at this extraction.

For Family 3.2, Belousov–Loginov [43] prove K-stability when the cubic fibre containing each double conic has only $A_1$ singularities along it. We use their corrected third version: its reduced-conic estimate is non-strict, and we retain the positive remainder in the first Abban–Zhuang step. Part XII treats the omitted double conics. Their weighted extraction has expected order $135/56$. An exact dimension count identifies its refined sections with monomial spaces. A second degeneration, with the different retained, gives refined threshold at least one. The finite-level equality at one weighted surface point is exact; the threefold discrepancy remainder $33/56$ gives strict K-stability. No Zariski decomposition on the weighted threefold is used.

For Family 3.5, Denisova [14] proves K-stability when all ramification is simple. The totally ramified cases are described in [3, Section 7.5]. Part XIII gives their explicit degeneration and its Abban–Zhuang input. It also computes $S_X(G)=(91r+25)/80$ for the weight-$(1,2,r-2)$ extraction at a fibre singularity of ramification index $r=3,4,5$. These first moments do not by themselves prove the required exceptional-surface inequalities. For $r=3$, an exact normal basis and an explicit surface log resolution now give $\delta(G;W^G)\ge160/151$ and $\delta_p(X)\ge160/151$. For $r=4$, a conic refinement with its different gives $\delta(G,\Delta;W^G)\ge80/77$ and the exact local threshold $\delta_p(X)=400/389$. Explicit line refinements cover the other centres outside the joining line of a mixed fibre. Ordinary point extractions then give $240/221$ at its $A_1$ endpoint and $32/29$ at its interior points. Their exact initial algebras and conic or cubic pencil resolutions complete the proof for every smooth member without a totally ramified point. The fibre-refinement calculation at an $A_2$ point is audited separately: the displayed surface volumes in [14] give $91/80$, rather than the stated $83/80$. The general semistability assertion for Family 3.5 and its complete smooth classification are included in the main theorem.

Section 102 applies the CM and GIT calculations in Liu’s moduli-continuity approach. The graph parameter space has anticanonical CM line $\mathcal O(22)$, and a smooth graph is GIT-stable exactly when its degree-five map has no totally ramified point. Liu–Zhao’s volume theorem applies to K-semistable limits of volume $20$; the anticanonical K3 lattice on a smooth member is $U\oplus\langle-28\rangle$. The independent local proof above establishes the smooth classification. Reconstructing contractions on arbitrary limits remains a question for the K-moduli compactification.

For Family 3.7, [3, Section 4.3] treats symmetric examples, while [23, Lemma 4.3] imposes an additional group-action hypothesis. Part XV treats every such centre. The matrix pencil has only smooth fibres and fibres with one $A_1$ or $A_2$ singularity. Fibre refinements reduce the problem to the two lines through these singularities. Quintic del Pezzo sections treat their open parts. Ordinary and weighted point extractions treat the singular points, the intersections with the original exceptional divisor, and the remaining smooth corner. Their actual initial section spaces are proved complete by exact Hilbert counts. No Zariski decomposition on an extracted threefold is used.

For Family 2.20, [3, Section 7.2] gives GIT-semistable smooth parameters whose orbits are not closed and whose associated threefolds are strictly K-semistable. Its Proposition 5.43 proves K-polystability of the unique member with infinite automorphism group. Part XI rewrites the parameter classification intrinsically using binary quartics and a sextic Wronskian, constructs the smooth special degenerations, and gives Abban–Zhuang reductions to singular conic fibres. The complete weighted refinement treats all double fibres on the GIT-stable locus. Component extractions treat reduced fibres away from finitely many nodes on $E$. The full point refinements at these nodes give $\delta_p(X)\ge104/103$ and prove the GIT-stable converse. This completes the smooth classification.

For Family 3.8, [3, Section 7.6] formulates the classification in terms of the degree-four cover and proves the total ramification cases. Part XVI completes the K-stable case. The quartic pencil can have fibres of types $(2,1,1)$, $(2,2)$ or $(3,1)$, and each type is included. Exact initial algebras treat the $A_1$ and $A_2$ points. Weighted curve extractions, surface flags and a marked specialization treat the exceptional rulings and the joining lines, including their endpoints. The final criterion is stated in Section 128.

Part I collects the Abban–Zhuang theory, the shared definitions and computational tools, and examples illustrating their later applications. The thirteen universal K-stability assertions of Theorem 1.1 are proved in Parts ??, II, III, IV, V, VI, VII, VIII, IX, X, XII, XIV and XV, respectively. The complete classification for Family 2.20 is proved in Part XI and summarized in Section 82. The classifications for Families 3.5 and 3.8 are treated in Parts XIII and XVI, respectively. The local theorems in those parts record the stronger estimates used in the proofs. In Part ?? we write $H=-K_X$ and use $L$ for a line. In the other parts we write $L=-K_X$; the remaining geometric notation is introduced separately in each part. In particular, the letters denoting fibre and exceptional classes are local to that part. Every local stability threshold takes the infimum over divisors whose centres contain the specified point, including positive-dimensional centres.