Complete the classification of three-dimensional extremal curve germs with reducible central curve: the semistable case 12o, Reid's general elephant conjecture there, sharp bounds on the number of components, and the existence of case 10o

The local building blocks of the minimal model program for complex threefolds are extremal curve germs: germs $(X, C)$ of a threefold with terminal singularities along a reduced curve $C$, contracted by a morphism $f\colon (X, C) \to (Z, o)$ with $C = f^{-1}(o)_{\mathrm{red}}$ and $-K_X$ $f$-ample. Kollár and Mori (1992) classified the case of irreducible $C$ into six types and proved Reid's general elephant conjecture for these germs, but left the reducible case open. In their rough classification of germs with reducible $C$, Mori–Prokhorov (2026) reduced the open problem to four questions: (i) classify the germs of case 12o, in which the components of $C$ are all semistable, of types $n \times (k2A) + k \times (k1A)$ with $n \geq 1$; (ii) prove or disprove that a general member of $|-K_X|$ of such a germ is a normal surface with only Du Val singularities — the last open case of Reid's general elephant conjecture for extremal curve germs; (iii) determine the sharp bounds on the number of components of $C$ (the known bounds are stated to be non-optimal); (iv) decide whether germs of case 10o exist at all.

Background

A complex variety has terminal singularities if $K_X$ is $\mathbb{Q}$-Cartier and on a log resolution $f\colon Y \to X$ one can write $K_Y = f^*K_X + \sum a_E E$ with all discrepancies $a_E > 0$; equivalently, $X$ has no 'worse than canonical' points, and $K_X$ behaves like the canonical class of a smooth variety for the purposes of the minimal model program (MMP). The MMP for a terminal threefold decomposes birational maps into elementary contractions of extremal rays; when such a contraction is small (exceptional locus of codimension at least two) it is replaced by a flip, whose existence in dimension three is due to Mori (1988), with the log-terminal case settled by Shokurov (1993).

The local objects encoding these contractions are extremal curve germs: a germ $(X, C)$ of a threefold with only terminal singularities along a reduced complete connected curve $C$ (if reducible, $C = C_1 \cup \dots \cup C_N$ with every component $C_i \simeq \mathbb{P}^1$), equipped with a proper morphism $f\colon (X, C) \to (Z, o)$ such that $C = f^{-1}(o)_{\mathrm{red}}$ and $-K_X$ is $f$-ample. According to the dimension of the exceptional locus, a germ is flipping (exceptional locus exactly $C$; its flip is a germ $X^+$ with terminal singularities and $K_{X^+}$ $f^+$-ample, computed as $X^+ = \mathrm{Proj}_Z \bigoplus_{m \geq 0} f_* \mathcal{O}_X(-mK_X)$), divisorial (exceptional locus a divisor $E \supset C$, mapped onto a curve in $Z$ passing through $o$; the target point $o \in Z$ need not be $\mathbb{Q}$-Gorenstein in this case), or a $\mathbb{Q}$-conic bundle germ (the target $Z$ is a surface and all fibers of $f$ are one-dimensional). A point of $X$ where $K_X$ is not Cartier is called non-Gorenstein; its index is the least positive integer making $K_X$ Cartier there. Surface singularities of type $A_n$, $D_n$, $E_6$, $E_7$, $E_8$ are the Du Val (rational double point) singularities, the mildest canonical surface singularities; the singularity terminology follows Reid's guide (1987).

A general elephant is a general member of the anticanonical linear system $|-K_X|$. Reid's general elephant conjecture predicts that, in the MMP context, a general elephant of a threefold with terminal (more generally log terminal) singularities is a normal surface with only Du Val singularities; it is the key that turns the study of a threefold germ into the study of deformations of surfaces, since $X \to Z$ then acts as a one-parameter $\mathbb{Q}$-Gorenstein smoothing of its general elephant (a deformation over a smooth curve whose total space has $\mathbb{Q}$-Cartier canonical class restricting to the canonical class of each fiber).

For irreducible $C$, the classification is complete: Kollár–Mori (1992) classified the extremal neighborhoods into the six types $k1A$, $k2A$, $cD/3$, $IIA$, $IC$, $kAD$ — the first two semistable, the last four exceptional — treating the exceptional types completely by reconstructing the neighborhood from the deformation space of a general elephant, and proving (their Theorem 1.7) that a general elephant is a normal surface with only Du Val singularities. The completion of the irreducible picture by Mori and Prokhorov (2008–2019) uses the extended list of types $(III)$, $(k1A)$, $(k2A)$, $(cD/2)$, $(cAx/2)$, $(cE/2)$, $(cD/3)$, $(IIA)$, $(II^{\vee})$, $(IE^{\vee})$, $(ID^{\vee})$, $(IC)$, $(IIB)$, $(kAD)$, $(k3A)$. Semistability is defined via elephants (Hacking–Tevelev–Urzúa 2017, following Kollár–Mori 1992): an extremal neighborhood $f\colon (C \subset X) \to (Q \in Y)$ is semistable if the general elephant $E_Y \in |-K_Y|$ has a Du Val singularity of type $A$ at $Q$; its type is $k1A$ or $k2A$ according as the general elephant $E_X \in |-K_X|$ has one or two singular points. For irreducible semistable germs, Mori (2002) gave the division-algorithm computation of $k2A$ flips and Hacking–Tevelev–Urzúa (2017) extended it to more general $k1A$ germs, showing both types form a single deformation family and constructing its universal family (for neighborhoods whose nearby surface fiber has second Betti number 1).

For reducible $C$, each $(X, C_i)$ is again an extremal curve germ, so the problem is a gluing problem: determine which combinations of component types can share one contraction. Kawamata (1988) showed that some such germs factor analytically into germs with irreducible central curve, but this does not classify the possible gluings. Mori–Prokhorov (2026) proved that every extremal curve germ with reducible central curve belongs to one of twelve cases $1o$–$12o$ of their Table 1, recording, for the cases $1o$–$11o$, the nature of $f$ and a bound on the number $N$ of components (a 'rough classification') and leaving case $12o$ described but unresolved in these two respects, showed that all cases except possibly $10o$ occur, and proved that a general elephant is normal with only Du Val singularities for every germ not of type $12o$. The case $12o$ consists of the germs with two or more non-Gorenstein points whose components comprise $n \geq 1$ semistable components of type $k2A$ and $k$ components of type $k1A$; the case $10o$ consists of the germs with exactly two non-Gorenstein points of component type $(k3A) + (N-1) \times (k1A)$. In the cases $7o$, $8o$, $10o$–$12o$ the base locus of $|-K_X|$ is one-dimensional, and a general elephant contains every component of type $(IIB)$, $(IC)$, $(k2A)$, $(k3A)$, or $(kAD)$ — this is exactly what makes both the classification and the general elephant question harder in the remaining cases than in the settled ones, where $|-K_X|$ meets $C$ in a single non-Gorenstein point.

Problem Statement

Complete the classification of extremal curve germs of threefolds with reducible central curve, in the form left open by Kollár–Mori (1992) and reduced by Mori–Prokhorov (2026) to the following tasks. Throughout, an extremal curve germ is a germ $(X, C)$ as defined in the background: $X$ a threefold with only terminal singularities, $C = C_1 \cup \dots \cup C_N$ a reduced complete connected curve, $f\colon (X, C) \to (Z, o)$ a proper morphism with $C = f^{-1}(o)_{\mathrm{red}}$ and $-K_X$ $f$-ample; component types refer to the (completed) classification of germs with irreducible central curve.

1. (Case 12o.) Classify the extremal curve germs of type 12o: those with two or more non-Gorenstein points whose components consist of $n \geq 1$ semistable components of type $k2A$ and $k \geq 0$ components of type $k1A$. Determine which configurations $(n, k)$ with their singularity data occur; give explicit equations (normal forms) up to analytic isomorphism and deformation for each occurring configuration; determine the nature of the contraction $f$ (flipping, divisorial, or $\mathbb{Q}$-conic bundle) in each; and where $f$ is flipping, compute the flip explicitly.

2. (General elephant in case 12o.) Prove, or disprove, that for every extremal curve germ of type 12o a general member $D \in |-K_X|$ is a normal surface with only Du Val singularities — the last open case of Reid's general elephant conjecture for extremal curve germs. Note that in this case the base locus of $|-K_X|$ is one-dimensional and $D$ contains all the $k2A$ components, so the reduction to a single non-Gorenstein point used in the settled cases is unavailable.

3. (Sharp bounds.) Determine the sharp upper bound on the number $N$ of components of the central curve for each case of the Mori–Prokhorov table (the published bounds are stated by its authors to be non-optimal in many cases); in particular, decide whether $N$ is bounded in case 12o and by what, proving unboundedness by explicit infinite families if it is not.

4. (Case 10o.) Decide whether extremal curve germs of type 10o — with exactly two non-Gorenstein points, of component type $(k3A) + (N-1) \times (k1A)$ — exist: either construct explicit examples or prove that no such germ exists.

Any complete classification of these germs must be consistent with, and subsume, the rough classification (Table 1, cases $1o$–$11o$) of Mori–Prokhorov (2026).

References

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