Preservation of divisorial contraction type under deformation of extremal rays: the exceptional (3.3.3)/(3.3.4) swap in Mori's lifting theorem for smooth families of threefolds

Mori's 1982 Annals paper "Threefolds whose canonical bundles are not numerically effective" proved the cone theorem for smooth projective threefolds, classified their extremal contractions, and recorded four limitations of its methods. Three of the four have since been settled in the later literature; one remains open, and it is the subject of this task. Mori's deformation theorem (3.47) shows that in a smooth projective family of threefolds over a discrete valuation ring, every extremal ray on the special fiber lifts uniquely to the geometric generic fiber with contraction type preserved, except that a divisorial ray whose exceptional divisor is a singular quadric cone (type (3.3.4), "E4") on the special fiber may a priori lift to a divisorial ray whose exceptional divisor is $\mathbb{P}^1\times\mathbb{P}^1$ (type (3.3.3), "E3") on the generic fiber. The problem is to settle whether this swap occurs, and to determine what non-numerical data governs the divisorial subtype in deformation.

Background

The minimal model program recasts the classification of complex projective varieties through the geometry of the canonical bundle. The canonical bundle $K_X$ of a smooth variety $X$ is the determinant line bundle of its cotangent bundle; a divisor or line bundle $D$ is called numerically effective (nef) if $D\cdot C\ge 0$ for every irreducible curve $C\subset X$, that is, if it has non-negative degree on every curve. When $K_X$ is not nef, Mori's 1982 cone theorem asserts that the closed cone $\overline{NE}(X)$ of effective curve classes is locally polyhedral in the region where $K_X$ has negative degree, and that each extremal ray $R$ of that cone (an edge half-line of $\overline{NE}(X)$) with $K_X\cdot R<0$ is spanned by rational curves and carries a contraction morphism $f=\mathrm{cont}_R:X\to Y$ sending exactly the curves whose classes lie in $R$ to points. Mori classified these contractions completely for smooth projective threefolds over an algebraically closed field of characteristic zero. When $R$ is not nef, the contraction is birational with a single irreducible exceptional divisor $E$, and exactly five cases occur, traditionally labeled E1-E5 and corresponding to the paper's types (3.3.1)-(3.3.5): E1 blows up a smooth curve ($E$ is a $\mathbb{P}^1$-bundle over a smooth curve and $Y$ is smooth); E2 blows up a smooth point ($E\simeq\mathbb{P}^2$ with $\mathcal{O}_E(E)\simeq\mathcal{O}_{\mathbb{P}^2}(-1)$); E3 contracts $E\simeq\mathbb{P}^1\times\mathbb{P}^1$ with $\mathcal{O}_E(E)\simeq\mathcal{O}(-1,-1)$ onto an ordinary double point (a node, locally $xy-zt=0$; "Gorenstein" means the canonical sheaf is invertible, "terminal" denotes the mildest singularity class of the minimal model program, characterized by the requirement that on a resolution $g:Y\to X$ of singularities the canonical bundles decompose as $K_Y=g^*K_X+\sum_i a_iE_i$ with all discrepancy coefficients $a_i>0$, and ordinary double points are the simplest Gorenstein terminal threefold singularities); E4 contracts a quadric cone $E\subset\mathbb{P}^3$ with $\mathcal{O}_E(E)\simeq\mathcal{O}_E\otimes\mathcal{O}_{\mathbb{P}^3}(-1)$ onto a double compound Du Val point (a hypersurface double point, locally $xy-z^2-t^3=0$; a compound Du Val point is a threefold hypersurface singularity whose general hyperplane section through the point is a Du Val surface singularity, the Du Val points or rational double points being the surface singularities of types $A_n$, $D_n$ and $E_n$; for this point the general hyperplane section is again a node, of type $A_1$, while the special hyperplane section $z=0$ is the surface singularity $xy-t^3$, of type $A_2$; the isolated compound Du Val points are precisely the Gorenstein terminal threefold singularities, so both the E3 and the E4 target points are terminal); and E5 contracts $E\simeq\mathbb{P}^2$ with $\mathcal{O}_E(E)\simeq\mathcal{O}_{\mathbb{P}^2}(-2)$ onto a non-Gorenstein point. When $R$ is nef the contraction is instead a conic bundle (a morphism onto a smooth surface whose fibers are plane conics), a del Pezzo fibration (a morphism onto a smooth curve whose fibers are del Pezzo surfaces, a del Pezzo surface being a surface whose anticanonical bundle is ample), or the structure morphism to a point, in which case $X$ is a Fano threefold ($-K_X$ ample). The Picard number $\rho(X)$, the rank of the Neron-Severi group of divisor classes modulo numerical equivalence, drops by exactly one in the birational cases.

Two auxiliary results of the paper frame the question. First, Lemma (3.9) establishes only $\chi(\mathcal{O}_S)\ne 0$ for a Gorenstein reduced projective surface $S$ with $\omega_S^{-1}$ ample (a possibly singular del Pezzo surface), where $\chi(\mathcal{O}_S)=\sum_i(-1)^i h^i(S,\mathcal{O}_S)$ is the holomorphic Euler characteristic; the exact value was left open. Second, and central to the present question, Theorem (3.47) studies deformations: let $T=\mathrm{Spec}\,R_0$ be the spectrum of a complete discrete valuation ring (a DVR is a one-dimensional regular local ring, the algebraic model of a small disc, with a unique closed point $t$ giving the "special fiber" and a generic point $\tau$ giving the "nearby general fiber"; "complete" means that the ring equals the inverse limit of its quotients by the powers of its maximal ideal, so that formal power series manipulations are valid in it) with algebraically closed residue field of characteristic zero, and let $\pi:\mathcal{X}\to T$ be a smooth projective morphism with connected geometric fibers of dimension three. Mori proved that every extremal ray $\mathfrak{R}_t$ on the special fiber $X_t$ lifts uniquely to an extremal ray $\mathfrak{R}_{\bar\tau}$ on the geometric generic fiber (the generic fiber with scalars extended from the fraction field of $R_0$ to an algebraic closure, so that it is again a variety over an algebraically closed field) whose specialization (reduction of numerical classes) is $\mathfrak{R}_t$, and that the contraction types of $\mathfrak{R}_t$ and $\mathfrak{R}_{\bar\tau}$ agree, with a single exception his method could not exclude: $\mathfrak{R}_t$ may be of type (3.3.4), whose exceptional divisor is a singular quadric cone, while $\mathfrak{R}_{\bar\tau}$ is of type (3.3.3), whose exceptional divisor is $\mathbb{P}^1\times\mathbb{P}^1$. In modern shorthand, E3 and E4 contractions have the same extremal-ray length $-K_X\cdot\ell=1$ and the same intersection numbers, and later work treats them as numerically equivalent; the pair is distinguished only by whether the exceptional quadric and the image singularity are of the smooth/ordinary-double-point kind (E3) or the singular cone/double-compound-Du-Val kind (E4). The open question is therefore exactly what data beyond numerical equivalence is preserved when the threefold deforms in a smooth family, and it connects to how the terminal singularity type of a contracted target behaves in specialization, a foundational input for relative versions of the minimal model program and for the moduli theory of threefolds equipped with Mori structures.

Problem Statement

Let $k$ be an algebraically closed field of characteristic zero, let $R_0$ be a complete discrete valuation ring (a one-dimensional regular local ring, equal to the inverse limit of its quotients by the powers of its maximal ideal) with residue field $k$, let $T=\mathrm{Spec}\,R_0$ with closed point $t$ and geometric generic point $\bar\tau$ (a point over the generic point of $T$ whose residue field is an algebraic closure of the fraction field of $R_0$), and let $\pi:\mathcal{X}\to T$ be a smooth projective morphism with connected geometric fibers of dimension three. For an extremal ray $\mathfrak{R}_t$ of the closed cone of curves $\overline{NE}(X_t)$ (the closure of the cone spanned by the classes of effective curves) on the special fiber, Mori's Theorem (3.47) (1982) produces a unique extremal ray $\mathfrak{R}_{\bar\tau}$ on the geometric generic fiber whose reduction (specialization of numerical classes) is $\mathfrak{R}_t$, and the contraction types of $\mathfrak{R}_t$ and $\mathfrak{R}_{\bar\tau}$ agree in all cases except possibly one: $\mathfrak{R}_t$ divisorial of type (3.3.4), with exceptional divisor an irreducible reduced singular quadric cone in $\mathbb{P}^3$ contracted to a double compound Du Val point (an "E4" contraction), while $\mathfrak{R}_{\bar\tau}$ is divisorial of type (3.3.3), with exceptional divisor isomorphic to $\mathbb{P}^1\times\mathbb{P}^1$ contracted to an ordinary double point (an "E3" contraction). The open problem is to settle this exceptional case, in either direction, at the same level of generality. (1) Existence: either construct a smooth projective family $\pi:\mathcal{X}\to\mathrm{Spec}\,R_0$ as above whose special fiber carries a type (3.3.4) divisorial extremal ray whose lift on the geometric generic fiber is of type (3.3.3), or prove that no such family exists, so that the divisorial contraction subtype E3/E4 is preserved under specialization in every smooth projective family of threefolds over a complete discrete valuation ring with algebraically closed residue field of characteristic zero. (2) Criterion: in the negative branch, exhibit the geometric or deformation-theoretic data, beyond numerical equivalence, that distinguishes the two divisorial subtypes and is preserved along smooth projective families of threefolds, so that preservation of the divisorial subtype is enforced; in the positive branch, identify, as part of the analysis of exactly which hypotheses of the lifting theorem are sharp, the data that govern which of the two divisorial subtypes appears on nearby fibers. By the observation of Cutrone-Marshburn that E3 and E4 contractions are numerically equivalent (same ray length and same intersection numbers), the numerical class of the ray cannot suffice, so the criterion must involve finer data.

References

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