Multiple fibers of del Pezzo fibrations: the unrealized types $I_{2,6}$ and $I_8$, the cyclic-quotient conjecture for multiplicity at least 5, and the local structure along a multiple fiber

Mori and Prokhorov (2009) proved that a three-dimensional del Pezzo fibration whose total space has only terminal singularities has no fiber of multiplicity $m_o\ge 7$, and classified in their Table 1 the possible baskets of singular points at which the multiple fiber fails to be Cartier; their explicit examples realize every classified type except possibly the two irregular types $I_{2,6}$ and $I_8$. Three problems from that paper remain open: (1) whether multiple fibers of types $I_{2,6}$ and $I_8$ actually occur — an explicit construction or an exclusion, either of which would sharpen Table 1; (2) Conjecture 6.2, asserting that a del Pezzo bundle germ with a multiple fiber of multiplicity $m_o\ge 5$ (types $I_{2,3,6}$ and $I_{5,5}$) is a quotient of a Gorenstein del Pezzo bundle by a cyclic group acting freely in codimension two, currently proved only under auxiliary hypotheses (the basket of the fiber equals the full basket of $X$, a Du Val general elephant exists, or the fiber is log terminal); and (3) a description of all local analytic realizations at points where the fiber is Cartier but the total space is non-Gorenstein — points that the basket data of Table 1 omits.

Background

The minimal model program for complex projective threefolds of negative Kodaira dimension terminates in a Mori fiber space: a contraction with connected fibers whose anticanonical class is relatively ample. When the base is a curve, this outcome is a del Pezzo fibration. Mori and Prokhorov study such fibrations locally: a weak del Pezzo fibration is a projective morphism $f\colon X\to Z$ with connected fibers from a threefold $X$ with only terminal singularities onto a smooth curve $Z$, such that $-K_X$ is $f$-nef and $f$-big near a general fiber; if $-K_X$ is $f$-ample one speaks of a del Pezzo bundle. Terminal is the strongest singularity class of the program (every exceptional divisor over $X$ has positive discrepancy); threefold terminal singularities are isolated and were classified by Mori. No $\mathbb{Q}$-factoriality or relative Picard number hypothesis is imposed. The general fiber $F_g$ is a smooth del Pezzo surface, that is, a rational surface with ample anticanonical class, of degree $K^2_{F_g}\in\{1,\dots,9\}$.

A special fiber $f^*(o)=m_oF_o$ with $m_o\ge 2$ ($F_o$ the reduced fiber) is called a multiple fiber. Multiplicity is a non-Gorenstein phenomenon: if $X$ is Gorenstein ($K_X$ Cartier), all fibers are reduced by a theorem of Kawamata (Lemma 2.2 of the source): at a terminal point every Weil $\mathbb{Q}$-Cartier divisor is locally linearly equivalent to a multiple of $K_X$, so on a Gorenstein total space $F_o$ is Cartier everywhere, whence $m_o=1$ by Proposition 2.11 of the source. The two-dimensional parallel is Kodaira's theory of elliptic fibrations: by Kodaira's canonical bundle formula, a relatively minimal elliptic fibration from a smooth surface with trivial canonical bundle has no multiple fibers, and the classification of singular fibers then pins down the fiber over the base point; this is the mechanism by which a good anticanonical member controls multiplicities in favorable cases, as in Proposition 6.3(ii) of the source, where a Du Val general elephant $S$ satisfies $K_S\sim 0$ and its minimal resolution is a relatively minimal elliptic surface over the base germ.

The key bookkeeping device is Reid's basket. A non-Gorenstein terminal threefold point deforms to a finite collection of cyclic quotient singularities of type $\frac{1}{r}(1,-1,b)$ with $0<b<r$ and $\gcd(b,r)=1$, and the multiset $\mathbf{B}(X)$ of these virtual points is the basket of $X$; for a Weil $\mathbb{Q}$-Cartier divisor $D$, the sub-basket $\mathbf{B}(D)\subset\mathbf{B}(X)$ records the points where $D$ is not Cartier. A multiple fiber is of type $I_{r_1,\dots,r_n}$ if $\mathbf{B}(F_o)=(r_1,\dots,r_n)$; it is regular if $K_X+F_o\sim 0$ near each point of $\mathbf{B}(F_o)$ (equivalently, its type is one of $I_{2,3,6}$, $I_{5,5}$, $I_{2,4,4}$, $I_{3,3,3}$, $I_{2,2,2,2}$) and irregular otherwise.

Mori–Prokhorov's Theorem 1.1, proved with Reid's orbifold Riemann–Roch formula (a Riemann–Roch formula for Weil $\mathbb{Q}$-Cartier divisors on terminal threefolds, with explicit local correction terms read off from the basket), states that $m_o\le 6$, that every multiplicity $1,\dots,6$ occurs, and gives Table 1 of possible baskets: $m_o=2$ allows $\mathbf{B}(F_o)=(8)$, $(2,6)$, $(4,4)$, $(2,2,4)$ or $(2,2,2,2)$; $m_o=3$ allows $(9)$, $(3,3,3)$ or $(3,6)$; $m_o=4$ only $(2,4,4)$; $m_o=5$ only $(5,5)$; and $m_o=6$ only $(2,3,6)$. Section 5 of the source constructs explicit examples realizing all of these types — except possibly $I_{2,6}$ and $I_8$.

The structural tool behind both the examples and the open conjecture is the cyclic-cover construction (Construction 2.9): after the base change $t\mapsto t^{m_o}$ and normalization one obtains a finite $\boldsymbol{\mu}_{m_o}$-cover $X'\to X$, étale outside the finite set where $F_o$ is not Cartier, with $X=X'/\boldsymbol{\mu}_{m_o}$ and $f'\colon X'\to Z'$ again a weak del Pezzo fibration whose central fiber has multiplicity one; conversely, a $\boldsymbol{\mu}_{m_o}$-equivariant weak del Pezzo fibration with central fiber of multiplicity one, whose action is free in codimension two and whose quotient has only terminal singularities, yields a weak del Pezzo fibration with a multiple fiber of multiplicity $m_o$ — and a del Pezzo bundle when $-K_{X'}$ is $f'$-ample. Conjecture 6.2 predicts that for the maximal multiplicities $m_o\in\{5,6\}$ this construction with Gorenstein $X'$ (Cartier canonical class, index one) accounts for every germ.

The state of the surrounding theory: a general elephant is a general member of $|{-K_X}|$, and Reid's General Elephant Conjecture predicts that for a Mori contraction with positive-dimensional base it has only Du Val singularities — the rational double points $A_n$, $D_n$, $E_6$, $E_7$, $E_8$, i.e. the canonical Gorenstein surface singularities. Locally at a single non-Gorenstein point this is a theorem of Reid; over a del Pezzo fibration germ it is known only under restrictions on the central fiber (Prokhorov proved it when the central fiber is reduced, irreducible, normal with quotient singularities), and the multiple-fiber case is open. This is exactly the gap between Proposition 6.3 and Conjecture 6.2. At points of $F_o$ that are Cartier while $X$ is non-Gorenstein, the fiber has — in the log terminal case treated by Proposition 6.5 of the source — singularities of type T, the two-dimensional quotient singularities admitting $\mathbb{Q}$-Gorenstein smoothings, studied by Kollár and Shepherd-Barron; Example 5.6 of the source shows that such points really occur on multiple fibers, and their complete local description is missing.

The sharp multiplicity table is consumed as an input elsewhere in three-dimensional birational geometry: in the study of conic-bundle structures on $\mathbb{Q}$-Fano threefolds (where a del Pezzo fibration of multiplicity $6$ forces a degree-$6$ general fiber and rationality), in the birational rigidity theory of singular del Pezzo fibrations, in the analysis of non-rational special fibers (Loginov), and as the terminal dimension-three precedent for Birkar's general boundedness theorems for multiplicities of fibers of Fano fibrations.

Problem Statement

Let $f\colon X\to(Z\ni o)$ be the germ of a del Pezzo bundle in the sense of Mori–Prokhorov: $X$ is a threefold with only terminal singularities, $Z$ is a smooth curve germ, $f$ is a projective morphism with connected fibers, $-K_X$ is $f$-ample, and no $\mathbb{Q}$-factoriality or relative Picard number condition is imposed. Let $f^*(o)=m_oF_o$ be a multiple special fiber and let $\mathbf{B}(F_o)$ be the basket of points of $X$ at which $F_o$ is not Cartier. Mori–Prokhorov's Theorem 1.1 gives $m_o\le 6$ and Table 1 of possible baskets. The following three problems, raised in that paper, remain open; resolving them completes the local structure theory of multiple fibers of three-dimensional del Pezzo fibrations.

(1) Realizability of types $I_{2,6}$ and $I_8$. Determine whether there exists a del Pezzo bundle germ as above with a fiber of multiplicity $m_o=2$ whose basket is $\mathbf{B}(F_o)=(2,6)$ (type $I_{2,6}$), and one with $\mathbf{B}(F_o)=(8)$ (type $I_8$).

(2) Conjecture 6.2 (cyclic-quotient structure for multiplicity at least five). Prove or disprove: if $m_o\ge 5$ — equivalently, $F_o$ is of type $I_{2,3,6}$ with $m_o=6$ or of type $I_{5,5}$ with $m_o=5$ — then $f$ is the quotient of a Gorenstein del Pezzo bundle by a cyclic group acting freely in codimension two; that is, in the base-change construction of the source, the normalization $X'$ of the $\boldsymbol{\mu}_{m_o}$-cover is Gorenstein. The known partial results to be extended or contradicted are: $\mathbf{B}(F_o)=\mathbf{B}(X)$ (Proposition 6.3(i)); existence of a general member of $|{-K_X}|$ with only Du Val singularities (Proposition 6.3(ii)); $F_o$ with only log terminal singularities (Proposition 6.5). A proof must remove these auxiliary hypotheses — in particular it must handle points of $F_o$ that are Cartier while $X$ is non-Gorenstein, and fibers $F_o$ that are not log terminal — without restricting the class of del Pezzo bundles; a disproof must exhibit a germ with $m_o\ge 5$ admitting no Gorenstein cyclic cover acting freely in codimension two at all.

(3) Local analytic types at Cartier points. Describe all possible local analytic forms of the germ $(X\ni P, F_o)$ at points $P\in F_o$ where $F_o$ is Cartier but $X$ is non-Gorenstein — the points omitted from $\mathbf{B}(F_o)$, cf. the Warning following Theorem 1.1 and Example 5.6, where such a point of type $\frac{1}{2}(1,1,1)$ occurs — so that Table 1, together with this description, yields a complete local classification of a del Pezzo bundle germ along a multiple fiber.

References

1. S. Mori and Yu. Prokhorov, Multiple fibers of del Pezzo fibrations, Proceedings of the Steklov Institute of Mathematics 264 (2009), no. 1, 131–145. DOI: 10.1134/S0081543809010167; arXiv:0806.3809; https://doi.org/10.1134/S0081543809010167
2. S. Mori and Yu. Prokhorov, On $\mathbb{Q}$-conic bundles, Publications of the Research Institute for Mathematical Sciences 44 (2008), no. 2, 315–369. DOI: 10.2977/prims/1210167329; https://doi.org/10.2977/prims/1210167329
3. S. Mori, On 3-dimensional terminal singularities, Nagoya Mathematical Journal 98 (1985), 43–66. DOI: 10.1017/S0027763000021358; https://doi.org/10.1017/S0027763000021358
4. M. Reid, Young person's guide to canonical singularities, in Algebraic Geometry, Bowdoin 1985, Proceedings of Symposia in Pure Mathematics 46, Part 1, American Mathematical Society, Providence, RI, 1987, pp. 345–414. DOI: 10.1090/pspum/046.1/927963; https://doi.org/10.1090/pspum/046.1/927963
5. K. Kodaira, On compact analytic surfaces. II, Annals of Mathematics (2) 77 (1963), 563–626. DOI: 10.2307/1970131; https://doi.org/10.2307/1970131
6. K. Kodaira, On the structure of compact complex analytic surfaces. I, American Journal of Mathematics 86 (1964), no. 4, 751–798. DOI: 10.2307/2373157; https://doi.org/10.2307/2373157
7. Y. Kawamata, Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces, Annals of Mathematics (2) 127 (1988), no. 1, 93–163. DOI: 10.2307/1971417; https://doi.org/10.2307/1971417
8. J. Kollár and N. I. Shepherd-Barron, Threefolds and deformations of surface singularities, Inventiones Mathematicae 91 (1988), no. 2, 299–338. DOI: 10.1007/BF01389370; https://doi.org/10.1007/BF01389370
9. Yu. Prokhorov, A note on degenerations of del Pezzo surfaces, Annales de l'Institut Fourier 65 (2015), no. 1, 369–388. DOI: 10.5802/aif.2934; arXiv:1108.5051; https://doi.org/10.5802/aif.2934
10. Yu. Prokhorov, Effective results in the three-dimensional minimal model program, in: Proceedings of the International Congress of Mathematicians — ICM 2022, Volume 3, EMS Press, Berlin, 2023, pp. 2324–2345. DOI: 10.4171/icm2022/30; arXiv:2108.13052; https://doi.org/10.4171/icm2022/30
11. Yu. Prokhorov, Conic bundle structures on $\mathbb{Q}$-Fano threefolds, Electronic Research Archive 30 (2022), no. 5, 1881–1897. DOI: 10.3934/era.2022095; arXiv:2111.01003; https://doi.org/10.3934/era.2022095
12. K. Loginov, Standard models of degree 1 del Pezzo fibrations, Moscow Mathematical Journal 18 (2018), no. 4, 721–737. DOI: 10.17323/1609-4514-2018-18-4-721-737; arXiv:1710.02482; https://doi.org/10.17323/1609-4514-2018-18-4-721-737
13. K. Loginov, On nonrational fibers of del Pezzo fibrations over curves, Mathematical Notes 106 (2019), no. 5–6, 930–939. DOI: 10.1134/S0001434619110294; arXiv:1811.04418; https://doi.org/10.1134/S0001434619110294
14. C. Birkar, Singularities on Fano fibrations and beyond, preprint (2023, revised 2025). arXiv:2305.18770; https://arxiv.org/abs/2305.18770
15. J. Han, C. Jiang and Y. Luo, Shokurov's conjecture on conic bundles with canonical singularities, Forum of Mathematics, Sigma 10 (2022), e38. DOI: 10.1017/fms.2022.32; arXiv:2104.15072; https://doi.org/10.1017/fms.2022.32
16. J. Han, J. Liu and Y. Luo, ACC for minimal log discrepancies of terminal threefolds, Advances in Mathematics 480 (2025), 110457. DOI: 10.1016/j.aim.2025.110457; arXiv:2202.05287; https://doi.org/10.1016/j.aim.2025.110457