Stein degree on non-Fano type fibrations

We construct examples showing that Stein degree of vertical divisors on non-Fano type log Calabi-Yau fibrations is unbounded.

💡 Research Summary

The paper investigates the behavior of the Stein degree—a numerical invariant measuring the degree of the Stein factorisation—of vertical divisors on log Calabi‑Yau fibrations that are not of Fano type. In the existing literature, notably Birkar (2022) and the authors’ earlier work (BQ25), it is shown that for a log Calabi‑Yau fibration (X,B)→Z the Stein degree of any horizontal lc centre is bounded solely in terms of the dimension of the total space, and that for vertical divisors the strong Stein degree (the degree after restricting to the image) is also bounded provided the fibration is of Fano type (i.e., X admits a big divisor Γ with (X,Γ) klt and K_X+Γ∼_R0/Z). These boundedness results are crucial for constructing moduli of semi‑log canonical stable minimal models.

The main contribution of this article is to demonstrate that the Fano‑type hypothesis cannot be dropped: the strong Stein degree of vertical divisors can be made arbitrarily large even when the coefficient of the divisor in B equals 1. The authors construct explicit families of log Calabi‑Yau fibrations (X,B)→Z over an algebraically closed field K of characteristic 0 for which the following holds: X is a normal projective threefold, Z a normal projective surface, the generic fibre of X→Z is a smooth genus‑one curve, and there exists a curve C⊂Z such that B contains n/d horizontal components over C (with n and d natural numbers satisfying d|n, d≠n). Each of these components S has coefficient 1 in B and its normalisation S^nor satisfies sdeg(S^nor/C)=d. Since d can be chosen arbitrarily large, the strong Stein degree is unbounded.

The construction rests on arithmetic surfaces with split multiplicative reduction of elliptic curves, as developed by Liu‑Liu‑Raynaud (LLR04). For any pair (n,d) with d|n and d≠n, one starts with a complete discrete valuation ring R whose residue field k admits a cyclic Galois extension k′/k of degree d. LLR04 provides a relatively minimal arithmetic surface X→Spec R whose generic fibre is a smooth elliptic curve and whose special fibre is a reduced cycle of n/d copies of ℙ¹_{k′}, arranged in a loop. Over the algebraic closure, this special fibre becomes a cycle of n projective lines, and each irreducible component S_k of the special fibre has Stein degree d over Spec k because after base change to k′ it splits into d disjoint lines.

Because this surface lives over a non‑finite‑type base (Spec R), the authors must “spread out” the construction to a genuine log Calabi‑Yau fibration over an algebraically closed field. This is achieved in three steps:

1. Algebraic approximation – Using Artin’s approximation theorem, the complete DVR R is approximated by an excellent DVR R′ that is essentially of finite type over K and admits an étale morphism Spec R′→Spec R. The arithmetic surface is lifted to a normal surface X′→Spec R′ preserving the required properties (Theorem 3.4).

2. Spreading out to a surface – The DVR Spec R′ is realized as the localisation of a smooth quasi‑projective surface Z° at the generic point of a smooth curve C°. By base‑changing the family X′→Spec R′ and using Hilbert scheme techniques, the authors construct a projective morphism (X°,B°)→Z° whose fibres over points of C° reproduce the arithmetic surface’s fibres. The horizontal components of B° over C° are precisely the n/d copies described above, each having strong Stein degree d (Theorem 3.6).

3. Projective compactification – Finally, applying the results of Hacon–Xu (HX13) and running a relative Minimal Model Program, the quasi‑projective fibration (X°,B°)→Z° is compactified to a projective log Calabi‑Yau fibration (X,B)→Z. The resulting pair (X,B) is log canonical, K_X+B∼_R0/Z, and retains the vertical divisor configuration with unbounded strong Stein degree (Theorem 3.7).

Thus the paper provides a concrete counterexample to any conjecture that the strong Stein degree of vertical divisors on log Calabi‑Yau fibrations is universally bounded without the Fano‑type assumption. It highlights a sharp dichotomy: while horizontal lc centres enjoy uniform boundedness, vertical divisors can exhibit arbitrarily large Stein degree once the ambient fibration ceases to be of Fano type. The work also showcases a fruitful blend of arithmetic geometry (Galois extensions, reduction types of elliptic curves) and birational techniques (approximation, Hilbert schemes, MMP) to produce sophisticated geometric examples.