Q-Cartierness versus pluricanonical base change

Definitions and mathematical statement

For a surface family $f:X\to B$, let $\omega_{X/B}^{[m]}=(\omega_{X/B}^{\otimes m})^{**}$ denote the $m$-th reflexive power. The original defines base-change compatibility by requiring, for every $\phi:B'\to B$, with $X'=X\times_BB'$ and induced $\phi_X:X'\to X$, \[ (\phi_X^*\omega_{X/B})^{[m]}\cong\phi_X^*(\omega_{X/B}^{[m]}). \] Kollár's condition requires this for every integer $m>0$; where the relative dualizing sheaf itself has the appropriate base-change identification, the usual expression is compatibility of $\omega_{X/B}^{[m]}$ with $\omega_{X'/B'}^{[m]}$. Relative ${\mathbb{Q}}$-Cartierness means a suitable positive reflexive power is locally invertible. The AIM question asks whether relative ${\mathbb{Q}}$-Cartierness is equivalent to Kollár's condition. The general family category and the relative-reflexive conventions on arbitrary test bases need to be specified.

The original says the equivalence is known for smoothings over a discrete valuation ring, reference [Hac], Proposition 10.14, and that it fails in positive characteristic, as Kollár showed at the conference. It attributes the question implicitly to Kollár, contrasting his stronger condition in [Kol90] with the weaker moduli-functor definition used with Shepherd-Barron in [KSB88]. These are source-specific historical qualifications.

The attached report claims that in characteristic zero, for a flat finite-type family of semi-log-canonical surfaces over a reduced original base, the two conditions are equivalent. It says Kollár's hull criterion gives compatibility with every test base change, including nonreduced tests. Reducedness is imposed on $B$, not on every allowed $B'$. It separately invokes Lee--Nakayama examples to say reducedness alone is insufficient without the slc hypothesis. The complete hull and family hypotheses are not supplied.

For a nonreduced original base, the report gives the Altmann--Kollár first-order deformation associated to the interval $[-2/5,2/5]$, in Section 5.7 of their referenced work. Over the first-order base $\operatorname{Spec}k[\varepsilon]/(\varepsilon^2)$, its asserted behavior is \[ \omega_{X/B}^{[5]}\text{ invertible},\qquad \omega_{X/B}^{[3]}\text{ fails base change to the closed fiber}. \] The source calls this deformation "VW-but-not-qG" and identifies its singularity as $\tfrac1{20}(1,11)$. These deformation labels must not be conflated with the weaker local invertible-power condition being tested.

Its candidate lattice witness is that $(1,2/5)$ lies in the relevant half-open zone and the exponent-three coset $M-(3/5)\overline R$, but not on the permitted diagonal. The report says this violates condition $(*)_3$, hence third-power base change, while the fifth power remains invertible. The lattice $M$, vector $\overline R$, zone, diagonal, and condition $(*)_3$ are not defined in the supplied summary; the numerical witness is preserved without pretending these data form a complete explicit counterexample here.

Short English statement

Compare relative invertibility of one canonical power with base-change compatibility of every positive reflexive power. The report claims equivalence for characteristic-zero slc families over reduced bases, but gives a nonreduced first-order example with invertible fifth power and failing third-power base change.

Sources

• AIM workshop "Compact moduli spaces and birational geometry," Problem 1.1: https://aimath.org/WWN/birational/birational.pdf. No PDF was accessed here.
• Local canonical registry aim-algebraic-geometry-notes.json, notes[319], tag problem; permanent UnsolvedMath ID $20000320$, source code catalogue entry. Original references are [Hac], Proposition 10.14, [Kol90], and [KSB88], with the historical Kollár/Shepherd-Barron attribution and positive-characteristic conference counterexample; their full bibliography is not supplied. The verification field claims displaced OCR subscripts and superscripts can be recovered. Original remarks and literature field are otherwise empty.
• Supplied research attempt 1 refers to Kollár's hull criterion, Lee--Nakayama examples, and Altmann--Kollár Section 5.7, and proposes the exponent-three lattice witness. No titles, identifiers, or full authoritative references are supplied. The report is machine generated, not peer reviewed, and requires independent verification before citation.