Brauer Groups of Smooth Models of Symmetric Powers

Definitions and mathematical statement

Let $X$ be a variety. For a nonnegative integer $m$, the symmetric power, when the relevant quotient exists, is \[ X_m=\operatorname{Sym}^{m}(X)=X^{m}/\mathfrak S_m, \] where $\mathfrak S_m$ acts by permuting the factors. This recalls the usual construction; the supplied record does not specify the intended range of $m$, the base field, or the class of varieties under consideration. Let $X'_m$ be a smooth model of $X_m$. Under the usual birational interpretation, this means a smooth variety birational to $X_m$; the record does not specify whether the model must be proper or projective, or must carry a resolution morphism to $X_m$.

The supplied question is: can the Brauer group \[ \operatorname{Br}(X'_m) \] be estimated in terms of $\operatorname{Br}(X)$?

The meaning of $\operatorname{Br}$ is not specified. Standard possibilities include the Azumaya Brauer group, consisting of Morita-equivalence classes of Azumaya algebras with group operation induced by tensor product, and the cohomological Brauer group \[ \operatorname{Br}'(Y)=H^2_{\mathrm{\acute et}}(Y,\mathbb G_m)_{\mathrm{tors}}. \] These conventions are listed as alternatives; their identification is not assumed here. The record also does not identify the invariant to be estimated, the form of the estimate, or its permitted dependence on $m$, the base field, the dimension of $X$, or the chosen smooth model. Thus the displayed question preserves the supplied target, but the missing qualifications prevent its conversion into a uniquely specified quantified bound or theorem. The supplied literature assessment additionally mentions a general vanishing theorem without stating its hypotheses or conclusion; no vanishing assertion is inferred.

Short English statement

The question asks how the Brauer group of a smooth model of a symmetric power of $X$ can be controlled by the Brauer group of $X$. The base field, hypotheses on $X$, range of $m$, meaning of smooth model and Brauer group, and desired estimate remain unspecified. The supplied open label is dated curation, not an independently verified status.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004489$, title: Brauer Groups of Smooth Models of Symmetric Powers; collection: Oberwolfach Reports Open Problems; exact source problem code: catalogue entry. No individual proposer is supplied; the proposed year is recorded as 2020.
• Supplied source citation: Komplexe Analysis -- Algebraicity and Transcendence (2021), Oberwolfach Reports, pp. 1263--1313, DOI 10.4171/owr/2020/24. https://doi.org/10.4171/owr/2020/24. The record describes this collection as extracted from workshop open-problem discussions. Its statement status is supplied as corrected_verified, with a supplied claim that the clean statement was checked against the official report and preserves its hypotheses, notation, and target. That verification has not been independently performed here.
• Supplied literature source: Cadorel--Campana--Rousseau, Hyperbolicity and specialness of symmetric powers. https://www.numdam.org/item/JEP_2022__9__381_0.pdf.
• Supplied literature source: OWR Complex Analysis report. https://ems.press/content/serial-article-files/46857.
• Literature assessment dated exactly 2026-08-22: the supplied status is open. The supplied assessment says that the Brauer-group question for smooth models of symmetric powers remains posed as open and that existing work on specialness and rationality does not provide the requested general estimate or vanishing theorem. This is supplied dated curation derived from the OWR record, not independently verified. The supplied background expressly calls for independent verification of time-sensitive status and cited claims.