Rationality of Ueno-Type Varieties

Definitions and mathematical statement

The supplied record does not specify a ground field. Let $k$ denote the intended ground field; this is a placeholder for missing source data, not a universal quantifier over fields. For $m\in\{4,6\}$, let $E_m$ and $g_m$ denote the elliptic curve and distinguished automorphism intended by the source. The record calls $E_4$ anharmonic and $E_6$ equianharmonic, and calls $g_4,g_6$ their standard automorphisms, without supplying equations or formulas. Assume these data and the quotients below are defined over $k$.

For a positive integer $n$, use the diagonal action \[ g_m(P_1,\ldots,P_n)=(g_m(P_1),\ldots,g_m(P_n)), \qquad Q_{n,m}=E_m^n/\langle g_m\rangle. \] The source denotes by $X_{n,m}$ the minimal resolution of $Q_{n,m}$. In dimensions four and five, the supplied record does not explain the intended meaning or choice of this minimal resolution. Whenever the intended resolution exists over $k$, it is birational to $Q_{n,m}$; the rationality questions therefore depend only on this birational class.

An integral $d$-dimensional variety over $k$ is $k$-rational if it is birational over $k$ to $\mathbb P_k^d$. Here $k$-unirational means that there exists a dominant rational map over $k$ from $\mathbb P_k^d$ to the variety.

Questions, conditional on the intended ground field and curve--automorphism data. Determine whether \[ X_{4,6}\dashrightarrow\mathbb P_k^4 \quad\text{is a birational equivalence for some such map,} \] and determine whether there exists a dominant rational map \[ \mathbb P_k^5\dashrightarrow X_{5,6}. \] Equivalently, these ask whether $Q_{4,6}$ is $k$-rational and whether $Q_{5,6}$ is $k$-unirational. The missing field and automorphism specifications must be resolved before these conditional questions constitute a fully specified formulation of the source problem.

Short English statement

For the diagonal cyclic quotients of powers of the source's equianharmonic elliptic curve, determine whether the four-dimensional resolution is rational and whether the five-dimensional resolution is unirational. The ground field, distinguished automorphism, and higher-dimensional meaning of minimal resolution require clarification. These are statement-only questions requiring human mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $30002829$, source problem code catalogue entry, title Rationality of Ueno-Type Varieties; supplied complete source record in the Oberwolfach Reports Open Problems collection. The proposed year is recorded as 2015; no individual proposer is supplied.
• Algebraic Geometry (2015), Oberwolfach Reports, pp. 783--836, DOI 10.4171/owr/2015/15: https://doi.org/10.4171/owr/2015/15. This is also the supplied literature reference labelled OWR Report 15/2015.
• Supplied literature reference: Catanese--Oguiso--Verra, https://arxiv.org/abs/1506.01925.
• Supplied literature reference: Mellit, Ueno-type manifolds, https://doi.org/10.1080/10586458.2019.1691088.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved, reports substantial unirationality and invariant-theoretic results, and states that rationality of $X_{4,6}$ and unirationality of $X_{5,6}$ remain unanswered. This is supplied dated curation, not an independently verified literature assessment; the record requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that it was checked against the official Oberwolfach report. That verification claim is supplied provenance and has not been independently checked for this draft.