Frobenius contraction and a log-canonical non-F-pure stress test for the symbolic cube

Definitions and mathematical statement

Takagi's question, identified as Problem 16 by the supplied extraction correction, is the following. Let $(R,\mathfrak m)$ be a regular local ring of characteristic $p>0$ and let $P\subset R$ be a height-two prime. If $\operatorname{Spec}(R/P)$ has log canonical singularities in the discrepancy sense, must \[ P^{(3)}\subset P^2? \] For a prime, $P^{(a)}=P^aR_P\cap R$ is the symbolic power, whereas $P^a$ is the ordinary power. Discrepancy-log-canonicity is not replaced by $F$-purity; its normality and ${\mathbb{Q}}$-Cartier conventions, the field and finiteness hypotheses are not stated in the original excerpt.

The supplied report first claims the unconditional regular-local Frobenius containment: for every height-$h$ prime, every $e\geq1$, and $q=p^e$, \[ P^{(h(q-1)+1)}\subset P^{[q]}\subset P^q, \qquad P^{[q]}=(a^q:a\in P). \] Its stated proof uses flatness of Frobenius to ensure that $P^{[q]}$ is $P$-primary, allowing contraction from $R_P$. In particular, in characteristic $2$, setting $h=2$, $q=2$ gives the requested symbolic-cube containment without any singularity hypothesis.

The report also cites published positive regimes: $F$-purity of $R/P$ gives the symbolic-cube containment in every characteristic, while a strongly $F$-regular quotient gives equality of all symbolic and ordinary powers for a height-two prime. Here $F$-pure means Frobenius is a pure ring map; strong $F$-regularity is the stronger Frobenius splitting condition for every nonzero test element in its usual finite setting. Exact finiteness and locality hypotheses and bibliographic references for these statements are not supplied.

For every algebraically closed field $k$ of characteristic $p\equiv2\pmod3$, the report gives the explicit height-two complete-intersection prime \[ R=k[x,y,z,w]_{(x,y,z,w)},\qquad P=(w,x^3+y^3+z^3)R. \] It claims that the quotient is a log-canonical elliptic cone but is not $F$-pure, yet \[ P^{(a)}=P^a\quad\text{for every integer }a\geq1 \] by the complete-intersection mechanism. This separates log canonicity from $F$-purity without refuting the desired containment. The report says that the fixed-characteristic problem with only discrepancy-log-canonicity remains open in odd characteristic; this is a supplied assessment with no date or full literature details, not independently verified current status.

Short English statement

For a height-two prime in a regular local ring of positive characteristic, ask whether log canonicity of the quotient forces symbolic-cube containment in the ordinary square. The report covers characteristic two unconditionally, cites Frobenius-singularity regimes, and supplies log-canonical non-$F$-pure complete-intersection examples where all powers still agree.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent ID $20000316$, source code catalogue entry; original extraction, supplied numbering correction and research summary.
• AIM workshop "Numerical invariants of singularities and higher-dimensional algebraic varieties", Takagi's Problem 16 according to supplied verification (extracted as 6): https://aimath.org/WWN/singularvariety/problems_final.pdf. Canonical location aim-algebraic-geometry-notes.json, notes[315]; no original remarks or literature entry. No PDF was opened here.
• The summary refers to published $F$-pure and strongly $F$-regular quotient results without supplying authors, titles, theorem numbers or full hypotheses.
• Supplied machine-generated research attempt 1: partial result, problem open at the run and currently partially solved; rigor metadata "proved", novelty confidence low. The non-peer-reviewed synthesis and explicit family require independent verification before citation.