A quadratic-to-cubic maximal-rank wall and its pure resolution

Definitions and mathematical statement

The exact extracted problem begins "Prove this for all $m$," but omits the antecedent map and hypotheses. Its binomial coefficient is damaged, and its discussion of "$m$-wise sums" is not defined. The verification says earlier PDF context recovers the setup, but that recovered context is not actually included in the supplied record. A precise unconditional all-$m$ theorem therefore cannot be reconstructed here.

A conditional interpretation consistent with the research report is the maximal-rank problem for section multiplication. For a smooth projective curve $X$ and line bundle $L$, put \[ \mu_m:\operatorname{Sym}^m H^0(X,L)\longrightarrow H^0(X,L^{\otimes m}). \] If $h^0(X,L)=r+1$, maximal rank means \[ \operatorname{rank}\mu_m =\min\left\{\binom{r+m}{m},h^0(X,L^{\otimes m})\right\}. \] The intended field, generality, genus, degree, embedding, and range of $m$ must be supplied before this is identified with "this." The source says the existing $m=2$ argument is more natural in the injective case because no choice of independent sections is needed, describes a combinatorial game, and suggests higher syzygies afterward; these motivations do not supply the missing hypotheses.

The research aid invokes Larson's maximal rank theorem for general complex curves and reports a transition strip with \[ g=\binom r2+a,\qquad d=g+r, \] where the quadratic cokernel has dimension $a$ and the cubic kernel has dimension $\binom{r+1}{3}-2a$. The strip's actual range and line-bundle/general-embedding assumptions are not given; in particular the dimensions cannot be asserted for arbitrary $a$.

At $a=0$, it reports an arithmetically Cohen--Macaulay homogeneous coordinate ring, an ideal with pure $3$-linear resolution, and Betti formula \[ \beta_i=\binom{r+1}{i+2}\binom{i+1}{2},\qquad g=\binom r2,\quad d=\binom{r+1}{2}. \] The index convention and full resolution are not supplied. It proposes the union of all coordinate lines in ${\mathbb{P}}^r$ as a characteristic-free monomial certificate with the same Hilbert series and Betti table, whose cubic basis is exactly the monomials supported on at most two variables. For $r=3$, it claims a flat Hilbert--Burch determinantal smoothing preserving the sixteen selected cubic monomials. These are unreviewed special-wall and certificate statements, not a replacement of the original all-$m$ question by quadratic/cubic cases alone.

Short English statement

The original all-$m$ assertion lacks its preceding definition and has damaged combinatorial notation. A multiplication-map maximal-rank interpretation is plausible from the report but remains conditional. The report supplies a complex-curve transition strip, a pure-resolution wall, and a rank-three monomial smoothing example, without the complete original hypotheses.

Sources

• Local UnsolvedMath registry, permanent ID $20000230$, source code catalogue entry; created and updated 2026-08-14. Status solved; classification SOLVED-IN-LITERATURE; difficulty L3. Proposer field null, while verification says preceding text suggests the problem by David Jensen.
• AIM Problem Lists, Algebraic geometry, workshop Degenerations in algebraic geometry, Problem 8.1, source type PDF, https://aimath.org/pastworkshops/degenalggeomproblems.pdf. Canonical location aim-algebraic-geometry-notes.json, notes[229], tag problem; statement status exact. Verification explicitly notes omitted preceding map, damaged binomial coefficient, run-together words, and a reported page-4 inspection, but does not include its promised recovered context. Original remarks and literature field empty.
• The research summary invokes Larson's maximal rank theorem and a Hilbert--Burch deformation without exact primary citations or the smoothing matrix.
• Supplied research attempt 1: result partial_result, run status solved, rigor label proved, low novelty confidence. Candidate contribution: derived wall theorem, Betti formula, characteristic-free coordinate-line certificate, and $r=3$ flat smoothing. Explicit caveat: machine-generated research aid, not peer reviewed; independently verify before citation.
• Dataset attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0; original problem attribution remains to the supplied AIM workshop collection.