The maximal-contact linear-space arrangement of a plane elliptic cubic

Definitions and mathematical statement

The entire original item is the setup sentence: "Problem 3 (Suggested by Michel) Let $X\subset{\mathbb{P}}^2$ be an elliptic curve." It contains no question, conclusion to prove, or requested quantity. An elliptic curve is a smooth projective geometrically connected genus-one curve with the appropriate field/point convention; the supplied embedding is the only additional datum. The field and chosen origin are not specified. No problem is invented from the later registry title.

The supplied research aid reports that the original PDF confirms this is setup for a separately numbered next problem. It nevertheless gives an auxiliary structural proposition, distinct from the original item. In that proposition take a smooth plane cubic $E$ with a flex origin $o$ and a positive degree $d$. It claims that degree-$d$ plane curves not containing $E$ and supported on a single intersection point $p$ exist precisely when $[3d](p-o)=0$, equivalently when the contact divisor is $3d[p]$ and $\mathcal O_E(d)\cong\mathcal O_E(3d[p])$ under the flex normalization. "$3d$-torsion" means killed by $3d$, not necessarily of exact order $3d$.

Writing $g_d=(d-1)(d-2)/2$, it reports that the projective closure of each fixed-$p$ parameter locus is a linear ${\mathbb{P}}^{g_d}$ and the non-containing locus is ${\mathbb{A}}^{g_d}$. For distinct eligible points, every pair of closures is claimed to intersect scheme-theoretically in the same core of forms divisible by the equation of $E$, denoted ${\mathbb{P}}^{g_d-1}$. When $d=1,2$, this notation requires the empty-core convention ${\mathbb{P}}^{-1}=\varnothing$, which the summary does not spell out.

Over an algebraically closed characteristic-zero field, the report counts $(3d)^2$ eligible points and hence a disjoint union of that many affine loci after deleting the core. It also claims an affine-bundle version over a smooth family of cubics, without specifying flex/torsion markings or base-change assumptions. It explicitly treats rationality as a separate delta-invariant condition and claims no enumerative count for the next record. All this is unreviewed auxiliary content, not a reconstructed missing question or a solution of the separately numbered problem.

Short English statement

This record supplies only an elliptic-curve setup and poses no mathematical question. Its research aid adds a maximal-contact parameter-space proposition with torsion and common-core conditions, but that proposition must remain separate from the absent original target and from any next-record curve count.

Sources

• Local UnsolvedMath registry, permanent ID $20000219$, source code catalogue entry; created and updated 2026-08-14. Status open; classification OPEN-TRIAGE; difficulty L2, also research-suggested L2. Proposer field null; original setup says "Suggested by Michel," without a surname or year.
• AIM Problem Lists, Algebraic geometry, workshop Degenerations in algebraic geometry, source item 3, source type PDF, https://aimath.org/pastworkshops/degenalggeomproblems.pdf. Canonical location aim-algebraic-geometry-notes.json, notes[218], tag problem; statement status exact. Original remarks and literature fields empty. The report says the PDF confirms that this sentence is setup for the separately numbered next problem; this draft does not claim a new source visit.
• Supplied research attempt 1: result context_only, run status not_a_problem, rigor label proved, low novelty confidence. Candidate contribution structural_proposition: common-core maximal-contact linear spaces and affine-family behavior. No exact primary references or detailed family construction are supplied. Explicit caveat: machine-generated research aid, not peer reviewed; independently verify before citation.
• Dataset attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0; original setup attribution remains to the supplied AIM workshop collection.