Low-degree integral curves on the Fermat fourfold

Definitions and mathematical statement

For a positive integer $d$, let \[ F_d=\{[x_1:\cdots:x_6]\in{\mathbb{P}}^5_{\mathbb{C}}:x_1^d+\cdots+x_6^d=0\}. \] This is the intended Fermat fourfold interpretation of the damaged exponent typography; the supplied record marks the reconstruction unverified. Heath-Brown's Question 26 asks to describe the parameter spaces of curves of low degree on $F_d$, with the curves taken over ${\mathbb{C}}$. The source gives no numerical meaning of "low degree" and no choice of parameter scheme or restrictions on reducibility.

For a precise integral-curve part of that question, let $C\subset F_d$ be a reduced irreducible projective curve and put $\delta=\deg_{\mathcal O(1)}C$. For each Hilbert polynomial $P(t)=\delta t+1-p_a$, one may ask to describe the integral-curve locus in $\operatorname{Hilb}^{P}(F_d)$, where $p_a$ is the arithmetic genus. A full answer to the original wording may require other curves and scheme structures too; those have not been silently excluded from its scope. The workshop remark says lines are known and expresses the hope of "none or very few" further curves, without explaining what range or exceptional families this refers to.

The attached aid proposes the following partial classification. Given a partition of $\{1,\ldots,6\}$ into three unordered pairs $\{a_i,b_i\}$, choose an ordering within each pair and numbers $\zeta_i\in{\mathbb{C}}$ with $\zeta_i^d=-1$. The equations \[ x_{a_i}=\zeta_i x_{b_i}\quad(i=1,2,3) \] define a standard coordinate-pairing plane $\Pi\subset F_d$. In the range in question there are $15d^3$ such planes. The aid claims that, for every integral curve of degree $\delta\geq2$, \[ d>10\delta-6\quad\Longrightarrow\quad C\text{ lies in a unique standard coordinate-pairing plane}. \] It describes the resulting set of closed points of the integral degree-$\delta$ parameter locus as the disjoint union, over these planes, of the sets of irreducible homogeneous degree-$\delta$ plane forms up to nonzero scalar. For each plane this is the irreducible-form open subset of ${\mathbb{P}}({\mathbb{C}}[u,v,w]_\delta)$. Such curves have arithmetic genus $(\delta-1)(\delta-2)/2$. Crucially, the aid makes no assertion identifying Hilbert-scheme nilpotents, embedded infinitesimal directions, or full parameter-scheme structures with those plane-form loci.

The same aid reports that every nonlinear integral rational curve is contained in a standard plane for $d\geq24$, and that a curve of genus $g\geq1$ is so contained if $d>20g+4$. Whether this latter $g$ means geometric or arithmetic genus is not specified in the summary and must be checked. For conics, $d>10\cdot2-6$ is $d\geq15$; the aid says that Reznick's published identity gives a nonstandard smooth conic on $F_{14}$, making that cutoff sharp. Neither the identity nor the conic is supplied. All of these classifications are reported unreviewed research-aid claims, not independently established results of this draft.

Short English statement

Describe the low-degree complex curves on the Fermat fourfold and their parameter spaces. The attached aid proposes that sufficiently low-degree integral curves lie in unique standard coordinate-pairing planes, while explicitly limiting its parameter description to closed points and leaving Hilbert-scheme infinitesimal structure untreated.

Sources

• Heath-Brown, Question 26, AIM workshop Rational and integral points on higher dimensional varieties: https://aimath.org/WWN/qptsurface2/qptsurface2.pdf. The supplied verification locates Question 26 on PDF p. 53 but does not append the promised restored statement.
• Local registry data/unsolvedmath/problems.json, permanent ID $20000459$, exact source code catalogue entry; canonical aim-algebraic-number-theory-notes.json, notes[110].
• The attached aid invokes Salberger's diagonal-curve inequality and Reznick's published identity without full bibliographic references. Attempt 1 is explicitly machine-generated, not peer reviewed, and requires independent verification before citation.