Gauss-inseparability strata and a Frobenius plane-curve family

Definitions and mathematical statement

Fix a prime $p$ and work over $\overline{{\mathbb{F}}}_p$. On the Hilbert locus $U$ of smooth geometrically integral nondegenerate curves $C\subset{\mathbb{P}}^r$, $r\geq2$, of degree $d$ and genus $g$, the Gauss map \[ \gamma_C:C\to\operatorname{Gr}(2,r+1) \] sends a point to its embedded tangent projective line. It is nonconstant on this locus and hence finite onto its image. Define \[ Z_e=\{[C]\in U:[k(C):k(\gamma_C(C))]_{\mathrm{insep}}=p^e\}, \qquad e\geq0. \] Here the degree means the inseparable part, not an unstated assumption that the total degree is purely inseparable; $e=0$ includes generically separable Gauss maps. These sets partition $U$. The original says only smooth curves and prints positive $e$ before taking a union over $e\geq0$; degenerate, disconnected, or constant-Gauss-map cases require additional conventions outside this reconstructed locus.

The problem asks what geometric properties the $Z_e$ have, whether this stratification can construct exotic Hilbert components existing only in characteristic $p$, and how $\operatorname{Gal}(\overline{{\mathbb{F}}}_p/{\mathbb{F}}_p)$ acts. It also suggests other stratifications, such as by the topological type of a ramification divisor. A precise use of exotic requires a chosen integral Hilbert model and a lifting/vertical-component criterion; the source does not specify one.

The attached aid reports that each fixed-degree set $Z_e$ is constructible, defined over ${\mathbb{F}}_p$, and Galois stable. It gives the necessary condition \[ p^e\mid 2d+2g-2, \] and says each geometric Hilbert component meeting the indicated locus has one generic exponent. These are set-theoretic/constructibility statements, not claims that every $Z_e$ is a closed subscheme or a Hilbert component.

For a concrete family take $e\geq1$, $q=p^e$, and an invertible three-by-three matrix $B$, up to nonzero scalar. In ${\mathbb{P}}^2$, put \[ F_B(X)=(X^{[q]})^TBX, \qquad X^{[q]}=(X_0^q,X_1^q,X_2^q)^T. \] The aid reports that these smooth degree-$q+1$ curves, of genus $q(q-1)/2$, form a subvariety $W_e\cong\operatorname{PGL}_3$ of dimension $8$, closed relative to the smooth plane-curve Hilbert locus, contained in $Z_e$. Their Gauss maps are exactly \[ [X]\longmapsto[B^TX^{[q]}]. \] The arithmetic-Frobenius convention $a\mapsto a^p$ acts on parameters by $[B]\mapsto[B^{(p)}]$. This concerns embedded curves, not a quotient by projective equivalence. The restriction $e\geq1$ is needed for this matrix parametrization; the separable stratum $e=0$ is part of the general question but is not asserted to have the same description.

The aid emphasizes that $W_e$ lies properly in the unique horizontal plane-curve Hilbert component. Consequently this explicit Frobenius family is not itself an exotic component. Constructibility, an inseparable Gauss map, or a characteristic-specific closed family alone does not prove existence of a new Hilbert component. These are supplied unreviewed partial results; the broader component-construction question remains unresolved by this family.

Short English statement

Study Hilbert-locus strata by the inseparable degree of a curve's Gauss map, their Galois action, and possible characteristic-specific components. The supplied aid gives constructibility and a Frobenius plane-curve family, while explicitly noting that the family is a proper subvariety of a horizontal component.

Sources

• Problem 3, Components of Hilbert Schemes, recorded by Izzet Coskun and edited by Li Li after the AIM workshop of 19--23 July 2010, according to the supplied verification: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf. The original remark also suggests ramification-divisor stratifications.
• Local registry data/unsolvedmath/problems.json, permanent ID $20000804$, exact source code catalogue entry; canonical aim-arithmetic-geometry-notes.json, notes[49].
• The attached aid supplies the constructibility, divisibility, and matrix-family results without full external citations. Attempt 2 is explicitly machine-generated, not peer reviewed, and requires independent verification before citation.