A stabilizer-aware speciality obstruction to rigid projective curves

Definitions and mathematical statement

The source attributes to Joe Harris the question whether a nondegenerate projectively rigid curve in ${\mathbb{P}}^n$ can be different from a rational normal curve. Nondegenerate means not contained in a hyperplane; the source defines rigid by requiring its deformations to be induced by automorphisms of ${\mathbb{P}}^n$. It does not explicitly require smoothness or irreducibility, fix a field, or distinguish actual deformations from all infinitesimal ones.

The attached aid treats smooth irreducible curves over an algebraically closed field of characteristic zero. For $C\subset{\mathbb{P}}^n$, $n\ge1$, write $g=h^1(C,\mathcal O_C)$, $d=\deg C$, and \[ s=\dim\operatorname{Stab}_{\operatorname{PGL}_{n+1}}(C). \] Projective rigidity in the actual Hilbert-deformation sense requires the projective orbit to account for the local reduced deformation locus. The aid states the necessary speciality inequality \[ (n+1)h^1(C,\mathcal O_C(1))\ge4g-3+s. \] Its underlying component-versus-orbit comparison uses the lower bound \[ \dim_{[C]}\operatorname{Hilb}\ge\chi(N_{C/{\mathbb{P}}^n})=(n+1)d+(n-3)(1-g) \] and orbit dimension $(n+1)^2-1-s$, together with Riemann--Roch and nondegeneracy. A positive dimension defect supplies actual non-projective Hilbert deformations, not merely surplus potentially obstructed tangent vectors.

The aid gives these consequences: every positive-genus nonspecial embedding, meaning $h^1(C,\mathcal O_C(1))=0$, has a non-projective deformation; no smooth elliptic curve is projectively rigid in any projective space; and among smooth rational nondegenerate curves only rational normal curves are rigid. A rational normal curve is the degree-$n$ image of ${\mathbb{P}}^1$ under the complete system $|\mathcal O_{{\mathbb{P}}^1}(n)|$. It further reports that these are the only smooth rigid curves for $n\le3$, leaving the special higher-genus case in $n\ge4$ open in its literature assessment. None of these restricted conclusions settles any additional singular or reducible cases allowed by the literal wording.

Short English statement

Ask whether Harris's projective-rigidity phenomenon extends beyond rational normal curves. The supplied aid derives a stabilizer-sensitive speciality obstruction for smooth curves and rules out several classes, while leaving special higher-genus embeddings in higher dimension outside its conclusion.

Sources

• Local UnsolvedMath registry, permanent ID $20000824$, source code catalogue entry; canonical aim-arithmetic-geometry-notes.json, notes[69].
• American Institute of Mathematics, workshop "Components of Hilbert schemes," question attributed to Joe Harris: https://aimath.org/WWN/hilbertschemes/hilbertschemes.pdf. The metadata gives source item 2, but the extracted number is split as "2" and "3" and may represent 23. The verification field reports an unambiguous statement on PDF page 2; no independent PDF inspection is claimed here. No original literature or remarks are supplied.
• Attached research attempt 1: partial result, recorded run status open, rigor proved, novelty confidence low; registry status partially solved, classification PARTIAL-PROGRESS. Its dimension-obstruction and low-dimensional claims are machine generated and not peer reviewed; independently verify before citation.