Finite Hilbert stability of (bi)canonical curves

TL;DR

This paper proves that generic smooth curves embedded via canonical or bicanonical maps have semistable or stable Hilbert points for all m, with specific stability results for special singular and hyperelliptic curves, advancing understanding of curve stability.

Contribution

It establishes finite Hilbert semistability and stability for generic curves and specific singular or hyperelliptic cases, extending previous results and providing new examples of stability thresholds.

Findings

Generic curves have semistable m-th Hilbert points for all m.

Bicanonically embedded curves have stable m-th Hilbert points for all m ≥ 3.

Examples show stability thresholds vary with m.

Abstract

We prove that a generic canonically or bicanonically embedded smooth curve has semistable m-th Hilbert points for all m. We also prove that a generic bicanonically embedded smooth curve has stable m-th Hilbert points for all m \geq 3. In the canonical case, this is accomplished by proving finite Hilbert semistability of special singular curves with G_m-action, namely the canonically embedded balanced ribbon and the canonically embedded balanced double A_{2k+1}-curve. In the bicanonical case, we prove finite Hilbert stability of special hyperelliptic curves, namely Wiman curves. Finally, we give examples of canonically embedded smooth curves whose m-th Hilbert points are non-semistable for low values of m, but become semistable past a definite threshold. (This paper subsumes the previous submission and arXiv:1110.5960).