A Nonlinear Version of Noether's Type Theorem

Abstract

Let L be a line bundle on a smooth curve C, which defines a birational morphism onto Φ(C) ⊂ P ^r . We prove that, under suitable assumptions on L, which are satisfied by Castelnuovo's curves, a generic section in H ⁰(C, L ²) can be written as α² + β² + γ², with α, β, γ ∈ H ⁰(C, L). If there are no quadrics of rank 3 containing Φ(C), this is true for any section. For canonical curves, this gives a non linear version of Noether's Theorem.

Key Word:

• Canonical sections

2000 Mathematics Subject Classification:

• 14N15
• 14H51

Acknowledgment

Partially supported by (1) PRIN 2003: Spazi di moduli e teoria di Lie. (Murst); (2) Gnsaga; (3) Far 2002 (Pavia): Varietà algebriche, calcolo algebrico, grafi orientati e topologici.

Notes

^#Communicated by L. Ein.