A strange example on Property Np

Abstract

 We exhibit an example of a line bundle M on a smooth complex projective variety Y s.t. M satisfies Property N ₁₀ , the 10-module of a minimal resolution of the ideal of the embedding of Y by M is nonzero and M ² does not satisfy Property N ₁₀ . Thus this is a completely convincing example showing that surprisingly it is not true that if a line bundle M satisfies Property N _p then any power of M satisfies Property N _p . We recall that in [Ru] we proved the following statement: if M is a line bundle on a smooth complex projective variety and M satisfies Property N _p then M ^s satisfies Property N _p if s≥p.