A Strange Example concerning Property N P

We exhibit an example of a line bundle M on a smooth complex projective variety Y s.t. M satisfies Property Np for some p, the p-module of a minimal resolution of the ideal of the embedding of Y by M is nonzero and M does not satisfy Property Np. Let M be a very ample line bundle on a smooth complex projective variety Y and let φM : Y → P(H (Y,M)) be the map associated to M . We recall the definition of Property Np of Green-Lazarsfeld (see [Gr1], [Gr2] and also [G-L], [Gr3]): let Y be a smooth complex projective variety and let L be a very ample line bundle on Y defining an embedding φL : Y →֒ P = P(H (Y,L)); set S = S(L) = SymH(L), the homogeneous coordinate ring of the projective space P, and consider the graded S-module G = G(L) = ⊕nH (Y,L); let E∗ 0 −→ En −→ En−1 −→ ... −→ E0 −→ G −→ 0 be a minimal graded free resolution of G; let Ep = ⊕q≥0Bp,q⊗S(−q); the line bundle L satisfies Property Np (p ∈ N) if and only if B0,0 = C B0,q = 0 for q > 0 Bi,q = 0 for q 6= i+ 1 and 1 ≤ i ≤ p. (Thus L satisfies Property N0 if and only if Y ⊂ P(H (L)) is projectively normal, i.e. L is normally generated; L satisfies Property N1 if and only if L satisfies Property N0 and the homogeneous ideal I of Y ⊂ P(H (L)) is generated by quadrics; L satisfies Property N2 if and only if L satisfies Property N1 and the module of syzygies among quadratic generators Qi ∈ I is spanned by relations of the form ∑ LiQi = 0, where Li are linear polynomials; and so on.) Address: Elena Rubei, Dipartimento di Matematica “U. Dini”, via Morgagni 67/A 50134 Firenze, Italia E-mail address: [email protected] 2000 Mathematical Subject Classification: 14C20, 13D02.