It is unknown if an Artinian level O-sequence of codimension 3 and type r (≥ 2) is unimodal, while it is known that any Gorenstein O-sequence of codimension 3 is unimodal. We show that some Artinian non-unimodal O-sequence of codimension 3 cannot be level. We also find another non-level case: if some Artinian algebra A of codimension 3 has the Hilbert function H : h0 h1 · · · hd−1 hd · · · hd }...

Let X be a hyperkähler manifold. Trianalytic subvarieties of X are subvarieties which are complex analytic with respect to all complex structures induced by the hyperkähler structure. Given a K3 surface M , the Hilbert scheme classifying zero-dimensional subschemes of M admits a hy-perkähler structure. We show that for M generic, there are no trianalytic subvarieties of the Hilbert scheme. This...

Gorenstein liaison seems to be the natural notion to generalize to higher codimension the well-known results about liaison o varieties of codimension 2 in projective space. In this paper we study points in P and cuves in P in an attempt to see how far typical codimension 2 results will extend. While the results are satisfactory for small degree, we find in each case examples where we cannot dec...

1. Write-up Wish List 2 2. To Do List 2 3. History 3 4. Basic definitions and notation 4 5. Codimension One Finite Length [Sarah] 5 6. Codimension Two Finite Length [Sarah] 5 7. Codimension Three Finite Length 7 8. Attempts at Codimension Four 8 9. Betti Death 8 10. Alternative diagram decompostions for complete intersections 9 11. Decomposition Via Separating the Betti Diagram 13 11.1. Heurist...