A construction of codimension three arithmetically Gorenstein subschemes of projective space

Abstract

Abstract. This paper presents a construction method for a class of codimension three arithmetically Gorenstein subschemes of projective space. These schemes are obtained from degeneracy loci of sections of certain specially constructed rank three reflexive sheaves. In contrast to the structure theorem of Buchsbaum and Eisenbud, we cannot obtain every arithmetically Gorenstein codimension three subscheme by our method. However, certain geometric applications are facilitated by the geometric aspect of this construction, and we discuss several examples of this in the final section.

The Weak and Strong Lefschetz properties for Artinian K-algebras

“…The bundle E used in this section is a Buchsbaum-Rim sheaf. The interested reader can find a much more extensive treatment of such sheaves and their properties in [17], [18] and [20].…”

A constructive approach to one-dimensional Gorenstein $k$-algebras

“…Note that the lack of a general structure of homogeneous Gorenstein ideals of higher codimension is the main obstacle to extending the Gorenstein liaison theory in codimension at least three; the codimension two Gorenstein liaison case is well understood, see [23]. See, for instance, [27], [24] and [22] for some constructions of particular families of Gorenstein algebras.…”

Pfaffian representations of cubic surfaces

“…As pointed out by the referee, codimension three AG subschemes have been considered also in [21], where they are obtained as zero loci of sections of certain rank-three sheaves. In the case of five points in general position in P 3 K, it turns out that all such sets are the zero loci of appropriate sections of the bundle Ω P 3 (3), which can be interpreted as four-tuple quadrics, that is, linear combinations (using linear forms as coefficients) of the syzygies of the map (x 0 x 1 x 2 x 3 ).…”