Syzygies capture intricate geometric properties of a subvariety in projective space. However, when the ambient space is a product of projective spaces or a more general smooth projective toric variety, minimal free resolutions over the Cox ring are too long and contain many geometrically superfluous summands. In this paper, we construct some much shorter free complexes that better encode the ge...

A tetrahedral curve is an unmixed, usually non-reduced, one-dimensional subscheme of projective 3-space whose homogeneous ideal is the intersection of powers of the ideals of the six coordinate lines. The second and third authors have shown that these curves have very nice combinatorial properties, and they have made a careful study of the even liaison classes of these curves. We build on this ...

A sparse generic matrix is a matrix whose entries are distinct variables and zeros. Such matrices were studied by Giusti and Merle who computed some invariants of their ideals of maximal minors. In this paper we extend these results by computing a minimal free resolution for all such sparse determinantal ideals. We do so by introducing a technique for pruning minimal free resolutions when a sub...

Let P be a finite partially ordered set with unique minimal element 0̂. We study the Betti poset of P , created by deleting elements q ∈ P for which the open interval (0̂, q) is acyclic. Using basic simplicial topology, we demonstrate an isomorphism in homology between open intervals of the form (0̂, p) ⊂ P and corresponding open intervals in the Betti poset. Our motivating application is that the...

Homogeneous bundles on P2 = SL(3)/P can be described by representations of the parabolic subgroup P . In 1966 Ramanan proved that if ρ is an irreducible representation of P then the induced bundle Eρ on P 2 is simple and even stable (see [Ram]). Since P is not a reductive group, there is a lot of indecomposable reducible representations of P and to classify homogeneous bundles on P2 and among t...

Let S = K[x1, . . . ,xn] be a polynomial ring and R = S/I where I ⊂ S is a graded ideal. The Multiplicity Conjecture of Herzog, Huneke, and Srinivasan states that the multiplicity of R is bounded above by a function of the maximal shifts in the minimal graded free resolution of R over S as well as bounded below by a function of the minimal shifts if R is Cohen–Macaulay. In this paper we study t...

This paper studies Ulrich ideals in hypersurface rings. A characterization of is given. Using this characterization, we construct a minimal free resolution an ideal concretely. We also explore ring the form R=k[[X,Y]]/(f).

Let $$S=K[x_1,\ldots ,x_n]$$ , where K is a field, and $$t_i$$ denotes the maximal shift in minimal graded free S-resolution of algebra S/I at degree i. In this paper, we prove:

In recent work, Clark and Tchernev introduced the notion of monomial resolution supported on a poset. In this talk, I will review this notion, and the related notion of monomial resolution supported on a CW complex. I will discuss a result of mine that shows that resolutions supported on a CW complex are also supported on the face poset of a CW complex. This relates to the work of Clark and Tch...