Let R := k[X1, . . . , XN ] be the polynomial ring in N indeterminates over a field k of characteristic 0 with deg(Xi) = 1 for i = 1, . . . , N , and let I be a homogeneous ideal of R . The Hilbert function of I is the function from N to N which associates to every natural number d the dimension of Id as a k -vectorspace. I has an essentially unique minimal graded free resolution 0 −→ Lm dm −→L...

Rank 2 indecomposable arithmetically Cohen-Macaulay bundles E on a nonsingular cubic surface X in P are classified, by means of the possible forms taken by the minimal graded free resolution of E over P. The admissible values of the Chern classes of E are listed and the vanishing locus of a general section of E is studied. Properties of E such as slope (semi) stability and simplicity are invest...

The main result, Theorem 1, provides an algorithm for determining the minimal free resolution of fat point subschemes of P involving up to 8 general points of arbitrary multiplicities; the resolutions obtained hold for any algebraically closed field, independent of the characteristic. The algorithm works by giving a formula in nice cases, and a reduction to the nice cases otherwise. The algorit...

The Riemann-Roch theorem on a graph G is closely related to Alexander duality in combinatorial commutive algebra. We study the lattice ideal given by chip firing on G and the initial ideal whose standard monomials are the G-parking functions. When G is a saturated graph, these ideals are generic and the Scarf complex is a minimal free resolution. Otherwise, syzygies are obtained by degeneration...

Consider the general n-gon with vertices at the points 1,2, . . . ,n. Then its suspension involves two more vertices, say at n+1 and n+2. Let R be the polynomial ring k[x1,x2, . . . ,xn], where k is any field. Then we can associate an ideal I to our suspension in the Stanley-Reisner sense. In this paper, we find a free minimal resolution and the Betti numbers of the R-module R/I.

In this note we develop some of the properties of separators of points in a multiprojective space. In particular, we prove multigraded analogs of results of Geramita, Maroscia, and Roberts relating the Hilbert function of X and X \ {P} via the degree of a separator, and Abrescia, Bazzotti, and Marino relating the degree of a separator to shifts in the minimal multigraded free resolution of the ...