Here we study rank 2 arithmetically Cohen-Macaulay vector bundles on a a ruled surface over a smooth genus q curve, essentially proving their non-existence if q ≥ 2 and the ruled surface is rather balanced.

The flag f-vectors of three-colored complexes are characterized. This also characterizes the flag h-vectors of balanced Cohen-Macaulay complexes of dimension two, as well as the flag h-vectors of balanced shellable complexes of dimension two.

Solving a system of polynomial equations is a ubiquitous problem in the applications of mathematics. Until recently, it has been hopeless to find explicit solutions to such systems, and mathematics has instead developed deep and powerful theories about the solutions to polynomial equations. Enumerative Geometry is concerned with counting the number of solutions when the polynomials come from a ...

Two formulas for the multiplicity of the fiber cone F (I) = ⊕∞n=0I /mI of an m-primary ideal of a d-dimensional Cohen-Macaulay local ring (R,m) are derived in terms of the mixed multiplicity ed−1(m|I), the multiplicity e(I) and superficial elements. As a consequence, the Cohen-Macaulay property of F (I) when I has minimal mixed multiplicity or almost minimal mixed multiplicity is characterized ...

We present a novel formulation, based on the latest advancement in polynomial system solving via linear algebra, for identifying limit cycles in general n-dimensional autonomous nonlinear polynomial systems. The condition for the existence of an algebraic limit cycle is first set up and cast into a Macaulay matrix format whereby polynomials are regarded as coefficient vectors of monomials. This...

A picture P of a graph G = (V, E) consists of a point P(v) for each vertex v ∈ V and a line P(e) for each edge e ∈ E, all lying in the projective plane over a field k and subject to containment conditions corresponding to incidence in G. A graph variety is an algebraic set whose points parametrize pictures of G. We consider three kinds of graph varieties: the picture space X (G) of all pictures...

We show that the linked Grassmannian scheme, which arises in a functorial compactification of spaces of limit linear series, and in local models of certain Shimura varieties, is Cohen-Macaulay, reduced, and flat. We give an application to spaces of limit linear series.