The multiplicity structure of a polynomial system at an isolated zero is identified with the dual space consisting of differential functionals vanishing on the entire polynomial ideal. Algorithms have been developed for computing dual spaces as the kernels of Macaulay matrices. These previous algorithms face a formidable obstacle in handling Macaulay matrices whose dimensions grow rapidly when ...

Conjecture, and the big and small Cohen Macaulay Conjectures. Many of these are shown to imply others. This document contains notes for a course taught by Tom Marley during the 2009 spring semester at the University of Nebraska-Lincoln. The notes loosely follow the treatment given in Chapters 8 and 9 of Cohen-Macaulay Rings, by W. Bruns and J. Herzog, although many other sources, including arti...

The Multiplicity conjecture of Herzog, Huneke, and Srinivasan states an upper bound for the multiplicity of any graded k-algebra as well as a lower bound for Cohen-Macaulay algebras. In this note we extend this conjecture in several directions. We discuss when these bounds are sharp, find a sharp lower bound in case of not necessarily arithmetically Cohen-Macaulay one-dimensional schemes of 3-s...

for some N ≥ k + 1, then the equality in (1.2) occurs only when a = 0. Kruskal-Macaulay functions are relevant for their applications to the study of antichains in multisets (see for example [11, 2]), posets, rings and polyhedral combinatorics (see [5] and the survey [3]). In particular, they play and important role in proving results, extensions and generalizations of classical problems concer...

We show that the algebraic invariants multiplicity and depth of a graded ideal in the polynomial ring are closely connected to the fan structure of its generic tropical variety in the constant coefficient case. Generically the multiplicity of the ideal is shown to correspond directly to a natural definition of multiplicity of cones of tropical varieties. Moreover, we can recover information on ...

Let (R,m) be a commutative Noetherian local ring. Suppose that M and N are finitely generated modules over R such that M has finite projective dimension and such that TorRi (M,N) = 0 for all i > 0. The main result of this note gives a condition on M which is necessary and sufficient for the tensor product of M and N to be a Cohen–Macaulay module over R, provided N is itself a Cohen–Macaulay mod...

A certain squarefree monomial ideal HP arising from a finite partially ordered set P will be studied from viewpoints of both commutative algbera and combinatorics. First, it is proved that the defining ideal of the Rees algebra of HP possesses a quadratic Gröbner basis. Thus in particular all powers of HP have linear resolutions. Second, the minimal free graded resolution of HP will be construc...

A certain squarefree monomial ideal HP arising from a finite partially ordered set P will be studied from viewpoints of both commutative algebra and combinatorics. First, it is proved that the defining ideal of the Rees algebra of HP possesses a quadratic Gröbner basis. Thus in particular all powers of HP have linear resolutions. Second, the minimal free graded resolution of HP will be construc...

The goal of this note is to study the equations of a curve C ⊂ P3 = PK that is contained in some quadric Q. By a curve, we mean a pure one-dimensional locally Cohen-Macaulay subscheme (i.e. without zero-dimensional components.) We assume that the field K is algebraically closed. If C ⊂ Q is arithmetically Cohen-Macaulay, then, by Dubreil’s Theorem, C is defined by at most 3 equations. The conve...

Let X be a doubling of a smooth surface F in a smooth threefold and let C ⊂ X be a locally Cohen-Macaulay curve. Then C gives rise to two effective divisors on F , namely the curve part P of C ∩ F and the curve R residual to C ∩ F in C. We show that a general deformation of R on F lifts to a deformation of C on X when a certain cohomology group vanishes and give applications to the study of Hil...