13 research outputs found
Syzygies and singularities of tensor product surfaces of bidegree (2,1)
Let U be a basepoint free four-dimensional subspace of the space of sections
of O(2,1) on P^1 x P^1. The sections corresponding to U determine a regular map
p_U: P^1 x P^1 --> P^3. We study the associated bigraded ideal I_U in
k[s,t;u,v] from the standpoint of commutative algebra, proving that there are
exactly six numerical types of possible bigraded minimal free resolution. These
resolutions play a key role in determining the implicit equation of the image
p_U(P^1 x P^1), via work of Buse-Jouanolou, Buse-Chardin, Botbol and
Botbol-Dickenstein-Dohm on the approximation complex. In four of the six cases
I_U has a linear first syzygy; remarkably from this we obtain all differentials
in the minimal free resolution. In particular this allows us to describe the
implicit equation and singular locus of the image.Comment: 35 pages 1 figur
Generalized multiplicities of edge ideals
We explore connections between the generalized multiplicities of square-free
monomial ideals and the combinatorial structure of the underlying hypergraphs
using methods of commutative algebra and polyhedral geometry. For instance, we
show the -multiplicity is multiplicative over the connected components of a
hypergraph, and we explicitly relate the -multiplicity of the edge ideal of
a properly connected uniform hypergraph to the Hilbert-Samuel multiplicity of
its special fiber ring. In addition, we provide general bounds for the
generalized multiplicities of the edge ideals and compute these invariants for
classes of uniform hypergraphs.Comment: 24 pages, 6 figures. The results of Theorem 4.6 and Theorem 9.2 are
now more general. To appear in Journal of Algebraic Combinatoric
Generalized Multiplicities of Edge Ideals
We explore connections between the generalized multiplicities of square-free monomial ideals and the combinatorial structure of the underlying hypergraphs using methods of commutative algebra and polyhedral geometry. For instance, we show that the j-multiplicity is multiplicative over the connected components of a hypergraph, and we explicitly relate the j-multiplicity of the edge ideal of a properly connected uniform hypergraph to the Hilbert–Samuel multiplicity of its special fiber ring. In addition, we provide general bounds for the generalized multiplicities of the edge ideals and compute these invariants for classes of uniform hypergraphs
Multiplicities of graded algebras
Let R be a universally catenary locally equidimensional Noetherian ring. We give a multiplicity based criterion for integral dependence of an arbitrary finitely generated R-module over a submodule. Our proof is self-contained and implies the previously known numerical criteria for integral dependence of ideals and modules
Generalized Multiplicities of Edge Ideals
We explore connections between the generalized multiplicities of square-free monomial ideals and the combinatorial structure of the underlying hypergraphs using methods of commutative algebra and polyhedral geometry. For instance, we show that the j-multiplicity is multiplicative over the connected components of a hypergraph, and we explicitly relate the j-multiplicity of the edge ideal of a properly connected uniform hypergraph to the Hilbert–Samuel multiplicity of its special fiber ring. In addition, we provide general bounds for the generalized multiplicities of the edge ideals and compute these invariants for classes of uniform hypergraphs