Javid Validashti

Numerical criteria for integral dependence

We study multiplicity based criteria for integral dependence of modules or of standard graded algebras, known as ‘Rees criteria’. Rather than using the known numerical invariants, we achieve this goal with a more direct approach by introducing a multiplicity defined as a limit superior of a sequence of normalized lengths; this multiplicity is a non-negative real number that can be irrational.

Multiplicities and Rees valuations

Let (R;m) be a local ring of Krull dimensiond andI ⊆R be an ideal with analytic spreadd. We show that thej-multiplicity ofI is determined by the Rees valuations ofI centered on m. We also discuss a multiplicity that is the limsup of a sequence of lengths that grow at anO(nd) rate.

Syzygies and singularities of tensor product surfaces of bidegree (2,1)

Let U ⊆ H(OP1×P1 (2, 1)) be a basepoint free four-dimensional vector space. The sections corresponding to U determine a regular map φU : P1 × P1 −→ P3. We study the associated bigraded ideal IU ⊆ 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 for φU (P1 × P1), via work of Buse-Jouanolou [5], Buse-Chardin [6], Botbol [2] and Botbol-DickensteinDohm [3] on the approximation complex Z. In four of the six cases IU has a linear first syzygy; remarkably from this we obtain all differentials in the minimal free resolution. In particular this allows us to explicitly describe the implicit equation and singular locus of the image.

On the equality of ordinary and symbolic powers of ideals

We consider the following question concerning the equality of ordinary and symbolic powers of ideals. In a regular local ring, if the ordinary and symbolic powers of a one-dimensional prime ideal are the same up to its height, then are they the same for all powers? We provide supporting evidence of a positive answer for classes of prime ideals defining monomial curves or rings of low multiplicities.

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.