Let G be a graph and let I= I (G) be its edge ideal. When G is unicyclic, we give a
decomposition of symbolic powers of I in terms of its ordinary powers. This allows us to …

The symbolic powers I (n) of a radical ideal I in a polynomial ring consist of the functions that
vanish up to order n in the variety defined by I. These do not necessarily coincide with the …

In a polynomial ring over a perfect field, the symbolic powers of a prime ideal can be
described via differential operators: a classical result by Zariski and Nagata says that the n …

Let G be a finite simple graph and J (G) denote its vertex cover ideal in a polynomial ring
over a field 𝕂. In this paper, we show that all symbolic powers of vertex cover ideals of …

This book contains reorganized and extended versions of our lectures at PRAGMATIC 2017,
held from June 19th to July 7th, 2017. PRAGMATIC (Promotion of Research in Algebraic …

Let k be a field of positive characteristic. Building on the work of the second named author,
we define a new class of k-algebras, called diagonally F-regular algebras, for which the so …

Let IX be the saturated homogeneous ideal defining a codimension two arithmetically
Cohen-Macaulay scheme X⊆ P n, and let IX (m) denote its m-th symbolic power. We are …

It has long been known how to read information about the socle degrees of the local
cohomology H m 0 (M) of a graded module over a polynomial ring R from the twists in …

We study the symbolic powers of determinantal ideals of generic, generic symmetric, and
Hankel matrices of variables, and of Pfaffians of generic skew-symmetric matrices, in prime …

Inspiredbytherecent workofLuand O'Rourke, westudytheai-invariantsof (symbolic)
powersofsomegradedideals. WhenIandJ aretwogradedidealsintwodistinct …