Symbolic powers of monomial ideals and vertex cover algebras

Asymptotic growth of algebras associated to powers of ideals

“…Set J k = (I ♯ ) k : s ∞ and I k = (I k ) * R. Then S J (I ♯ ) is finitely generated if and only if for some integer d > 0 one has (J k ) d = J dk for all k, and a corresponding statement holds for A(I), see for example [13,Theorem] or [24].…”

“…We use the criterion which says that S J (I) is finitely generated if and only if for some integer d > 0 the dth Veronese subalgebra S J (I) (d) is standard graded, see for example [13…”

“…Symbolic Rees algebras of squarefree monomial ideals can be identified with vertex cover algebras. This class of algebras is always finitely generated, as shown in [13].…”

Associated primes of monomial ideals and odd holes in graphs

“…This classification allows us to describe the irreducible decomposition of J 2 . The algebra of vertex covers of a (hyper)graph was first studied by Herzog, Hibi and Trung [19]. In [19], the authors use the terminology of decomposable and indecomposable covers instead of reducible and irreducible covers.…”

“…The algebra of vertex covers of a (hyper)graph was first studied by Herzog, Hibi and Trung [19]. In [19], the authors use the terminology of decomposable and indecomposable covers instead of reducible and irreducible covers. We choose to use reducible and irreducible covers to be consistent with the result of [6] that we require.…”