On the almost Gorenstein property in the Rees algebras of contracted ideals

We explore the question of when the Rees algebra R(I)=⊕n≥0In of I is an almost Gorenstein graded ring, where R is a 2-dimensional regular local ring and I is a contracted ideal of R. Recently, we showed that R(I) is an almost Gorenstein graded ring for every integrally closed ideal I of R. The main results of the present article show that if I is a contracted ideal with o(I)≤2, then R(I) is an almost Gorenstein graded ring, while if o(I)≥3, then R(I) is not necessarily an almost Gorenstein graded ring, even though I is a contracted stable ideal. Thus both affirmative and negative answers are given.