Abstract Fix a poset Q on $\{x_1,\ldots ,x_n\}$ . A -Borel monomial ideal $I \subseteq \mathbb {K}[x_1,\ldots ,x_n]$ is whose monomials are closed under the Borel-like moves induced by I principal ideal, denoted $I=Q(m)$ , if there m such that all minimal generators of can be obtained via from In this paper we study powers ideals. Among our results, show $Q(m)$ agree with their symbolic powers,...

We study the Mathias–Prikry and the Laver type forcings associated with filters and coideals. We isolate a crucial combinatorial property of Mathias reals, and prove that Mathias–Prikry forcings with summable ideals are all mutually bi-embeddable. We show that Mathias forcing associated with the complement of an analytic ideal does add a dominating real. We also characterize filters for which t...

We prove that the multi-Rees algebra $${\mathcal {R}}(I_1 \oplus \cdots I_r)$$ of a collection strongly stable ideals $$I_1, \ldots , I_r$$ is fiber type. In particular, we provide Gröbner basis for its defining ideal as union special and binomial syzygies. also study Koszulness based on parameters associated to collection. Furthermore, establish quadratic I_2)$$ where each has two quadric Bore...

Let I be an ideal of subsets of a Polish space X, containing all singletons and possessing a Borel basis. Assuming that I does not satisfy ccc, we consider the following conditions (B), (M) and (D). Condition (B) states that there is a disjoint family F ⊆ P (X) of size c, consisting of Borel sets which are not in I . Condition (M) states that there is a Borel function f : X → X with f[{x}] / ∈ ...

Let S = C[x1, . . . , xn+1] be the set of polynomials in n+1 variables over C (usually corresponding to the homogeneous coordinate ring of P). Let ≻ be the reverse lexicographical order on the monomials of S. Given a homogeneous ideal I in S, gin(I), the generic initial ideal of I, is the Borel-fixed monomial ideal associated to I. (For details see [B], [BM], [BS] or [Gr].) The generic initial ...