Given an ideal $I$, the containment problem is concerned with finding the values $m$ and $r$ such that the $m$-th symbolic power of $I$ is contained in its $r$-th ordinary power. A central issue related to this is determining the resurgence for ideals $I$ of fat points in projective space. In this paper we obtain complete results for the resurgence of fat point schemes $m_0{P}_0+m_1{P}_1+m_2{P}_2$ in ${\mathbb{P}}^N$ for any distinct points $P_0,P_1,P_2$, and, when the points $P_0,\dots ,P_n$ are collinear, we extend this result to fat point schemes $m_0{P}_0+\cdots +m_n{P}_n$. As a by-product of our determining the resurgence for all three points fat point ideals, we give new examples of ideals with symbolic defect zero. In case the points are noncollinear, a three fat points ideal can be regarded as a monomial ideal, but it is typically not square-free.

Given a surjective ring homomorphism, we study when the induced group homomorphism on unit groups is surjective. To this end, we introduce notions of generalized inverses and units, as well as a class of rings such that the set of closed points in the spectrum is a closed set. It is shown that any surjection out of such a ring induces a surjection on unit groups.

We give algebraic characterizations of the affine $2$-space and the affine $3$-space over an algebraically closed field of characteristic zero, using a variant of the Makar-Limanov invariant.

In this work, we study the Betti numbers of pinched Veronese rings, by means of the reduced homology of squarefree divisor complexes. We characterize when these rings are Cohen–Macaulay and we study the shape of the Betti tables for the pinched Veronese in the two variables. As a byproduct we obtain information on the linearity of such rings. Moreover, in the last section we compute the canonical modules of the Veronese modules.

Let $Q$ be a quiver with dimension vector $\alpha$ prehomogeneous under the action of the product of general linear groups $\text{GL}\left(\alpha \right)$ on the representation variety $\text{Rep}\left(Q,\alpha \right)$. We study geometric properties of zero sets of semi-invariants of this space. It is known that for large numbers $N$, the nullcone in $\text{Rep}\left(Q,N\cdot \alpha \right)$ becomes a complete intersection. First, we show that it also becomes reduced. Then, using Bernstein–Sato polynomials, we discuss some criteria for zero sets to have rational singularities. In particular, we show that for Dynkin quivers codimension $1$ orbit closures have rational singularities.

The Lefschetz question asks if multiplication by a power of a general linear form, $L$, on a graded algebra has maximal rank (in every degree). We consider a quotient by an ideal that is generated by powers of linear forms. Then the Lefschetz question is, for example, related to the problem whether a set of fat points imposes the expected number of conditions on a linear system of hypersurfaces of fixed degree. Our starting point is a result that relates Lefschetz properties in different rings. It suggests to use induction on the number of variables, $n$. If $n=3$, then it is known that multiplication by $L$ always has maximal rank. We show that the same is true for multiplication by $L^2$ if all linear forms are general. Furthermore, we give a complete description of when multiplication by $L^3$ has maximal rank (and its failure when it does not). As a consequence, for such ideals that contain a quadratic or cubic generator, we establish results on the so-called strong Lefschetz property for ideals in $n=3$ variables, and the weak Lefschetz property for ideals in $n=4$ variables.

We study the monoid of fractional ideals and the ideal class semigroup of an arbitrary given one dimensional normal domain $\mathfrak{𝔒}$ obtained by an infinite integral extension of a Dedekind domain. We introduce a notion of “upper semicontinuous functions” whose domain is the maximal spectrum of $\mathfrak{𝔒}$ equipped with the inverse topology introduced by Hochster, and whose codomain is a certain totally ordered monoid containing $ℝ$. We construct an isomorphism between a monoid consisting of such upper semicontinuous functions satisfying certain conditions and the monoid of fractional ideals of $\mathfrak{𝔒}$. This result can be regarded as a generalization of the theory of prime ideal factorization for Dedekind domains. By using such isomorphism, we study the Galois-monoid structure of the ideal class semigroup of $\mathfrak{𝔒}$.

The purpose of this paper is to extend the symmetry of maximals of the ring of a germ of a reducible plane curve proved by Delgado to a relation between the relative maximals of a fractional ideal and the absolute maximals of its dual for any admissible ring. In particular, it includes the case of germs of reduced reducible curve of any codimension. We then apply this symmetry to characterize the elements in the set of values of a fractional ideal from some of its projections and the irreducible absolute maximals of the dual ideal.

Let $H$ be a Krull monoid with infinite cyclic class group $G$ that we identify with $\mathbb{Z}$. Let $G_P\subseteq G$ denote the set of classes containing prime divisors. It was shown that $\rho \left(H\right)$, the elasticity of $H$, is finite if and only if $G_P$ is bounded above or below. By a result of Lambert, it is easy to show that $\rho \left(H\right)\le min\left\{-inf\left(G_P\right),sup\left(G_P\right)\right\}$. In this paper, we focus on $H$ with large elasticity. When $G_P$ is infinite but bounded above or below, we give necessary and sufficient conditions for that . When $G_P$ is a finite set, we give a better upper bound on $\rho \left(H\right)$, which is sharp in some sense.