Unexpected hypersurface is a name given an element to some particular linear system introduced by Cook, Harbourne, Migliore and Nagel, motivated by work of Di Gennaro, Ilardi and Vallès and of Faenzi and Vallès, and it is a field of great study since then. It attracts many people because of their close ties to various other areas of mathematics including vector bundles, arrangements of hyperplanes, geometry of projective varieties, etc. Harbourne, Migliore, Nagel and Teitler introduced the concept of unexpected hypersurfaces and explained the so-called BMSS duality showing that unexpected curves are in some sense dual to their tangent cones at their singular point. In this paper, we continue the study of BMSS duality. We revisit the configuration of points associated to Hesse arrangement and Hesse union dual Hesse arrangement, and we study the geometry of the associated varieties and their companions.

We reformulate a fundamental result due to Cook, Harbourne, Migliore and Nagel on the existence and irreducibility of unexpected plane curves of a set of points $Z$ in ${\mathbb{P}}^2$, using the minimal degree of a Jacobian syzygy of the defining equation for the dual line arrangement ${\mathcal{𝒜}}_Z$. Several applications of this new approach are given. In particular, we show that the irreducible unexpected quintics may occur only when the set $Z$ has the cardinality equal to 11 or 12, and describe five cases where this happens.

We consider a generalization of a problem raised by P. Griffith on abelian groups to modules over integral domains, and prove an analogue of a theorem of M. Dugas and J. Irwin. Torsion modules $T$ with the following property are characterized: if $M$ is a torsion-free module and $F$ is a projective submodule such that $M∕F\cong T$, then $M$ is projective. It is shown that for abelian groups whose cardinality is not cofinal with $\omega$ this is equivalent to being totally reduced in the sense of L. Fuchs and K. Rangaswamy. The problem for valuation domains is also discussed, with results similar to the case of abelian groups.

Let $R$ be a ring and $I$ a proper ideal of $R$. An ideal $J\subseteq I$ is a reduction of $I$ if $J{I}^n=I^{n+1}$ for some positive integer $n$; and $I$ is called basic if it has no proper reductions. The notion of reduction was introduced by Northcott and Rees with the initial purpose to contribute to the analytic theory of ideals in Noetherian (local) rings via reductions.

Two well-known results, due to Hays, assert that an integral domain is Prüfer if and only if every finitely generated ideal is basic, and it is one-dimensional Prüfer if and only if every ideal is basic. This paper investigates reductions of ideals in the family of Prüfer rings, with the aim to recover and generalize Hays’ results to classes of rings with zero-divisors subject to various Prüfer conditions.

We study Serre’s condition $(S_n)$ for tensor products of modules over a commutative noetherian local ring $R$. Specifically, we consider the following question. For finitely generated $R$-modules $M$ and $N$, either of which is $(n+1)$-Tor-rigid, if the tensor product $M{\otimes }_{R}N$ satisfies $(S_{n+1})$, then does ${\text{Tor}}_i^R(M,N)=0$ hold for all $i\ge 1$? The aim of this paper is to give an affirmative answer to this question if we assume local freeness and Serre’s condition on modules. As applications, we recover several known results.

For a discrete valuation domain $V$ with maximal ideal $\mathfrak{𝔪}$ such that the residue field $V∕\mathfrak{𝔪}$ is finite, there exists a sequence of polynomials ${(F_n(x))}_{n\ge 0}$ defined over the quotient field $K$ of $V$ that forms a basis of the $V$-module $\text{Int}(V)=\{f\in K[x]|f(V)\subseteq V\}$. This sequence of polynomials bears many resemblances to the classical binomial polynomials ${\left(\left(\genfrac{}{}{0.0pt}{}{x}{n}\right)\right)}_{n\ge 0}$. In this paper, we introduce a generating polynomial to account for the distribution of the $V$-values of the polynomials $F_n(x)$ modulo the maximal ideal $\mathfrak{𝔪}$, and prove a result that provides a method for counting exactly how many $V$-values of the polynomials ${(F_n(x))}_{n\ge 0}$ fall into each of the residue classes modulo $\mathfrak{𝔪}$. Our main theorem in this paper can be viewed as an analogue of the classical theorem of Garfield and Wilf in the context of discrete valuation domains.

Let $(R,\mathfrak{𝔪})$ be a local ring, $\mathcal{𝒵}$ a specialization closed subset of Spec $R$ and $X$ an $R$-complex with finitely generated homology and finite dimension. We show that

$${\text{Att}}_R{H}_{\mathcal{𝒵}}^{\text{dim}X}(X)=\{\mathfrak{𝔭}\in {\text{Supp}}_{R}X\mid \text{cd}(\mathcal{𝒵},R∕\mathfrak{𝔭})-\text{inf}{X}_{\mathfrak{𝔭}}={\text{dim}}_{R}X\}.$$

We also present a generalization of the Lichtenbaum–Hartshorne vanishing theorem for complexes of $R$-modules.

Let $G$ be a graph with edge ideal $I(G)$. We recall the notions, given by Seyed Fakhari and Yassemi (2018), of ${\text{min-match}}_{\{K_2,C_5\}}(G)$ and ${\text{ind-match}}_{\{K_2,C_5\}}(G)$. We show that

$$\text{reg}(I{(G)}^s)\le 2s+{\text{min-match}}_{\{K_2,C_5\}}(G)-1,$$

for all $s\ge 1$, which implies that

$$\text{reg}(I{(G)}^s)\le 2s+\text{min-match}(G)-1.$$

Moreover, we show that

$$\text{reg}(I{(G)}^s)\ge 2s+{\text{ind-match}}_{\{K_2,C_5\}}(G)-2,$$

and if ${\text{ind-match}}_{\{K_2,C_5\}}(G)$ is an odd integer, then

$$\text{reg}(I{(G)}^s)\ge 2s+{\text{ind-match}}_{\{K_2,C_5\}}(G)-1.$$

As a consequence, we estimate the regularity of powers of edge ideals of Cohen–Macaulay graphs with $\text{girth}(G)\ge 5$. Furthermore, it is shown that

$$\text{reg}(I{(G)}^s)\le 2s+\text{ord-match}(G)-1,$$

where $\text{ord-match}(G)$ denotes the ordered matching number of $G$.

One of the fundamental invariants connecting algebra and geometry is the degree of an ideal. In this paper we derive the probabilistic behavior of degree with respect to the versatile Erdős–Rényi-type model for random monomial ideals. We study the staircase structure associated to a monomial ideal, and show that in the random case the shape of the staircase diagram is approximately hyperbolic, and this behavior is robust across several random models. Since the discrete volume under this staircase is related to the summatory higher-order divisor function studied in number theory, we use this connection and our control over the shape of the staircase diagram to derive the asymptotic degree of a random monomial ideal. Another way to compute the degree of a monomial ideal is with a standard pair decomposition. This paper derives bounds on the number of standard pairs of a random monomial ideal indexed by any subset of the ring variables. The standard pairs indexed by maximal subsets give a count of degree, as well as being a more nuanced invariant of the random monomial ideal.

One of the most stunning results in the representation theory of Cohen–Macaulay rings is Auslander’s well-known theorem which states a CM local ring of finite CM type can have at most an isolated singularity. There have been some generalizations of this in the direction of countable CM type by Huneke and Leuschke. In this paper, we focus on a different generalization by restricting the class of modules. Here we consider modules which are high syzygies of MCM modules over noncommutative rings, exploiting the fact that noncommutative rings allow for finer homological behavior. We then generalize Auslander’s theorem in the setting of complete Gorenstein local domains by examining path algebras, which preserve finiteness of global dimension.

Let $R$ be a commutative ring with identity. The small finitistic dimension $\text{fPD}(R)$ of $R$ is defined to be the supremum of projective dimensions of $R$-modules with finite projective resolutions. In this paper, we characterize a ring $R$ with $\text{fPD}(R)\le n$ using finitely generated semiregular ideals, tilting modules, cotilting modules of cofinite type and vaguely associated prime ideals. As an application, we obtain that if $R$ is a Noetherian ring, then $\text{fPD}(R)=\text{sup}\{\text{grade}(\mathfrak{𝔪},R)|\mathfrak{𝔪}\in \text{Max}(R)\}$ where $\text{grade}(\mathfrak{𝔪},R)$ is the grade of $\mathfrak{𝔪}$ on $R$. We also show that a ring $R$ satisfies $\text{fPD}(R)\le 1$ if and only if $R$ is a $\text{DW}$ ring. As applications, we show that the small finitistic dimensions of strong Prüfer rings and $\text{LPVD}$s are at most one. Moreover, for any given $n\in \mathbb{N}$, we obtain a total ring of quotients $R$ satisfying $\text{fPD}(R)=n$.

Let $R$ be a ring and $S$ a multiplicative subset of $R$. An $R$-module $P$ is called uniformly $S$-projective provided that the induced sequence $0\to {\text{Hom}}_R(P,A)\to {\text{Hom}}_R(P,B)\to {\text{Hom}}_R(P,C)\to 0$ is $u$-$S$-exact for any $u$-$S$-short exact sequence $0\to A\to B\to C\to 0$. Some characterizations and properties of $u$-$S$-projective modules are obtained. The notion of $u$-$S$-semisimple modules is also introduced. A ring $R$ is called a $u$-$S$-semisimple ring provided that any free $R$-module is $u$-$S$-semisimple. Several characterizations of $u$-$S$-semisimple rings are provided in terms of $u$-$S$-semisimple modules, $u$-$S$-projective modules, $u$-$S$-injective modules and $u$-$S$-split $u$-$S$-exact sequences.