Deciding the presence of the weak Lefschetz property often is a challenging problem. Continuing studies of Brenner and Kaid (2007), Cook II and Nagel (2011) and Migliore, Miró-Roig, Murai and Nagel (2013) we carry out an in-depth study of Artinian monomial ideals with four generators in three variables. We use a connection to lozenge tilings to describe semistability of the syzygy bundle of such an ideal, to determine its generic splitting type, and to decide the presence of the weak Lefschetz property. We provide results in both characteristic zero and positive characteristic.

Let $\mathbb{𝕏}$ be a set of $K$-rational points in ${\mathbb{P}}^1\times {\mathbb{P}}^1$ over a field $K$ of characteristic zero, let $\mathbb{𝕐}$ be a fat point scheme supported at $\mathbb{𝕏}$, and let $R_{\mathbb{𝕐}}$ be the bihomogeneous coordinate ring of $\mathbb{𝕐}$. In this paper we investigate the module of Kähler differentials ${\mathrm{\Omega }}_{R_{\mathbb{𝕐}}∕K}^1$. We describe this bigraded $R_{\mathbb{𝕐}}$-module explicitly via a homogeneous short exact sequence and compute its Hilbert function in a number of special cases, in particular when the support $\mathbb{𝕏}$ is a complete intersection or an almost complete intersection in ${\mathbb{P}}^1\times {\mathbb{P}}^1$. Moreover, we introduce a Kähler different for $\mathbb{𝕐}$ and use it to characterize ACM reduced schemes in ${\mathbb{P}}^1\times {\mathbb{P}}^1$ having the Cayley–Bacharach property.

Boij–Söderberg theory describes the Betti diagrams of graded modules over a polynomial ring as a linear combination of pure diagrams with positive coefficients. In this paper, we focus on the Betti diagrams of lexicographic ideals. Mainly, we characterize the Boij–Söderberg decomposition of the Betti table of a lexicographic ideal in the polynomial ring with three variables, and show a nice connection between its Boij–Söderberg decomposition and the ones of other related lexicographic ideals.

We develop a local cohomology theory for ${\mathcal{ℱ}\mathcal{ℐ}}^m$-modules, and show that it in many ways mimics the classical theory for multigraded modules over a polynomial ring. In particular, we define an invariant of ${\mathcal{ℱ}\mathcal{ℐ}}^m$-modules using this local cohomology theory which closely resembles an invariant of multigraded modules over Cox rings defined by Maclagan and Smith. It is then shown that this invariant behaves almost identically to the invariant of Maclagan and Smith.

We show that the derived category of Brauer–Severi curves satisfies the Jordan–Hölder property and cannot have quasi-phantoms, phantoms or universal phantoms. In this way we obtain that quasi-phantoms, phantoms or universal phantoms cannot exist in the derived category of smooth projective curves over a field $k$. Moreover, we show that a $n$-dimensional Brauer–Severi variety is completely characterized by the existence of a full weak exceptional collection consisting of pure vector bundles of length $n+1$, at least in characteristic zero. We conjecture that Brauer–Severi varieties $X$ satisfy ${rdim}_{cat}\left(X\right)=ind\left(X\right)-1$, provided period equals index, and prove this in the case of curves, surfaces and for Brauer–Severi varieties of index at most three. We believe that the results for curves are known to the experts. We nevertheless give the proofs, adding to the literature.

It was shown by Kang (1989) that a domain $R$ is a Krull domain if and only if $R$ is a Mori domain and a P$v$MD. In this paper, we extend this result to Gorenstein multiplicative ideal theory. To do this, we introduce the concepts of $FT$-domains and G-P$v$MDs, and study them by a new star-operation, i.e., the $f$-operation. We prove that (1) a domain $R$ is an integrally closed $FT$-domain if and only if $R$ is a $P$-domain; (2) a domain $R$ is a G-P$v$MD if and only if $R$ is a $g$-coherent $FT$-domain; (3) a domain $R$ is a G-Krull domain if and only if $R$ is a Mori domain and a G-P$v$$v$MD.

Given a polarized variety $\left(X,D\right)$, we can associate a graded ring and a Hilbert series. Assume $D$ is an ample $\mathbb{Q}$ Cartier divisor, and $\left(X,D\right)$ is quasi smooth and projectively Gorenstein, we give a parsing formula for the Hilbert series according to their singularities. Here we allow the variety to have singularities of dimension $\le 1$, that is, both singularities of dimension $1$ and singular points, extending a 2013 result of Buckley, Reid and the author about varieties with only isolated singularities.