Let $I\subset R=\mathbb{𝔽}[x_1,x_2]$ be a height two ideal minimally generated by three homogeneous polynomials of the same degree $d$, where $\mathbb{𝔽}$ is a field of characteristic zero. We use the theory of $D$-modules to deduce information about the defining equations of the Rees algebra of $I$. Let $\mathcal{𝒦}$ be the kernel of the canonical map $\alpha :\mathrm{\text{ Sym}}(I)\to \mathrm{\text{ Rees}}(I)$ from the symmetric algebra of $I$ onto the Rees algebra of $I$. We prove that $\mathcal{𝒦}$ can be described as the solution set of a system of differential equations, that the whole bigraded structure of $\mathcal{𝒦}$ is characterized by the integral roots of certain $b$-functions, and that certain de Rham cohomology groups can give partial information about $\mathcal{𝒦}$.

Using an old example of Nagata, we construct a Noetherian ring of prime characteristic $p$, whose Frobenius morphism is locally finite, but not finite.

We consider homogeneously licci ideals in a polynomial ring and focus on the degrees of the forms generating the regular sequences. Using a sequentially bounded condition on these degrees, E. Chong discovered a large class of licci ideals satisfying the Eisenbud–Green–Harris conjecture (among them, grade $3$ Gorenstein ideals). He raised the question of whether these sequentially bounded links were possible for all homogeneously licci ideals. We answer his question in the negative, and in doing so answer a question of C. Huneke and B. Ulrich about strongly licci ideals. The structure of certain Betti tables plays a central role in our proof.

Let $\mathfrak{𝔮}$ be an ideal of a Noetherian local ring $(A,\mathfrak{𝔪})$ and $M$ a nonzero finitely generated $A$-module. We present a criterion of Cohen–Macaulayness of the form module $G_M(\mathfrak{𝔮})$ in terms of (non)vanishing of a variation of local cohomology introduced by P. Schenzel and the author (J. Commut. Algebra 12:3 (2020), 353-370).

Let $P$ be a commutative Noetherian ring, $\mathfrak{𝔠}$ be an ideal of $P$ which is generated by a regular sequence of length four, $f$ be a regular element of $P$, and $\overline{P}$ be the hypersurface ring $P∕(f)$. Assume that $\mathfrak{𝔠}:f$ is a grade four Gorenstein ideal of $P$. We give a resolution $N$ of $\overline{P}∕\mathfrak{𝔠}\overline{P}$ by free $\overline{P}$-modules. The resolution $N$ is built from a differential graded algebra resolution of $P∕(\mathfrak{𝔠}:f)$ by free $P$-modules, together with one homotopy map. In particular, we give an explicit form for the matrix factorization which is the infinite tail of the resolution $N$.

We prove that the divisor class group of any open Richardson variety in the Grassmannian is trivial. Our proof uses Nagata’s criterion, localizing the coordinate ring at a suitable set of Plücker coordinates. We prove that these Plücker coordinates are prime elements by showing that the subscheme they define is an open subscheme of a positroid variety. Our results hold over any field and over the integers.

We establish a link between trace modules and rigidity in modules over Noetherian rings. We identify classes of modules which must have self-extensions and use the theory of trace ideals to verify the Auslander–Reiten conjecture for syzygies of ideals over Artinian Gorenstein rings.

Let $G$ be a simple graph. We show that if $G$ is connected and $R(I(G))$ is normal, $\mathrm{reg}(R(I(G)))\le {\alpha }_0(G)$, where ${\alpha }_0(G)$ is the vertex cover number of $G$. As a consequence, for every normal König connected graph $G$, $\mathrm{reg}(R(I(G)))=\mathrm{mat}(G)$, the matching number of $G$. For a gap-free graph $G$, we give various combinatorial upper bounds for $\mathrm{reg}(R(I(G)))$. As a consequence we give various sufficient conditions for the equality of $\mathrm{reg}(R(I(G)))$ and $\mathrm{mat}(G)$. Finally we show that if $G$ is a chordal graph such that $K[G]$ has $q$-linear resolution ($q\ge 4$), then $K[G]$ is a hypersurface, which proves the conjecture of Hibi, Matsuda and Tsuchiya [10, Conjecture 0.2] affirmatively for chordal graphs.

We provide a direct computation of the $F$-pure threshold of degree four homogeneous polynomials in two variables and, more generally, of certain homogeneous polynomials with four distinct roots. The computation depends on whether the cross ratio of the roots satisfies a specific Möbius transformation of a Legendre polynomial. We then make a connection between a long lasting open question, involving the relationship between the $F$-pure and the log canonical threshold, and roots of Legendre polynomials over ${\mathbb{𝔽}}_p$.