Let $A$ be an artin algebra. An $A$-module $M$ will be said to be semi-Gorenstein-projective provided that ${Ext}^i\left(M,A\right)=0$ for all $i\ge 1$. All Gorenstein-projective modules are semi-Gorenstein-projective and only few and quite complicated examples of semi-Gorenstein-projective modules which are not Gorenstein-projective have been known. One of the aims of the paper is to provide conditions on $A$ such that all semi-Gorenstein-projective left modules are Gorenstein-projective (we call such an algebra left weakly Gorenstein). In particular, we show that in case there are only finitely many isomorphism classes of indecomposable left modules which are both semi-Gorenstein-projective and torsionless, then $A$ is left weakly Gorenstein. On the other hand, we exhibit a 6-dimensional algebra $\mathrm{\Lambda }$ with a semi-Gorenstein-projective module $M$ which is not torsionless (thus not Gorenstein-projective). Actually, also the $\mathrm{\Lambda }$-dual module $M^{\ast }$ is semi-Gorenstein-projective. In this way, we show the independence of the total reflexivity conditions of Avramov and Martsinkovsky, thus completing a partial proof by Jorgensen and Şega. Since all the syzygy-modules of $M$ and $M^{\ast }$ are 3-dimensional, the example can be checked (and visualized) quite easily.

Recently, a density result for the $16$-rank of $\text{ Cl}\left(\mathbb{Q}\left(\sqrt{-p}\right)\right)$ was established when $p$ varies among the prime numbers, assuming a short character sum conjecture. We prove the same density result unconditionally.

Let $F$ be a local field of mixed characteristic $\left(0,p\right)$, let $k$ be a finite extension of its residue field, let $\mathcal{\mathscr{H}}$ be the pro-$p$-Iwahori Hecke $k$-algebra attached to ${GL}_{d+1}\left(F\right)$ for some $d\ge 1$. We construct an exact and fully faithful functor from the category of supersingular $\mathcal{\mathscr{H}}$-modules to the category of $Gal\left(\bar{F}∕F\right)$-representations over $k$. More generally, for a certain $k$-algebra ${\mathcal{\mathscr{H}}}^{♯}$ surjecting onto $\mathcal{\mathscr{H}}$ we define the notion of $♯$-supersingular modules and construct an exact and fully faithful functor from the category of $♯$-supersingular ${\mathcal{\mathscr{H}}}^{♯}$-modules to the category of $Gal\left(\bar{F}∕F\right)$-representations over $k$.

Let $R$ be a (not necessarily commutative) ring whose additive group is finitely generated and let $U_n\left(R\right)\subset {GL}_n\left(R\right)$ be the group of upper-triangular unipotent matrices over $R$. We study how the homology groups of $U_n\left(R\right)$ vary with $n$ from the point of view of representation stability. Our main theorem asserts that if for each $n$ we have representations $M_n$ of $U_n\left(R\right)$ over a ring $k$ that are appropriately compatible and satisfy suitable finiteness hypotheses, then the rule $\left[n\right]\mapsto H_i\left(U_n\left(R\right),M_n\right)$ defines a finitely generated $OI$-module. As a consequence, if $k$ is a field then $dim{H}_i\left(U_n\left(R\right),k\right)$ is eventually equal to a polynomial in $n$. We also prove similar results for the Iwahori subgroups of ${GL}_n\left(\mathcal{𝒪}\right)$ for number rings $\mathcal{𝒪}$.

We characterize all pairs of completely multiplicative functions $fg:\mathbb{N}\to \mathbb{𝕋}$, where $\mathbb{𝕋}$ denotes the unit circle, such that

$$\overline{{\left\{\left(f\left(n\right),g\left(n+1\right)\right)\right\}}_{n\ge 1}}\ne \mathbb{𝕋}\times \mathbb{𝕋}.$$

In so doing, we settle an old conjecture of Zoltán Daróczy and Imre Kátai.

We determine birational superrigidity for a quasismooth prime Fano $3$-fold of codimension $4$ with no projection centers. In particular we prove birational superrigidity for Fano $3$-folds of codimension $4$ with no projection centers which were recently constructed by Coughlan and Ducat. We also pose some questions and a conjecture regarding the classification of birationally superrigid Fano $3$-folds.

We construct two small resolutions of singularities of the Coble fourfold (the double cover of the four-dimensional projective space branched over the Igusa quartic). We use them to show that all ${\mathfrak{𝔖}}_6$-invariant three-dimensional quartics are birational to conic bundles over the quintic del Pezzo surface with the discriminant curves from the Wiman–Edge pencil. As an application, we check that ${\mathfrak{𝔖}}_6$-invariant three-dimensional quartics are unirational, obtain new proofs of rationality of four special quartics among them and irrationality of the others, and describe their Weil divisor class groups as ${\mathfrak{𝔖}}_6$-representations.