Let $G$ be a finite group, and let $\{B_{ℂ}G\}$ the class of its classifying stack $B_{ℂ}G$ in Ekedahl’s Grothendieck ring of algebraic $ℂ$-stacks $K_0({\text{Stacks}}_{ℂ})$. We show that if $B_{ℂ}G$ has the mixed Tate property, the invariants $H^i(\{B_{ℂ}G\})$ defined by Ekedahl are zero for all $i\ne 0$. We also extend Ekedahl’s construction of these invariants to fields of positive characteristic.

We study the Gross–Schoen diagonal cycle on a triple product of Shimura curves at a place of bad reduction. We relate the image of the diagonal cycle under the Abel–Jacobi map to a certain period integral that governs the central critical value of the Garrett–Rankin type triple product $L$-function via level raising congruences. As an application we prove the rank $0$ case of the Bloch–Kato conjecture for the symmetric cube motive of a weight $2$ modular form.

This paper investigates effectivity problems of pluricanonical systems on varieties of general type in positive characteristic. In practice, we will consider a sublinear system $|S_{-}^0(X,K_X+n{K}_X)|\subseteq |H^0(X,K_X+n{K}_X)|$ generated by certain Frobenius stable sections, and prove that for a minimal terminal threefold $X$ of general type with either or Gorenstein singularities, if $n\ge 28$ then $|S_{-}^0(X,K_X+n{K}_X)|\ne \varnothing$; and if $n\ge 42$ then the linear system $|S_{-}^0(X,K_X+n{K}_X)|$ defines a birational map.

We associate a certain tensor product lattice to any primitive integer lattice and ask about its typical shape. These lattices are related to the tangent bundle of Grassmannians and their study is motivated by Peyre’s program on “freeness” for rational points of bounded height on Fano varieties.

The strong Bombieri–Lang conjecture postulates that, for every variety $X$ of general type over a field $k$ finitely generated over $\mathbb{Q}$, there exists an open subset $U\subset X$ such that $U(K)$ is finite for every finitely generated extension $K∕k$. The weak Bombieri–Lang conjecture postulates that, for every positive dimensional variety $X$ of general type over a field $k$ finitely generated over $\mathbb{Q}$, the rational points $X(k)$ are not dense. Furthermore, Lang conjectured that every variety of general type $X$ over a field of characteristic $0$ contains an open subset $U\subset X$ such that every subvariety of $U$ is of general type, this statement is usually called geometric Lang conjecture.

We reduce the strong Bombieri–Lang conjecture to the case $k=\mathbb{Q}$. Assuming the geometric Lang conjecture, we reduce the weak Bombieri–Lang conjecture to $k=\mathbb{Q}$, too.

Let $(X,\mathrm{\Delta })$ be a strictly lc log Fano pair, we show that every lc place of complements of $(X,\mathrm{\Delta })$ is dreamy and there exists a correspondence between weakly special test configurations of $(X,\mathrm{\Delta };-K_X-\mathrm{\Delta })$ and lc places of complements of $(X,\mathrm{\Delta })$.

We introduce a method for associating a chain complex to a module over a combinatorial category such that if the complex is exact then the module has a rational Hilbert series. We prove homology-vanishing theorems for these complexes for several combinatorial categories including the category of finite sets and injections, the opposite of the category of finite sets and surjections, and the category of finite-dimensional vector spaces over a finite field and injections.

Our main applications are to modules over the opposite of the category of finite sets and surjections, known as ${\mathit{F}\mathit{S}}^{\text{op}}$ modules. We obtain many constraints on the sequence of symmetric group representations underlying a finitely generated ${\mathit{F}\mathit{S}}^{\text{op}}$ module. In particular, we describe its character in terms of functions that we call character exponentials. Our results have new consequences for the character of the homology of the moduli space of stable marked curves, and for the equivariant Kazhdan–Lusztig polynomial of the braid matroid.

We construct (in significant generality) moduli spaces representing (on the category of geometrically reduced schemes over the base field) the functor of morphisms from a scheme into a solvable algebraic group.