We generalize the classical Chevalley–Shephard–Todd theorem to the case of finite linearly reductive group schemes. As an application, we prove that every scheme $X$ which is étale-locally the quotient of a smooth scheme by a finite linearly reductive group scheme is the coarse space of a smooth tame Artin stack (as defined by Abramovich, Olsson, and Vistoli), whose stacky structure is supported on the singular locus of $X$.

Mustaţă (1997) stated a generalized version of the minimal resolution conjecture for a set $Z$ of general points in arbitrary projective varieties and he predicted the graded Betti numbers of the minimal free resolution of $I_Z$. In this paper, we address this conjecture and we prove that it holds for a general set $Z$ of points on any (not necessarily normal) del Pezzo surface $X\subseteq {\mathbb{P}}^d$ — up to three sporadic cases — whose cardinality $\left|Z\right|$ sits into the interval $\left[P_X\left(r-1\right),m\left(r\right)\right]$ or $\left[n\left(r\right),P_X\left(r\right)\right]$, $r\ge 4$, where $P_X\left(r\right)$ is the Hilbert polynomial of $X$, $m\left(r\right):=\frac{1}{2}d{r}^2+\frac{1}{2}r\left(2-d\right)$ and $n\left(r\right):=\frac{1}{2}d{r}^2+\frac{1}{2}r\left(d-2\right)$. As a corollary we prove: (1) Mustaţă’s conjecture for a general set of $s\ge 19$ points on any integral cubic surface in ${\mathbb{P}}^3$; and (2) the ideal generation conjecture and the Cohen–Macaulay type conjecture for a general set of cardinality $s\ge 6d+1$ on a del Pezzo surface $X\subseteq {\mathbb{P}}^d$.

We develop the notion of stratifiability in the context of derived categories and the six operations for stacks. Then we reprove the Lefschetz trace formula for stacks, and give the meromorphic continuation of $L$-series (in particular, zeta functions) of ${\mathbb{F}}_q$-stacks. We also give an upper bound for the weights of the cohomology groups of stacks, and an “independence of $\ell$” result for a certain class of quotient stacks.

We study exponential sums whose coefficients are completely multiplicative and belong to the complex unit disc. Our main result shows that such a sum has substantial cancellation unless the coefficient function is essentially a Dirichlet character. As an application we improve current bounds on odd-order character sums. Furthermore, conditionally on the generalized Riemann hypothesis we obtain a bound for odd-order character sums which is best possible.

Quiver Grassmannians are varieties parametrizing subrepresentations of a quiver representation. It is observed that certain quiver Grassmannians for type A quivers are isomorphic to the degenerate flag varieties investigated earlier by Feigin. This leads to the consideration of a class of Grassmannians of subrepresentations of the direct sum of a projective and an injective representation of a Dynkin quiver. It is proved that these are (typically singular) irreducible normal local complete intersection varieties, which admit a group action with finitely many orbits and a cellular decomposition. For type A quivers, explicit formulas for the Euler characteristic (the median Genocchi numbers) and the Poincaré polynomials are derived.