Fifth volume of Épijournal de Géométrie Algébrique - 2021
Kahn, Bruno ; Miyazaki, Hiroyasu ; Saito, Shuji ; Yamazaki, Takao.
We develop a theory of modulus sheaves with transfers, which generalizes
Voevodsky's theory of sheaves with transfers. This paper and its sequel are
foundational for the theory of motives with modulus, which is developed in
[KMSY20].
Kahn, Bruno ; Miyazaki, Hiroyasu ; Saito, Shuji ; Yamazaki, Takao.
We develop a theory of sheaves and cohomology on the category of proper
modulus pairs. This complements [KMSY21], where a theory of sheaves and
cohomology on the category of non-proper modulus pairs has been developed.
Bonnafé, Cédric ; Sarti, Alessandra.
We construct here many families of K3 surfaces that one can obtain as
quotients of algebraic surfaces by some subgroups of the rank four complex
reflection groups. We find in total 15 families with at worst
$ADE$--singularities. In particular we classify all the K3 surfaces that can be
obtained as quotients by the derived subgroup of the previous complex
reflection groups. We prove our results by using the geometry of the weighted
projective spaces where these surfaces are embedded and the theory of Springer
and Lehrer-Springer on properties of complex reflection groups. This
construction generalizes a previous construction by W. Barth and the second
author.
Craw, Alastair ; Heuberger, Liana ; Amador, Jesus Tapia.
Reid's recipe for a finite abelian subgroup $G\subset
\text{SL}(3,\mathbb{C})$ is a combinatorial procedure that marks the toric fan
of the $G$-Hilbert scheme with irreducible representations of $G$. The
geometric McKay correspondence conjecture of Cautis--Logvinenko that describes
certain objects in the derived category of $G\text{-Hilb}$ in terms of Reid's
recipe was later proved by Logvinenko et al. We generalise Reid's recipe to any
consistent dimer model by marking the toric fan of a crepant resolution of the
vaccuum moduli space in a manner that is compatible with the geometric
correspondence of Bocklandt--Craw--Quintero-Vélez. Our main tool
generalises the jigsaw transformations of Nakamura to consistent dimer models.