Fifth volume of Épijournal de Géométrie Algébrique - 2021
Bruno Kahn ; Hiroyasu Miyazaki ; Shuji Saito ; Takao Yamazaki.
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].
Bruno Kahn ; Hiroyasu Miyazaki ; Shuji Saito ; Takao Yamazaki.
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.
Cédric Bonnafé ; Alessandra Sarti.
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.
Alastair Craw ; Liana Heuberger ; Jesus Tapia Amador.
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.
Sébastien Boucksom ; Simone Diverio.
It was conjectured by Lang that a complex projective manifold is Kobayashi
hyperbolic if and only if it is of general type together with all of its
subvarieties. We verify this conjecture for projective manifolds whose
universal cover carries a bounded, strictly plurisubharmonic function. This
includes in particular compact free quotients of bounded domains.
Nicolas Tholozan ; Jérémy Toulisse.
We prove that some relative character varieties of the fundamental group of a
punctured sphere into the Hermitian Lie groups $\mathrm{SU}(p,q)$ admit compact
connected components. The representations in these components have several
counter-intuitive properties. For instance, the image of any simple closed
curve is an elliptic element. These results extend a recent work of Deroin and
the first author, which treated the case of $\textrm{PU}(1,1) =
\mathrm{PSL}(2,\mathbb{R})$. Our proof relies on the non-Abelian Hodge
correspondance between relative character varieties and parabolic Higgs
bundles. The examples we construct admit a rather explicit description as
projective varieties obtained via Geometric Invariant Theory.
Laurent Evain ; Mathias Lederer.
The Bialynicki-Birula strata on the Hilbert scheme $H^n(\mathbb{A}^d)$ are
smooth in dimension $d=2$. We prove that there is a schematic structure in
higher dimensions, the Bialynicki-Birula scheme, which is natural in the sense
that it represents a functor. Let $\rho_i:H^n(\mathbb{A}^d)\rightarrow {\rm
Sym}^n(\mathbb{A}^1)$ be the Hilbert-Chow morphism of the ${i}^{th}$
coordinate. We prove that a Bialynicki-Birula scheme associated with an action
of a torus $T$ is schematically included in the fiber $\rho_i^{-1}(0)$ if the
${i}^{th}$ weight of $T$ is non-positive. We prove that the monic functors
parametrizing families of ideals with a prescribed initial ideal are
representable.
Daniele Faenzi ; Joan Pons-Llopis.
We show that all reduced closed subschemes of projective space that have a
Cohen-Macaulay graded coordinate ring are of wild Cohen-Macaulay type, except
for a few cases which we completely classify.
Tom Bachmann ; Elden Elmanto ; Marc Hoyois ; Adeel A. Khan ; Vladimir Sosnilo ; Maria Yakerson.
We obtain geometric models for the infinite loop spaces of the motivic
spectra $\mathrm{MGL}$, $\mathrm{MSL}$, and $\mathbf{1}$ over a field. They are
motivically equivalent to $\mathbb{Z}\times
\mathrm{Hilb}_\infty^\mathrm{lci}(\mathbb{A}^\infty)^+$, $\mathbb{Z}\times
\mathrm{Hilb}_\infty^\mathrm{or}(\mathbb{A}^\infty)^+$, and $\mathbb{Z}\times
\mathrm{Hilb}_\infty^\mathrm{fr}(\mathbb{A}^\infty)^+$, respectively, where
$\mathrm{Hilb}_d^\mathrm{lci}(\mathbb{A}^n)$ (resp.
$\mathrm{Hilb}_d^\mathrm{or}(\mathbb{A}^n)$,
$\mathrm{Hilb}_d^\mathrm{fr}(\mathbb{A}^n)$) is the Hilbert scheme of lci
points (resp. oriented points, framed points) of degree $d$ in $\mathbb{A}^n$,
and $+$ is Quillen's plus construction. Moreover, we show that the plus
construction is redundant in positive characteristic.
Katharina Heinrich ; Roy Skjelnes ; Jan Stevens.
We consider the Cohen-Macaulay compactification of the space of twisted
cubics in projective n-space. This compactification is the fine moduli scheme
representing the functor of CM-curves with Hilbert polynomial 3t+1. We show
that the moduli scheme of CM-curves in projective 3-space is isomorphic to the
twisted cubic component of the Hilbert scheme. We also describe the
compactification for twisted cubics in n-space.
Samuel Boissière ; Enrica Floris.
Let $G$ be a connected algebraic group. We study $G$-equivariant extremal
contractions whose centre is a codimension three $G$-simply connected orbit. In
the spirit of an important result by Kawakita in 2001, we prove that those
contractions are weighted blow-ups.