Fourth volume of Épijournal de Géométrie Algébrique - 2020
Michael Wibmer.
Replacing finite groups by linear algebraic groups, we study an
algebraic-geometric counterpart of the theory of free profinite groups. In
particular, we introduce free proalgebraic groups and characterize them in
terms of embedding problems. The main motivation for this endeavor is a
differential analog of a conjecture of Shafarevic.
Vance Blankers.
We show that the class of the locus of hyperelliptic curves with $\ell$
marked Weierstrass points, $m$ marked conjugate pairs of points, and $n$ free
marked points is rigid and extremal in the cone of effective codimension-($\ell
+ m$) classes on $\overline{\mathcal{M}}_{2,\ell+2m+n}$. This generalizes work
of Chen and Tarasca and establishes an infinite family of rigid and extremal
classes in arbitrarily-high codimension.
Alina Marian ; Xiaolei Zhao.
For a moduli space of Bridgeland-stable objects on a K3 surface, we show that
the Chow class of a point is determined by the Chern class of the corresponding
object on the surface. This establishes a conjecture of Junliang Shen, Qizheng
Yin, and the second author.
Boris Pasquier.
We classify all smooth projective horospherical varieties of Picard group $\mathbb{Z}^2$ and we give a first description of their geometry via the Log Minimal Model Program.
Marco Antei ; Michel Emsalem ; Carlo Gasbarri.
Let $S$ be a Dedekind scheme, $X$ a connected $S$-scheme locally of finite
type and $x\in X(S)$ a section. The aim of the present paper is to establish
the existence of the fundamental group scheme of $X$, when $X$ has reduced
fibers or when $X$ is normal. We also prove the existence of a group scheme,
that we will call the quasi-finite fundamental group scheme of $X$ at $x$,
which classifies all the quasi-finite torsors over $X$, pointed over $x$. We
define Galois torsors, which play in this context a role similar to the one of
Galois covers in the theory of étale fundamental group.
Cong Xue.
In this paper we prove that the cohomology groups with compact support of
stacks of shtukas are modules of finite type over a Hecke algebra. As an
application, we extend the construction of excursion operators, defined by V.
Lafforgue on the space of cuspidal automorphic forms, to the space of
automorphic forms with compact support. This gives the Langlands
parametrization for some quotient spaces of the latter, which is compatible
with the constant term morphism.
Martin Schwald.
This paper concerns different types of singular complex projective varieties
generalizing irreducible symplectic manifolds. We deduce from known results
that the generalized Beauville-Bogomolov form satisfies the Fujiki relations
and has the same rank as in the smooth case. This enables us to study
fibrations of these varieties; imposing the newer definition from [GKP16,
Definition 8.16.2] we show that they behave much like irreducible symplectic
manifolds.
Frank Gounelas ; John Christian Ottem.
Using recent results of Bayer-Macrì, we compute in many cases the
pseudoeffective and nef cones of the projectivised cotangent bundle of a smooth
projective K3 surface. We then use these results to construct explicit families
of smooth curves on which the restriction of the cotangent bundle is not
semistable (and hence not nef). In particular, this leads to a counterexample
to a question of Campana-Peternell.
Dario Beraldo.
We strengthen the gluing theorem occurring on the spectral side of the
geometric Langlands conjecture. While the latter embeds $IndCoh_N(LS_G)$ into a
category glued out of 'Fourier coefficients' parametrized by standard
parabolics, our refinement explicitly identifies the essential image of such
embedding.
Lev A. Borisov ; Sai-Kee Yeung.
We construct explicit equations of Cartwright-Steger and related surfaces.
Andrea Fanelli ; Stefan Schröer.
We introduce and study the maximal unipotent finite quotient for algebraic
group schemes in positive characteristics. Applied to Picard schemes, this
quotient encodes unusual torsion. We construct integral Fano threefolds where
such unusual torsion actually appears. The existence of such threefolds is
surprising, because the torsion vanishes for del Pezzo surfaces. Our
construction relies on the theory of exceptional Enriques surfaces, as
developed by Ekedahl and Shepherd-Barron.
Rahul Pandharipande ; Johannes Schmitt.
While the Chow groups of 0-dimensional cycles on the moduli spaces of
Deligne-Mumford stable pointed curves can be very complicated, the span of the
0-dimensional tautological cycles is always of rank 1. The question of whether
a given moduli point [C,p_1,...,p_n] determines a tautological 0-cycle is
subtle. Our main results address the question for curves on rational and K3
surfaces. If C is a nonsingular curve on a nonsingular rational surface of
positive degree with respect to the anticanonical class, we prove
[C,p_1,...,p_n] is tautological if the number of markings does not exceed the
virtual dimension in Gromov-Witten theory of the moduli space of stable maps.
If C is a nonsingular curve on a K3 surface, we prove [C,p_1,...,p_n] is
tautological if the number of markings does not exceed the genus of C and every
marking is a Beauville-Voisin point. The latter result provides a connection
between the rank 1 tautological 0-cycles on the moduli of curves and the rank 1
tautological 0-cycles on K3 surfaces. Several further results related to
tautological 0-cycles on the moduli spaces of curves are proven. Many open
questions concerning the moduli points of curves on other surfaces (Abelian,
Enriques, general type) are discussed.
Frank Grosshans ; Hanspeter Kraft.
Let $k$ be an algebraically closed field of characteristic 0, and let $V$ be
a finite-dimensional vector space. Let $End(V)$ be the semigroup of all
polynomial endomorphisms of $V$. Let $E$ be a subset of $End(V)$ which is a
linear subspace and also a semi-subgroup. Both $End(V)$ and $E$ are
ind-varieties which act on $V$ in the obvious way. In this paper, we study
important aspects of such actions. We assign to $E$ a linear subspace $D_{E}$
of the vector fields on $V$. A subvariety $X$ of $V$ is said to $D_{E}$
-invariant if $h(x)$ is in the tangent space of $x$ for all $h$ in $D_{E}$ and
$x$ in $X$. We show that $X$ is $D_{E}$ -invariant if and only if it is the
union of $E$-orbits. For such $X$, we define first integrals and construct a
quotient space for the $E$-action. An important case occurs when $G$ is an
algebraic subgroup of $GL(V$) and $E$ consists of the $G$-equivariant
polynomial endomorphisms. In this case, the associated $D_{E}$ is the space the
$G$-invariant vector fields. A significant question here is whether there are
non-constant $G$-invariant first integrals on $X$. As examples, we study the
adjoint representation, orbit closures of highest weight vectors, and
representations of the additive group. We also look at finite-dimensional
irreducible representations of SL2 and its nullcone.
Bertrand Toën.
The objective of this work is to reconsider the schematization problem of
[6], with a particular focus on the global case over Z. For this, we prove the
conjecture [Conj. 2.3.6][15] which gives a formula for the homotopy groups of
the schematization of a simply connected homotopy type. We deduce from this
several results on the behaviour of the schematization functor, which we
propose as a solution to the schematization problem.
Lothar Göttsche.
We compute generating functions for elliptic genera with values in line
bundles on Hilbert schemes of points on surfaces. As an application we also
compute generating functions for elliptic genera with values in determinant
line bundles on moduli spaces of sheaves on K3 surfaces.
Remy van Dobben de Bruyn.
We consider several conjectures on the independence of $\ell$ of the étale
cohomology of (singular, open) varieties over $\bar{\mathbf F}_p$. The main
result is that independence of $\ell$ of the Betti numbers
$h^i_{\text{c}}(X,\mathbf Q_\ell)$ for arbitrary varieties is equivalent to
independence of $\ell$ of homological equivalence $\sim_{\text{hom},\ell}$ for
cycles on smooth projective varieties. We give several other equivalent
statements. As a surprising consequence, we prove that independence of $\ell$
of Betti numbers for smooth quasi-projective varieties implies the same result
for arbitrary separated finite type $k$-schemes.
Kirti Joshi ; Christian Pauly.
In this paper we continue our study of the Frobenius instability locus in the
coarse moduli space of semi-stable vector bundles of rank $r$ and degree $0$
over a smooth projective curve defined over an algebraically closed field of
characteristic $p>0$. In a previous paper we identified the "maximal" Frobenius
instability strata with opers (more precisely as opers of type $1$ in the
terminology of the present paper) and related them to certain Quot-schemes of
Frobenius direct images of line bundles. The main aim of this paper is to
describe for any integer $q \geq 1$ a conjectural generalization of this
correspondence between opers of type $q$ (which we introduce here) and
Quot-schemes of Frobenius direct images of vector bundles of rank $q$. We also
give a conjectural formula for the dimension of the Frobenius instability
locus.
Aleksandr V. Pukhlikov.
We show that for a Zariski general hypersurface $V$ of degree $M+1$ in
${\mathbb P}^{M+1}$ for $M\geqslant 5$ there are no Galois rational covers
$X\dashrightarrow V$ of degree $d\geqslant 2$ with an abelian Galois group,
where $X$ is a rationally connected variety. In particular, there are no
rational maps $X\dashrightarrow V$ of degree 2 with $X$ rationally connected.
This fact is true for many other families of primitive Fano varieties as well
and motivates a conjecture on absolute rigidity of primitive Fano varieties.
Olivier Debarre ; Alexander Kuznetsov.
We describe intermediate Jacobians of Gushel-Mukai varieties $X$ of
dimensions 3 or 5: if $A$ is the Lagrangian space associated with $X$, we prove
that the intermediate Jacobian of $X$ is isomorphic to the Albanese variety of
the canonical double covering of any of the two dual Eisenbud-Popescu-Walter
surfaces associated with $A$. As an application, we describe the period maps
for Gushel-Mukai threefolds and fivefolds.
Ana-Maria Castravet ; Jenia Tevelev.
We construct an $S_2\times S_n$ invariant full exceptional collection on
Hassett spaces of weighted stable rational curves with $n+2$ markings and
weights $(\frac{1}{2}+\eta, \frac{1}{2}+\eta,\epsilon,\ldots,\epsilon)$, for
$0<\epsilon, \eta\ll1$ and can be identified with symmetric GIT quotients of
$(\mathbb{P}^1)^n$ by the diagonal action of $\mathbb{G}_m$ when $n$ is odd,
and their Kirwan desingularization when $n$ is even. The existence of such an
exceptional collection is one of the needed ingredients in order to prove the
existence of a full $S_n$-invariant exceptional collection on
$\overline{\mathcal{M}}_{0,n}$. To prove exceptionality we use the method of
windows in derived categories. To prove fullness we use previous work on the
existence of invariant full exceptional collections on Losev-Manin spaces.