# Volume 4

Fourth volume of Épijournal de Géométrie Algébrique - 2020

### 1. Free Proalgebraic Groups

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.

### 2. Hyperelliptic classes are rigid and extremal in genus two

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.

### 3. On the group of zero-cycles of holomorphic symplectic varieties

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.

### 4. Smooth projective horospherical varieties of Picard group $\mathbb{Z}^2$

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.

### 5. Sur l'existence du schéma en groupes fondametal

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.

### 6. Finiteness of cohomology groups of stacks of shtukas as modules over Hecke algebras, and applications

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.

### 7. Fujiki relations and fibrations of irreducible symplectic varieties

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.

### 8. Remarks on the positivity of the cotangent bundle of a K3 surface

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.

### 9. The spectral gluing theorem revisited

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.

### 10. Explicit equations of the Cartwright-Steger surface

We construct explicit equations of Cartwright-Steger and related surfaces.

### 11. The maximal unipotent finite quotient, unusual torsion in Fano threefolds, and exceptional Enriques surfaces

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.

### 12. Zero cycles on the moduli space of curves

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.

### 13. Covariants, Invariant Subsets, and First Integrals

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.