Tautological classes on low-degree Hurwitz spacesCanning, Samir; Larson, Hannah
doi: 10.1093/imrn/rnac360pmid: N/A
Let ${\mathcal {H}}_{k,g}$ be the Hurwitz stack parametrizing degree $k$, genus $g$ covers of ${\mathbb {P}}^{1}$. We define the tautological ring of ${\mathcal {H}}_{k,g}$ and we show that all Chow classes, except possibly those supported on the locus of “factoring covers,” are tautological up to codimension roughly $g/k$ when $k \leq 5$. The set-up developed here is also used in our subsequent work [6], wherein we prove new results about the structure of the Chow ring for $k \leq 5$.
Cosets of Free Field Algebras via Arc SpacesLinshaw, Andrew R; Song, Bailin
doi: 10.1093/imrn/rnac367pmid: N/A
Using the invariant theory of arc spaces, we find minimal strong generating sets for certain cosets of affine vertex algebras inside free field algebras that are related to classical Howe duality. These results have several applications. First, for any vertex algebra ${{\mathcal {V}}}$, we have a surjective homomorphism of differential algebras $\mathbb {C}[J_{\infty }(X_{{{\mathcal {V}}}})] \rightarrow \text {gr}^{F}({{\mathcal {V}}})$; equivalently, the singular support of ${{\mathcal {V}}}$ is a closed subscheme of the arc space of the associated scheme $X_{{{\mathcal {V}}}}$. We give many new examples of classically free vertex algebras (i.e., this map is an isomorphism), including $L_{k}({{\mathfrak {s}}}{{\mathfrak {p}}}_{2n})$ for all positive integers $n$ and $k$. We also give new examples where the kernel of this map is nontrivial but is finitely generated as a differential ideal. Next, we prove a coset realization of the subregular ${{\mathcal {W}}}$-algebra of ${{\mathfrak {s}}}{{\mathfrak {l}}}_{n}$ at a critical level that was previously conjectured by Creutzig, Gao, and the 1st author. Finally, we give some new level-rank dualities involving affine vertex superalgebras.
Rationality of Real Conic Bundles With Quartic Discriminant CurveJi, Lena; Ji, Mattie
doi: 10.1093/imrn/rnad003pmid: N/A
We study real double covers of $\mathbb P^{1}\times \mathbb P^{2}$ branched over a $(2,2)$-divisor, which are conic bundles with smooth quartic discriminant curve by the second projection. In each isotopy class of smooth plane quartics, we construct examples where the total space is $\mathbb R$-rational. For five of the six isotopy classes, we construct $\mathbb C$-rational examples with obstructions to rationality over $\mathbb R$, and for the sixth class, we show that the models we consider are all rational. Moreover, for three of the five classes with irrational members, we characterize rationality using the real locus and the intermediate Jacobian torsor obstruction of Hassett–Tschinkel and Benoist–Wittenberg. These double cover models were introduced by Frei, Sankar, Viray, Vogt, and the first author, who determined explicit descriptions for their intermediate Jacobian torsors.
The Fundamental Fiber Sequence in Étale Homotopy TheoryHaine, Peter J; Holzschuh, Tim; Wolf, Sebastian
doi: 10.1093/imrn/rnad018pmid: N/A
Let $ k $ be a field with separable closure $ \bar {k} \supset k $, and let $ X $ be a qcqs $ k $-scheme. We use the theory of profinite Galois categories developed by Barwick–Glasman–Haine to provide a quick conceptual proof that the sequences $ \Pi _{<\infty }^{\'{e}\textrm {t}}(X_{\bar {k}}) \to \Pi _{<\infty }^{\'{e}\textrm {t}}(X) \to \text {BGal}(\bar {k}/k) $ and $ \widehat {\Pi }_{\infty }^{\'{e}\textrm {t}}(X_{\bar {k}}) \to \widehat {\Pi }_{\infty }^{\'{e}\textrm {t}}(X) \to \text {BGal}(\bar {k}/k) $ of protruncated and profinite étale homotopy types are fiber sequences. This gives a common conceptual reason for the following two phenomena: first, the higher étale homotopy groups of $ X $ and the geometric fiber $ X_{\bar {k}} $ are isomorphic, and second, if $ X_{\bar {k}} $ is connected, then the sequence of profinite étale fundamental groups $ 1 \to \hat {\pi }^{\'{e}\textrm {t}}_1(X_{\bar {k}}) \to \hat {\pi }^{\'{e}\textrm {t}}_1(X) \to \text {Gal}(\bar {k}/k) \to 1 $ is exact. It also proves the analogous results for the groupe fondamental élargi of SGA3.
A Mixing Property for the Action of SL(3,ℤ) × SL(3,ℤ) on the Stone–Čech Boundary of SL(3,ℤ)Bassi, J; Rădulescu, F
doi: 10.1093/imrn/rnad014pmid: N/A
By analogy with the construction of the Furstenberg boundary, the Stone–Čech boundary of $\textrm {SL}(3,\mathbb {Z})$ is a fibered space over products of projective matrices. The proximal behaviour on this space is exploited to show that the preimages of certain sequences have accumulation points that belong to specific regions, defined in terms of flags. We show that the $\textrm {SL}(3,\mathbb {Z})\times \textrm {SL}(3,\mathbb {Z})$-quasi-invariant Radon measures supported on these regions are tempered. Thus, every quasi-invariant Radon boundary measure for $\textrm {SL}(3,\mathbb {Z})$ is an orthogonal sum of a tempered measure and a measure having matrix coefficients belonging to a certain ideal $c^{\prime}_0 ((\textrm {SL}(3,\mathbb {Z}) \times \textrm {SL}(3,\mathbb {Z}))$, slightly larger than $c_0 ((\textrm {SL}(3,\mathbb {Z}) \times \textrm {SL}(3,\mathbb {Z}))$. Hence, the left–right representation of $C^*(\textrm {SL}(3,\mathbb {Z}) \times \textrm {SL}(3,\mathbb {Z}))$ in the Calkin algebra of $\textrm {SL}(3,\mathbb {Z})$ factors through $C^*_{c^{\prime}_0} (\textrm {SL}(3,\mathbb {Z}) \times \textrm {SL}(3,\mathbb {Z}))$ and the centralizer of every infinite subgroup of $\textrm {SL}(3,\mathbb {Z})$ is amenable.
Irrationality of Generic Quotient Varieties via Bogomolov MultipliersJezernik, Urban; Sánchez, Jonatan
doi: 10.1093/imrn/rnad012pmid: N/A
The Bogomolov multiplier of a group is the unramified Brauer group associated with the quotient variety of a faithful representation of the group. This object is an obstruction for the quotient variety to be stably rational. The purpose of this paper is to study these multipliers associated with nilpotent pro-$p$ groups by transporting them to their associated Lie algebras. Special focus is set on the case of $p$-adic Lie groups of nilpotency class $2$, where we analyse the moduli space. This is then applied to give information on asymptotic behaviour of multipliers of finite images of such groups of exponent $p$. We show that with fixed $n$ and increasing $p$, a positive proportion of these groups of order $p^n$ have trivial multipliers. On the other hand, we show that by fixing $p$ and increasing $n$, log-generic groups of order $p^n$ have non-trivial multipliers. Whence quotient varieties of faithful representations of log-generic $p$-groups are not stably rational, applications in non-commutative Iwasawa theory aredeveloped.
Mixed Hodge Structure on Local Cohomology with Support in Determinantal VarietiesPerlman, Michael
doi: 10.1093/imrn/rnad019pmid: N/A
We employ the inductive structure of determinantal varieties to calculate the mixed Hodge module structure of local cohomology modules with determinantal support. We show that the weight of a simple composition factor is uniquely determined by its support and cohomological degree. As a consequence, we obtain the equivariant structure of the Hodge filtration on each local cohomology module. Finally, as an application, we provide a formula for the generation level of the Hodge filtration on these modules.