arXiv subjects
Publications and source records attributed to Fabio Tonini.
Berthelot's conjecture predicts that under a proper and smooth morphism of schemes in characteristic $p$, the higher direct images of an overconvergent $F$-isocrystal are overconvergent $F$-isocrystals. In this paper we prove that this is true for crystals up to isogeny. As an application we prove a Künneth formula for the crystalline fundamental group.
In the present paper, we study a purely inseparable counterpart of Abhyankar's conjecture for the affine line in positive characteristic, and prove its validity for all the finite local non-abelian simple group schemes in characteristic $p>5$. The crucial point is how to deal with finite local group schemes which cannot be realized as the Frobenius kernel of a smooth algebraic group. Such group schemes appear as the ones associated with Cartan type Lie algebras. We settle the problem for such Lie algebras by making use of natural gradations or filtrations on them.
The aim of this paper is to study the geometry of the stack of $S_{3}$-covers. We show that it has two irreducible components $\mathcal{Z}_{S_{3}}$ and $\mathcal{Z}_{2}$ meeting in a "degenerate" point $\{0\}$, $\mathcal{Z}_{2}-\{0\}\simeq \rm B\rm{GL}_{2}$, while $(\mathcal{Z}_{S_{3}}-\{0\})$, which contains $\rm B S_{3}$ as open substack, is a smooth and universally closed algebraic stack. More precisely we show that $\mathcal{Z}_{S_{3}}-\{0\}\simeq[X/\rm{GL}_{2}]$, where $X$ is an explicit smooth non degenerate projective surface inside $\mathbb{P}^{7}$ intersection of five quadrics. All these results are based on the description of certain families of $S_{3}$-covers in terms of "building data".
Let ${\mathcal X}$ be a category fibered in groupoids over a finite field $\mathbb{F}_q$, and let $k$ be an algebraically closed field containing $\mathbb{F}_q$. Denote by $\phi_k\colon {\mathcal X}_k\to {\mathcal X}_k$ the arithmetic Frobenius of ${\mathcal X}_k/k$ and suppose that ${\mathcal M}$ is a stack over $\mathbb{F}_q$ (not necessarily in groupoids). Then there is a natural functor $\alpha_{{\mathcal M},{\mathcal X}}\colon{\mathcal M}({\mathcal X})\to{\mathcal M}({\mathbf D_k}({\mathcal X}))$, where ${\mathcal M}({\mathbf D_k}({\mathcal X}))$ is the category of $\phi_k$-invariant maps ${\mathcal X}_k\to {\mathcal M}$. A version of Drinfeld's lemma states that if ${\mathcal X}$ is a projective scheme and ${\mathcal M}$ is the stack of quasi-coherent sheaves of finite presentation, then $\alpha_{{\mathcal M},{\mathcal X}}$ is an equivalence. We extend this result in several directions. For proper algebraic stacks or affine gerbes ${\mathcal X}$, we prove Drinfeld's lemma and deduce that $\alpha_{{\mathcal M},{\mathcal X}}$ is an equivalence for very general algebraic stacks ${\mathcal M}$. For arbitrary ${\mathcal X}$, we show that $\alpha_{{\mathcal M},{\mathcal X}}$ is an equivalence when ${\mathcal M}$ is the stack of immersions, the stack of quasi-compact separated \'etale morphisms or any quasi-separated Deligne-Mumford stack with separated diagonal.
We introduce ``sheafification'' functors from categories of (lax monoidal) linear functors to categories of quasi-coherent sheaves (of algebras) of stacks. They generalize the homogeneous sheafification of graded modules for projective schemes.
Given a quasi-compact category fibered in groupoids $\mathcal{X}$ and a monoidal subcategory $\mathcal{C}$ of its category of locally free sheaves $\text{Vect}(\mathcal{X})$, we are going to introduce the stack of fiber functors $\text{Fib}_{\mathcal{X},\mathcal{C}}$ with source $\mathcal{C}$, which comes equipped with a map $\mathcal{P}_{\mathcal{C}}\colon\mathcal{X}\to\text{Fib}_{\mathcal{X},\mathcal{C}}$ and a functor $\mathcal{G}\colon\mathcal{C}\to\text{Vect}(\text{Fib}_{\mathcal{X},\mathcal{C}})$. If $\mathcal{C}$ generates $\text{QCoh}(\mathcal{X})$ and $\mathcal{X}$ is an fpqc stack with quasi-affine diagonal, we show that $\mathcal{P}_{\mathcal{C}}\colon\mathcal{X}\to\text{Fib}_{\mathcal{X},\mathcal{C}}$ is an equivalence, as it happens by Tannaka's reconstruction when $\mathcal{X}$ is an affine gerbe over a field. In general, under mild assumption on $\mathcal{C}$, e.g. $\mathcal{C}=\text{Vect}(\mathcal{X})$, we show that $\text{Fib}_{\mathcal{X},\mathcal{C}}$ is a quasi-compact fpqc stack with affine diagonal and that the image $\mathcal{G}(\mathcal{C})$ generates $\text{QCoh}(\text{Fib}_{\mathcal{X},\mathcal{C}})$.
We give a proper definition of the multiplicative structure of the following rings: the Cox ring of invertible sheaves on a general algebraic stack; and the Cox ring of rank one reflexive sheaves on a normal and excellent algebraic stack. We show that such Cox rings always exist and establish their (non-)uniqueness in terms of an Ext-group. Moreover, we compare our definition with the classical construction of a Cox ring on a variety. Finally, we give an application to the theory of Mori dream stacks.
Applying the authors' preceding work, we construct a version of the moduli space of $G$-torsors over the formal punctured disk for a finite group $G$. To do so, we introduce two Grothendieck topologies, the sur (surjective) and luin (locally universally injective) topologies, and define P-schemes using them as variants of schemes. Our moduli space is defined as a P-scheme approximating the relevant moduli functor. We then prove that Fr\"ohlich's module resolvent gives a locally constructible function on this moduli space, which implies that motivic integrals appearing the wild McKay correspondence are well-defined.
We formulate a conjecture on the motivic McKay correspondence for the group scheme $ \alpha_{p}$ in characteristic $p>0$ and give a few evidences. The conjecture especially claims that there would be a close relation between quotient varieties by $\alpha_{p}$ and ones by the cyclic group of order $p$.
We construct the moduli stack of torsors over the formal punctured disk in characteristic p > 0 for a finite group isomorphic to the semidirect product of a p-group and a tame cyclic group. We prove that the stack is a limit of separated Deligne-Mumford stacks with finite and universally injective transition maps.
We study the stack B_{h,g,n} of uniform cyclic covers of degree n between smooth curves of genus h and g and, for h >> g, present it as an open substack of a vector bundle over the universal Jacobian stack of M_g. We use this description to compute the integral Picard group of B_{h,g,n}, showing that it is generated by tautological classes of B_{h,g,n}.
Let $X$ be a normal, connected and projective variety over an algebraically closed field $k$. It is known that a vector bundle $V$ on $X$ is essentially finite if and only if it is trivialized by a proper surjective morphism $f:Y\to X$. In this paper we introduce a different approach to this problem which allows to extend the results to normal, connected and strongly pseudo-proper algebraic stack of finite type over an arbitrary field $k$.
In this paper we extend the generalized algebraic fundamental group constructed by Esnault and Hogadi to general fibered categories using the language of gerbes. As an application we obtain a Tannakian interpretation for the Nori fundamental gerbe defined by Borne and Vistoli for non smooth non pseudo-proper algebraic stacks.
We interpret the regularity of a finite and flat extension of a discrete valuation ring in terms of the trace map of the extension.
By a result of Biswas and Dos Santos, on a smooth and projective variety over an algebraically closed field, a vector bundle trivialized by a proper and surjective map is essentially finite, that is it corresponds to a representation of the Nori fundamental group scheme. In this paper we obtain similar results for non-proper non-smooth algebraic stacks over arbitrary fields of characteristic $p>0$. As by-product we have the following partial generalization of the Biswas-Dos Santos' result in positive characteristic: on a pseudo-proper and inflexible stack of finite type over $k$ a vector bundle which is trivialized by a proper and flat map is essentially finite.
We interpret Galois covers in terms of particular monoidal functors, extending the correspondence between torsors and fiber functors. As applications we characterize tame $G$-covers between normal varieties for finite and étale group schemes and we prove that, if $G$ is a finite, flat and finitely presented nonabelian and linearly reductive group scheme over a ring, then the moduli stack of $G$-covers is reducible.
Given a finite, flat and finitely presented group scheme $G$ over some base $S$, we introduce the notion of ramified $G$-covers and study the moduli stack $G$-Cov they form. The thesis is divided in three parts. The first one concerns the case when $G$ is a diagonalizable group scheme and it essentially coincides with arxiv:1106.2347. In the second part I deal with the general case. Assuming that the base S is affine and given an $S$-scheme $T$, I interpret $G$-covers of $T$ as particolar (lax) monoidal functors from the category of finite, $G$-equivariant locally free sheaves over $S$ to the category of finite locally free sheaves over $T$, extending the classical Tannakian correspondence between $G$-torsors and strong monoidal functors as above. Using this point of view, I prove that $G$-Cov is always reducible if $G$ is a non-abelian linearly reductive group. When $G$ is constant and tame I also give a criterion to detect when a $G$-cover of a regular in codimension one, integral scheme has regular in codimension one total space in terms of the functor associated with the cover. In the last part I focus on the case $G=S_3$, prove that $S_3$-Cov has exactly two irreducible components and describe the principal one. I also describe particular open loci of $S_3$-Cov, that is particular families of $S_3$-covers, classify $S_3$-covers of regular schemes whose total space is regular and compute the invariants of $S_3$-covers of smooth surfaces.
Given a flat, finite group scheme G finitely presented over a base scheme we introduce the notion of ramified Galois cover of group G (or simply G-cover), which generalizes the notion of G-torsor. We study the stack of G-covers, denoted with G-Cov, mainly in the abelian case, precisely when G is a finite diagonalizable group scheme over Z. In this case we prove that G-Cov is connected, but it is irreducible or smooth only in few finitely many cases. On the other hand, it contains a 'special' irreducible component Z_G, which is the closure of BG and this reflects the deep connection we establish between G-Cov and the equivariant Hilbert schemes. We introduce 'parametrization' maps from smooth stacks, whose objects are collections of invertible sheaves with additional data, to Z_G and we establish sufficient conditions for a G-cover in order to be obtained (uniquely) through those constructions. Moreover a toric description of the smooth locus of Z_G is provided.