Let $X$ be an irreducible smooth projective curve, defined over an algebraically closed field $k$, of genus at least three and $L$ a line bundle on $X$. Let ${\mathcal M}_X(r,L)$ be the moduli space of stable vector bundles on $X$ of rank $r$ and determinant $L$ with $r\geq 2$. We prove that the Brauer group ${\rm Br}(\mathcal{M}_X(r,L))$ is cyclic of order ${\rm g.c.d.}(r,{\rm degree}(L))$. We also prove that ${\rm Br}(\mathcal{M}_X(r,L))$ is generated by the class of the projective bundle obtained by restricting the universal projective bundle. These results were proved earlier in \cite{BBGN} under the assumption that $k=\mathbb C$.
We compute the Brauer group of the moduli stack of stable PGL(r)-bundles on a curve $X$ over an algebraically closed field of characteristic zero. We also show that the Brauer group of such a moduli stack coincides with the Brauer group of the smooth locus of the corresponding coarse moduli space of stable PGL(r)-bundles on $X$.
Let $X$ be a smooth projective curve over the complex numbers. We compute the Brauer group of the moduli stack of Bruhat-Tits group scheme $\mathcal{G}$-torsors on $X$. When $g(X) \geq 3$ we compute the Brauer group of the regularly stable locus of the coarse moduli space of semi-stable $\mathcal{G}$-torsors.
We prove that the moduli spaces of framed bundles over a smooth projective curve are rational. We compute the Brauer group of these moduli spaces to be zero under some assumption on the stability parameter.
Let $G$ be a semisimple linear algebraic group over the field $\mathbb C$, and let $C$ be an irreducible smooth complex projective curve of genus at least three. We compute the Brauer group of the smooth locus of the moduli space of semistable principal $G$--bundles over $C$. We also compute the Brauer group of the moduli stack of principal $G$--bundles over $C$.
We compute the Brauer group of the universal moduli stack of vector bundles on (possibly marked) smooth curves of genus at least three over the complex numbers. As consequence, we obtain an explicit description of the Brauer group of the smooth locus of the associated moduli space of semistable vector bundles, when the genus is at least four.
We determine the Brauer group and Picard group of the moduli space $U^{' s}_{L,par}$ of stable parabolic vector bundles of rank $r$ with determinant $L$ on a real curve $Y$ of arithmetic genus $g \,\ge\, 2$ with at most nodes as singularities.
We compute the Brauer group of the moduli stack of hyperelliptic curves $\mathcal{H}_g$ over any field of characteristic zero. In positive characteristic, we compute the part of the Brauer group whose order is prime to the characteristic of the base field.
Let U^{'s}_{L}(n,d) be the moduli space of stable vector bundles of rank $n$ and fixed determinant L of degree d on a nodal curve Y. The moduli space of semistable vector bundles of rank n and degree $d$ will be denoted by U'_Y(n,d). We calculate the Brauer groups of U^{'s}_{L}(n,d)$. We study the question of rationality of $U^{'s}_{L}(n,d)$ and $U'_Y(n,d)$.
Let $X$ be an irreducible smooth complex projective curve of genus at least two. Let $N$ be a connected component of the moduli space of semistable principal ${\rm PGL}_r({\mathbb C})$- bundles over $X$; it is a normal unirational complex projective variety. We prove that the Brauer group of a desingularization of $N$ is trivial.
Let C be an irreducible smooth projective curve, of genus at least two, defined over an algebraically closed field of characteristic zero. For a fixed line bundle L on C, let M_C(r,L) be the coarse moduli space of semistable vector bundles E over C of rank r and determinant L. We show that the Brauer group of any desingularization of M_C(r, L)$ is trivial.
In this paper we consider the stratification on the moduli space of principally polarized abelian surfaces in characteristic $p>0$ defined by the height of the formal group associated to $H^2(X,O_X)$. We compute the cycle classes of the strata and compare these with those of the $p$-rank stratification.
Brauer group of moduli of parabolic symplectic bundles
Shiv Nadar University（希夫纳德大学）；Indian Institute of Science Education and Research Tirupati（印度科学教育与研究学院提鲁帕蒂分校）；Indian Institute of Technology Chennai（印度理工学院金奈分校）
In this paper we consider the Brauer groups of algebraic stacks and GIT quotients: the only algebraic stacks we consider in this paper are quotient stacks [X/G], where X is a smooth scheme of finite type over a field k, and G is a linear algebraic group over k and acting on X, as well as various moduli stacks of principal G-bundles on a smooth projective curve X, associated to a reductive group G. We also consider the Brauer groups of the corresponding coarse moduli spaces, which most often identify with the corresponding GIT-quotients. One conclusion that we seem to draw then is that the Brau…
This paper proves a result on the existence of finite flat scheme covers of Deligne-Mumford stacks. This result is used to prove that a large class of smooth Deligne-Mumford stacks with affine moduli space are quotient stacks, and in the case of quasi-projective moduli space, to reduce the question to one concerning Brauer groups of schemes.
Let ${\mathcal P}{\mathcal M}^α_s$ be a moduli space of stable parabolic vector bundles of rank $n \geq 2$ and fixed determinant of degree $d$ over a compact connected Riemann surface $X$ of genus $g(X) \geq 2$. If $g(X) = 2$, then we assume that $n > 2$. Let $m$ denote the greatest common divisor of $d$, $n$ and the dimensions of all the successive quotients of the quasi-parabolic filtrations. We prove that the cohomological Brauer group ${\rm Br}({\mathcal P}{\mathcal M}^α_s)$ is isomorphic to the cyclic group ${\mathbb Z}/ m{\mathbb Z}$. We also show that ${\rm Br}({\mathcal P}{\mathcal M}^…
We classify products of symmetric powers of a Severi-Brauer variety, up-to stable birational equivalence. The description also includes Grassmannians and moduli spaces of genus 0 stable maps.
Given a compact Riemann surface $X$ and a semisimple affine algebraic group $G$ defined over $\mathbb C$, there are moduli spaces of Higgs bundles and of connections associated to $(X,\, G)$. We compute the Brauer group of the smooth locus of these varieties.
We prove that the Brauer group of TMF is isomorphic to the Brauer group of the derived moduli stack of elliptic curves. Then, we compute the local Brauer group, i.e., the subgroup of the Brauer group of elements trivialized by some étale cover of the moduli stack, up to a finite 2-torsion group.
We prove that the Brauer group of the moduli stack of elliptic curves $\mathscr{M}_{1,1,k}$ over an algebraically closed field $k$ of characteristic $2$ is isomorphic to $\mathbb{Z}/(2)$. We also compute the Brauer group of $\mathscr{M}_{1,1,k}$ where $k$ is a finite field of characteristic $2$.
We compute the Brauer group of the moduli stack of elliptic curves over the integers, localizations of the integers, finite fields of odd characteristic, and algebraically closed fields of characteristic not $2$. The methods involved include the use of the parameter space of Legendre curves and the moduli stack of curves with full (naive) level $2$ structure, the study of the descent spectral sequence in étale cohomology and the Leray spectral sequence in fppf cohomology, the computation of the group cohomology of $S_3$ in a certain integral representation, the classification of cubic Galois e…
Let S be a complex Enriques surface; it is the quotient of a K3 surface X by a fixed-point-free involution. The Brauer group Br(S) has a unique nonzero element. We describe its pull-back in Br(X), and show that the surfaces S for which it is trivial form a countable union of hypersurfaces in the moduli space of Enriques surfaces.
Let M be the moduli space of semistable rank 2 Higgs pairs (V,ϕ) with trivial determinant over a smooth projective curve X of genus g>1. We compute the stringy E-function of M and prove that there does not exist a symplectic desingularization of M for g>2.
Let $k$ be a field and $X/k$ be a smooth quasiprojective orbifold. Let $X\to \underline{X}$ be its coarse moduli space. In this paper we study the Brauer group of $X$ and compare it with the Brauer group of the smooth locus of $\underline{X}$.
Let k be a perfect field of characteristic p > 2. In this note, we show that the local moduli space of a non-supersingular K3 surface over k with trivial deformation of the associated enlarged formal Brauer group admits a natural p-divisible formal group structure.
Let $\text{M}_C( 2, \mathcal{O}_C) \cong \mathbb{P}^3$ denote the coarse moduli space of semistable vector bundles of rank $2$ with trivial determinant over a smooth projective curve $C$ of genus $2$ over $\mathbb{C}$. Let $β_C$ denote the natural Brauer class over the stable locus. We prove that if $f^*( β_{C'}) = β_C$ for some birational map $f$ from $\text{M}_C( 2, \mathcal{O}_C)$ to $\text{M}_{C'}( 2, \mathcal{O}_{C'})$, then the Jacobians of $C$ and of $C'$ are isomorphic as abelian varieties. If moreover these Jacobians do not admit real multiplication, then the curves $C$ and $C'$ are i…
Let $R$ be a complete discrete valuation ring with fraction field of characteristic $0$ and algebraically closed residue field of characteristic $p>0$. Let $X_R \to \mathrm{Spec}(R)$ be a smooth projective morphism of relative dimension $1$. We prove that, given a line bundle $\mathcal{L}_R$ the moduli space of Gieseker stable torsion-free sheaves of rank $r\geq 2$ over $X_R$, with determinant $\mathcal{L}_R$, is smooth over $R$.
Let SU_X(3) be the moduli space of semi-stable vector bundles of rank 3 and trivial determinant on a curve X of genus 2. It maps onto P^8 and the map is a double cover branched over a sextic hypersurface called the Coble sextic. In the dual P^8 there is a unique cubic hypersurface, the Coble cubic, singular exactly along the abelian surface of degree 1 line bundles on X. We give a new proof that these two hypersurfaces are dual. As an immediate corollary, we derive a Torelli-type result.
We propose a three-step program for the classification of stable rank 2 bundles on the projective space $\mathbb{P}^3$ inspired by an article by Hartshorne and Rao. While this classification program has been successfully completed for stable rank 2 bundles with even determinant and $c_2\le5$, much less is known for bundles with odd determinant. After revising the known facts about these objects, we list all possible spectra and minimal monads for stable rank 2 bundles with odd determinant and $c_2\le8$. We provide a full classification of all bundles with positive minimal monads, provide a neg…
Let $M$ be the moduli space of generalized parabolic bundles (GPBs) of rank $r$ and degree $d$ on a smooth curve $X$. Let $M_{\bar L}$ be the closure of its subset consisting of GPBs with fixed determinant ${\bar L}$. We define a moduli functor for which $M_{\bar L}$ is the coarse moduli scheme. Using the correspondence between GPBs on $X$ and torsion-free sheaves on a nodal curve $Y$ of which $X$ is a desingularization, we show that $M_{\bar L}$ can be regarded as the compactified moduli scheme of vector bundles on $Y$ with fixed determinant. We get a natural scheme structure on the closure o…
Harvard University（哈佛大学）；Carleton College（卡尔顿学院）；Boston University（波士顿大学）；University of New South Wales（新南威尔士大学）；University of California Berkeley（加州大学伯克利分校）
Let $Y$ be an integral nodal projective curve of arithmetic genus $g\ge 2$ with $m$ nodes defined over an algebraically closed field $k$ and $x$ a nonsingular closed point of $Y$. Let $n$ and $d$ be coprime integers with $n\ge 2$. Fix a line bundle $L$ of degree $d$ on $Y$. Let $U_Y(n,d,L)$ denote the (compactified) "fixed determinant moduli space". We prove that the restriction $\mathcal{U}_{L,x}$ of the Poincare bundle to $x \times U_Y(n,d,L)$ is stable with respect to the polarisation $θ_L$ and its restriction to $x \times U'_Y(n,d,L)$, where $U'_Y(n,d,L)$ is the moduli space of vector bund…
Let $X$ be any smooth simply connected projective surface. We consider some moduli space of pure sheaves of dimension one on $X$, i.e. $\mhu$ with $u=(0,L,χ(u)=0)$ and $L$ an effective line bundle on $X$, together with a series of determinant line bundles associated to $r[\mo_X]-n[\mo_{pt}]$ in Grothendieck group of $X$. Let $g_L$ denote the arithmetic genus of curves in the linear system $\ls$. For $g_L\leq2$, we give a upper bound of the dimensions of sections of these line bundles by restricting them to a generic projective line in $\ls$. Our result gives, together with Göttsche's computati…
We give an explicit description of $\overline{\mathcal{M}}_{1,2}$ as a weighted blow-up of a weighted projective stack. We use this description to compute the Brauer group of $\overline{\mathcal{M}}_{1,2;S}$ over any base scheme $S$ where 6 is invertible, and the integral Chow rings of $\overline{\mathcal{M}}_{1,2}$ and $\mathcal{M}_{1,2}$.
Let $\mhu$ be the moduli space of semi-stable pure sheaves of class $u$ on a smooth complex projective surface $X$. We specify $u=(0,L,χ(u)=0),$ i.e. sheaves in $u$ are of dimension $1$. There is a natural morphism $π$ from the moduli space $\mhu$ to the linear system $\ls$. We study a series of determinant line bundles $\lcn$ on $\mhu$ via $π.$ Denote $g_L$ the arithmetic genus of curves in $\ls.$ For any $X$ and $g_L\leq0$, we compute the generating function $Z^r(t)=\sum_{n}h^0(\mhu,\lcn)t^n$. For $X$ being $\mathbb{P}^2$ or $\mathbb{P}(\mo_{\pone}\oplus\mo_{\pone}(-e))$ with $e=0,1$, we com…
In this paper we introduce two new ways to split ramification of Brauer classes on surfaces using stacks. Each splitting method gives rise to a new moduli space of twisted stacky vector bundles. By studying the structure of these spaces we prove new results on the standard period-index conjecture. The first yields new bounds on the period-index relation for classes on curves over higher local fields, while the second can be used to relate the Hasse principle for forms of moduli spaces of stable vector bundles on pointed curves over global fields to the period-index problem for Brauer groups of…
We consider the problem of counting the number of rational points of bounded height in the zero-loci of Brauer group elements on semi-simple algebraic groups over number fields. We obtain asymptotic formulae for the counting problem for wonderful compactifications using the spectral theory of automorphic forms. Applications include asymptotic formulae for the number of matrices over Q whose determinant is a sum of two squares. These results provide a positive answer to some cases of a question of Serre concerning such counting problems.
We describe new irreducible components of the moduli space of rank $2$ semistable torsion free sheaves on the three-dimensional projective space whose generic point corresponds to non-locally free sheaves whose singular locus is either 0-dimensional or consists of a line plus disjoint points. In particular, we prove that the moduli spaces of semistable sheaves with Chern classes $(c_1,c_2,c_3)=(-1,2n,0)$ and $(c_1,c_2,c_3)=(0,n,0)$ always contain at least one rational irreducible component. As an application, we prove that the number of such components grows as the second Chern class grows, an…
We constructed a projective moduli space of semistable torsion free sheaves with `fixed determinant' on a reducible curve. When a family of smooth curves degenerates to the reducible curve, our moduli space is a degeneration of the moduli spaces of semistable vector bundles with a fixed determinant on the smooth curves. Moreover, the moduli space is reduced and seminormal.
Let X be a smooth projective curve of genus at least two over the complex numbers. A pair (E,ϕ) over X consists of an algebraic vector bundle E over X and a holomorphic section ϕof E. There is a concept of stability for pairs which depends on a real parameter τ. Here we prove that the third cohomology groups of the moduli spaces of τ-stable pairs with fixed determinant and rank at least two are polarised pure Hodge structures, and they are isomorphic to H^1(X) with its natural polarisation (except in very few exceptional cases). This implies a Torelli theorem for such moduli spaces. We recover…
