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Register a new userPrime decomposition for the index of a Brauer class
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We prove that the index of a Brauer class satisfies prime decomposition over a general base scheme. This contrasts with our previous result that there is no general prime decomposition of Azumaya algebras.
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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 $beta_C$ denote the natural Brauer class over the stable locus. We prove that if $f^*( beta_{C}) = beta_C$ 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 isomorphic. Similar statements hold for Kummer surfaces in $mathbb{P}^3$ and for quadratic line complexes.
Let $k$ be a field and $X/k$ be a smooth quasiprojective orbifold. Let $Xto underline{X}$ 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 field finitely generated over the finite field $mathbb F_p$ of odd characteristic $p$. For any K3 surface $X$ over $k$ we prove that the prime to $p$ component of the cokernel of the natural map $Br(k)to Br(X)$ is finite.
Classifying elements of the Brauer group of a variety X over a p-adic field according to the p-adic accuracy needed to evaluate them gives a filtration on Br X. We show that, on the p-torsion, this filtration coincides with a modified version of that defined by Katos Swan conductor, and that the refined Swan conductor controls how the evaluation maps vary on p-adic discs, giving a geometric characterisation of the refined Swan conductor. We give applications to the study of rational points on varieties over number fields.
Let ${P_i}_{1 leq i leq r}$ and ${Q_i}_{1 leq i leq r}$ be two collections of Brauer Severi surfaces (resp. conics) over a field $k$. We show that the subgroup generated by the $P_is$ in $Br(k)$ is the same as the subgroup generated by the $Q_is$ iff $Pi P_i $ is birational to $Pi Q_i$. Moreover in this case $Pi P_i$ and $Pi Q_i$ represent the same class in $M(k)$, the Grothendieck ring of $k$-varieties. The converse holds if $char(k)=0$. Some of the above implications also hold over a general noetherian base scheme.
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