Essential dimension of simple algebras with involutions

Let $1\leq m \leq n$ be integers with $m|n$ and \cat{Alg}_{n,m} the class of central simple algebras of degree $n$ and exponent dividing $m$. In this paper, we find new, improved upper bounds for the essential dimension and 2-dimension of \cat{Alg}_{n,2}. In particular, we show that \ed_{2}(\cat{Alg}_{16,2})=24 over a field $F$ of characteristic different from 2.Comment: Sections 1 and 3 are rewritte

Essential dimension of simple algebras in positive characteristic

Let $p$ be a prime integer, $1\leq s\leq r$ integers, $F$ a field of characteristic $p$. Let \cat{Dec}_{p^r} denote the class of the tensor product of $r$ $p$-symbols and \cat{Alg}_{p^r,p^s} denote the class of central simple algebras of degree $p^r$ and exponent dividing $p^s$. For any integers $s<r$, we find a lower bound for the essential $p$-dimension of \cat{Alg}_{p^r,p^s}. Furthermore, we compute upper bounds for \cat{Dec}_{p^r} and \cat{Alg}_{8,2} over $\ch(F)=p$ and $\ch(F)=2$, respectively. As a result, we show \ed_{2}(\cat{Alg}_{4,2})=\ed(\cat{Alg}_{4,2})=\ed_{2}(\gGL_{4}/\gmu_{2})=\ed(\gGL_{4}/\gmu_{2})=3 and 3\leq \ed(\cat{Alg}_{8,2})=\ed(\gGL_{8}/\gmu_{2})\leq 10 over a field of characteristic 2.Comment: Any comments are welcom

Essential dimension of semisimple groups of type BB

We determine the essential dimension of an arbitrary semisimple group of type $B$ of the form $$G=\big(\operatorname{\mathbf{Spin}}(2n_{1}+1)\times\cdots \times \operatorname{\mathbf{Spin}}(2n_{m}+1)\big)/\boldsymbol{\mu}$$ over a field of characteristic $0$, for all $n_{1},\ldots, n_{m}\geq 7$, and a central subgroup $\boldsymbol{\mu}$ of $\operatorname{\mathbf{Spin}}(2n_{1}+1)\times\cdots \times \operatorname{\mathbf{Spin}}(2n_{m}+1)$ not containing the center of $\operatorname{\mathbf{Spin}}(2n_i+1)$ as a direct factor. We also find the essential dimension of $G$ for each of the following cases, where either $n_{i}=1$ for all $i$ or $m=2$, $n_{1}=1$, $2\leq n_{2}\leq 3$, $\boldsymbol{\mu}$ is the diagonal central subgroup for both cases

Counter-examples to a conjecture of Karpenko for spin groups

Consider the canonical morphism from the Chow ring of a smooth variety $X$ to the associated graded ring of the topological filtration on the Grothendieck ring of $X$. In general, this morphism is not injective. However, Nikita Karpenko conjectured that these two rings are isomorphic for a generically twisted flag variety $X$ of a semisimple group $G$. The conjecture was first disproved by Nobuaki Yagita for $G=\mathop{\mathrm{Spin}}(2n+1)$ with $n=8, 9$. Later, another counter-example to the conjecture was given by Karpenko and the first author for $n=10$. In this note, we provide an infinite family of counter-examples to Karpenko's conjecture for any $2$-power integer $n$ greater than $4$. This generalizes Yagita's counter-example and its modification due to Karpenko for $n=8$.Comment: 28 page