arXiv subjects

Vlad Roman

Publications and source records attributed to Vlad Roman.

3 recordsLinked to original sources

arXiv2026

Commuting varieties in bad characteristic

Let $k$ be an algebraically closed field of characteristic $2$. We consider the commuting variety and the commuting nilpotent variety of the Lie algebra $\mathfrak{sp}_{2n}$, namely the sets $\mathcal{C}_2(\mathfrak{sp}_{2n})=\{ (x,y) \in \mathfrak{sp}_{2n} \times \mathfrak{sp}_{2n} \mid [x,y]=0\}$ and $\mathcal{C}_2^{\text{nil}}(\mathfrak{sp}_{2n})=\{ (x,y) \in \mathfrak{sp}_{2n} \times \mathfrak{sp}_{2n} \mid x,y \text{ nilpotent, } [x,y]=0\}$ and prove that they are both irreducible, of dimensions $\dim(\mathfrak{sp}_{2n}) + 2n$ and $\dim(\mathfrak{sp}_{2n}) + n-1$, respectively.

math.AG↗

arXiv2025

The variety of nilpotent pairs $(A,B)$ with $[A,B] = \lambda I$

Let $k$ be an algebraically closed field of characteristic $p >0$. We consider the variety of nilpotent pairs $(A,B)$ with $[A,B]=\lambda I$, namely the set of pairs $ X = \{ (A,B) \in M_n(k) \times M_n(k) \mid A,B \text{ nilpotent}, [A,B]=\lambda I, \lambda \in k \}$. We prove that if $n=pr$, then $X$ is irreducible of dimension $n^2$.

math.AG↗

arXiv2024

The commuting variety of $\mathfrak{pgl}_n$

We are considering the commuting variety of the Lie algebra $\mathfrak{pgl}_n$ over an algebraically closed field of characteristic $p >0$, namely the set of pairs $ \{ (A,B) \in \mathfrak{pgl}_n \times \mathfrak{pgl}_n \mid [A,B]=0 \} $. We prove that if $n=pr$, then there are precisely two irreducible components, of dimensions $n^2+r-1$ and $n^2+n-2$. We also prove that the variety $\{ (x,y) \in GL_n(k) \times GL_n(k) \mid [x,y]=\zeta I \}$ is irreducible of dimension $n^2 +n/d$, where $\zeta$ is a root of unity of order $d$ with $d$ dividing $n$.

math.AG↗