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Grzegorz Zwara

Publications and source records attributed to Grzegorz Zwara.

9 recordsLinked to original sources
arXiv2021

Tangent spaces of orbit closures for representations of Dynkin quivers of type D

Let $\Bbbk$ be an algebraically closed field, $Q$ a finite quiver, and denote by $\mathop{\mathrm{rep}}_Q^{\mathbf{d}}$ the affine $\Bbbk$-scheme of representations of $Q$ with a fixed dimension vector ${\mathbf{d}}$. Given a representation $M$ of $Q$ with dimension vector ${\mathbf{d}}$, the set ${\mathcal{O}}_M$ of points in $\mathop{\mathrm{rep}}_Q^{\mathbf{d}}(\Bbbk)$ isomorphic as representations to $M$ is an orbit under an action on $\mathop{\mathrm{rep}}_Q^{\mathbf{d}}(\Bbbk)$ of a product of general linear groups. The orbit ${\mathcal{O}}_M$ and its Zariski closure $\overline{\mathcal{O}}_M$, considered as reduced subschemes of $\mathop{\mathrm{rep}}_Q^{\mathbf{d}}$, are contained in an affine scheme ${\mathcal{C}}_M$ defined by rank conditions on suitable matrices associated to $\mathop{\mathrm{rep}}_Q^{\mathbf{d}}$. For all Dynkin and extended Dynkin quivers, the sets of points of $\overline{\mathcal{O}}_M$ and ${\mathcal{C}}_M$ coincide, or equivalently, $\overline{\mathcal{O}}_M$ is the reduced scheme associated to ${\mathcal{C}}_M$. Moreover, $\overline{\mathcal{O}}_M={\mathcal{C}}_M$ provided $Q$ is a Dynkin quiver of type ${\mathbb{A}}$, and this equality is a conjecture for the remaining Dynkin quivers (of type ${\mathbb{D}}$ and ${\mathbb{E}}$). Let $Q$ be a Dynkin quiver of type ${\mathbb{D}}$ and $M$ a finite dimensional representation of $Q$. We show that the equality $T_N\overline{\mathcal{O}}_M=T_N{\mathcal{C}}_M$ of Zariski tangent spaces holds for any closed point $N$ of $\overline{\mathcal{O}}_M$. As a consequence, we describe the tangent spaces to $\overline{\mathcal{O}}_M$ in representation theoretic terms.

math.AG↗
arXiv2006

Regular orbit closures in module varieties

Let A be a finitely generated associative algebra over an algebraically closed field. We characterize the finite dimensional modules over A whose orbit closures are regular varieties.

math.AG↗
arXiv2006

Codimension two singularities for representations of extended Dynkin quivers

Let M and N be two representations of an extended Dynkin quiver such that the orbit O_N of N is contained in the orbit closure \bar{O_M} and has codimension two. We show that the pointed variety $(\bar{O_M},N)$ is smoothly equivalent to a simple surface singularity of type A_n, or to the cone over a rational normal curve.

math.AG↗
arXiv2004

Regularity in codimension one of orbit closures in module varieties

Let M_d(k) denote the space of dxd-matrices with coefficients in an algebraically closed field k. Let X be an orbit closure in the product [M_d(k)]^t equipped with the action of the general linear group GL_d(k) by simultaneous conjugation. We show that X is regular at any its point y such that the orbit of y has codimension one in X. The proof uses mainly the representation theory of associative algebras.

math.AG↗