Affine pavings for moduli spaces of pure sheaves on ${\mathbb{P}}^2$ with degree $\le 5$

Abstract

Let $M(d,r)$ be the moduli space of semistable sheaves of rank 0, Euler characteristic $r$ and first Chern class $dH(d>0)$, with $H$ the hyperplane class in ${\mathbb{P}}^2$. By previous work, we gave an explicit description of the class $[M(d,r)]$ of $M(d,r)$ in the Grothendieck ring of varieties for $d\le 5$ and $g.c.d(d,r)=1$. In this paper we compute the fixed locus of $M(d,r)$ under some $({\mathbb{C}}^{\ast }{)}^2$-action and show that $M(d,r)$ admits an affine paving for $d\le 5$ and $g.c.d(d,r)=1$. We also pose a conjecture that for any $d$ and $r$ coprime to $d$, $M(d,r)$ would admit an affine paving.

22 pages

References in corpus (1)

• Torus action on the moduli spaces of plane sheaves