Affine pavings for moduli spaces of pure sheaves on $$\mathbb {P}^2$$ P 2 with degree $$\le 5$$ ≤ 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 \(\mathbb {P}^2\). In [14] 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 \((\mathbb {C}^{*})^2\)-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.