Moduli Spaces of Semistable Sheaves of Dimension $1$ on $\mathbb{P}^2$

Let $M(d, \chi)$ be the moduli space of semistable sheaves of rank $0$, Euler characteristic $\chi$ and first Chern class $dH (d \gt 0)$, with $H$ the hyperplane class in $\mathbb{P}^2$. We give a description of $M(d, \chi)$, viewing each sheaf as a class of matrices with entries in $\bigoplus_{i \geq 0} H^0 (\mathcal{O}_{\mathbb{P}^2}(i))$. We show that there is a big open subset of $M(d, 1)$ isomorphic to a projective bundle over an open subset of a Hilbert scheme of points on $\mathbb{P}^2$. Finally we compute the classes of $M(4, 1)$, $M(5, 1)$ and $M(5, 2)$ in the Grothendieck ring of varieties, especially we conclude that $M(5, 1)$ and $M(5, 2)$ are of the same class.

Keywords

moduli spaces, 1-dimensional sheaves on surfaces, projective plan, BPS states of weight 0

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Published 27 September 2023