Blow ups of $\mathbb{P^n}$ as quiver moduli for exceptional collections

Abstract

Suppose P n m is the blow up of P n at a linear subspace of dimension m, L = {L 1 , . . . , Lr} is a (not necessarily full) strong exceptional collection of line bundles on P n m . Let Q be the quiver associated to this collection. One might wonder when is P n m the moduli space of representations of Q with dimension vector (1, . . . , 1) for a suitably chosen stability condition θ: S ∼ = M θ . In this paper, we achieve such isomorphism using L of length 3. As a result, P n m is the moduli space of representations of a very simple quiver. Moreover, we realize the blow up as morphism of moduli spaces for the same quiver.

Quivers and moduli of their thin sincere representations in Macaulay2

“…Therefore, the geometry of C(Q) describes the birational geometry of these compactifications. Particular examples of algebraic varieties constructed via our package include the blow up of P n along a linear subspace, see [Qin18], and toric compactifications of the moduli space of n labeled points in P 1 , see [BH20].…”

Moduli of quiver representations for exceptional collections on surfaces

“…The curious readers are referred to the extended version of the present paper [20]. The techniques also apply to some varieties in higher dimension, see [19].…”