Moduli of quiver representations for exceptional collections on surfaces

Abstract

Suppose S is a smooth projective surface over an algebraically closed field k, L = {L 1 , . . . , Ln} is a full strong exceptional collection of line bundles on S. Let Q be the quiver associated to this collection. One might hope that S is 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 show that this is the case for del Pezzo surfaces. Furthermore, we show the blow-up at a point can be recovered from an augmentation of exceptional collections (in the sense of L. Hille and M.Perling) via morphism between moduli of quiver representations.

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

“…It is natural to wonder when T is a morphism and the relation between M θ and X for various θ. [QZ18] provided some answers for this problem for rational surfaces. On the other hand, one can also consider the (n + 1)-Kronecker quiver, which can be thought of as the quiver associated to the (not full) strong exceptional collection {O P n , O P n (1)} on P n .…”

“…The proof of the theorem follows the idea in [QZ18] by thinking of {O P n m , O P n m (H − E), O P n m (H)} as an 'augmentation' of {O P n , O P n (1)} on P n . However, many arguments were simplified because the quiver in the present case is nicer and has no relations.…”