In this paper, we study the birational geometry of the Hilbert scheme P 2[n] of n-points on P 2 . We discuss the stable base locus decomposition of the effective cone and the corresponding birational models. We give modular interpretations to the models in terms of moduli spaces of Bridgeland semi-stable objects. We construct these moduli spaces as moduli spaces of quiver representations using G.I.T. and thus show that they are projective. There is a precise correspondence between wall-crossings in the Bridgeland stability manifold and wall-crossings between Mori cones. For n ≤ 9, we explicitly determine the walls in both interpretations and describe the corresponding flips and divisorial contractions.

The Bounded Negativity Conjecture predicts that for any smooth complex surface X there exists a lower bound for the selfintersection of reduced divisors on X. This conjecture is open. It is also not known if the existence of such a lower bound is invariant in the birational equivalence class of X. In the present note we introduce certain constants H(X) which measure in effect the variance of the lower bounds in the birational equivalence class of X. We focus on rational surfaces and relate the value of H(P 2 ) to certain line arrangements. Our main result is Theorem 3.3 and the main open challenge is Problem 3.10. Problem 1.2 (Birational invariance of the BNC). Let X and Y be birationally equivalent projective surfaces. Does BNC hold for X if and only if it holds for Y ? In other words: is the bounded negativity property a birational invariant? Remark 1.3. Note, that a solution to the above problem is not known even if Y is the blow-up of X in a single point.Of course, if BNC is true in general, then the above problem has an affirmative solution. However, even in that situation, it is still of interest to know how the bounds b(X) and b(Y ) are related in terms of the complexity of a birational map between X and Y .

Let ξ be the Chern character of a stable coherent sheaf on P 2 . For every ξ, we compute the cone of effective divisors on the moduli space M (ξ) of semistable sheaves on P 2 with Chern character ξ. The computation hinges on finding a good resolution of the general sheaf in M (ξ). This resolution is determined by Bridgeland stability and arises from a well-chosen Beilinson spectral sequence. The existence of a good choice of spectral sequence depends on remarkable numbertheoretic properties of the slopes of exceptional bundles.

Let ξ be the Chern character of a stable sheaf on P 2 . Assume either rk(ξ) 6 and that there are no strictly semistable sheaves with character ξ, or that rk(ξ) and c 1 (ξ) are coprime and the discriminant ∆(ξ) is sufficiently large. We use recent results of Bayer and Macrì on Bridgeland stability to compute the ample cone of the moduli space M (ξ) of Gieseker-semistable sheaves on P 2 . We recover earlier results, such as those by Strømme and Yoshioka, as special cases.

We compute the cone of effective divisors on the Hilbert scheme of n points in the projective plane. We show the sections of many stable vector bundles satisfy a natural interpolation condition, and that these bundles always give rise to the edge of the effective cone of the Hilbert scheme. To do this, we give a generalization of Gaeta's theorem on the resolution of the ideal sheaf of a general collection of n points in the plane. This resolution has a natural interpretation in terms of Bridgeland stability, and we observe that ideal sheaves of collections of points are destabilized by exceptional bundles. By studying the Bridgeland stability of exceptional bundles, we also show that our computation of the effective cone of the Hilbert scheme is consistent with a conjecture in [ABCH] which predicts a correspondence between Mori and Bridgeland walls for the Hilbert scheme.