We study birational geometry of the moduli space of stable sheaves on a quadric surface with Hilbert polynomial 5m + 1 and c1 = (2, 3). We describe a birational map between the moduli space and a projective bundle over a Grassmannian as a composition of smooth blow-ups/downs.

We give further counterexamples to the conjectural construction of Bridgeland stability on threefolds due to Bayer, Macrì, and Toda. This includes smooth projective threefolds containing a divisor that contracts to a point, and Weierstraß elliptic Calabi-Yau threefolds. Furthermore, we show that if the original conjecture, or a minor modification of it, holds on a smooth projective threefold, then the space of stability conditions is non-empty on the blow up at an arbitrary point. More precisely, there are stability conditions on the blow up for which all skyscraper sheaves are semistable.

We prove that the "Thaddeus flips" of L-twisted sheaves appearing in [MW97] can be obtained via Bridgeland wall-crossing. Similarly, we realize the change of polarization for moduli spaces of 1-dimensional Gieseker semistable sheaves on a surface by varying a family of stability conditions. 1 arXiv:1505.07091v1 [math.AG]

Let v d (P 2 ) ⊂ |O P 2 (d)| denote the d-uple Veronese surface. After studying some general aspects of the wall-crossing phenomena for stability conditions on surfaces, we are able to describe a sequence of flips of the secant varieties of v d (P 2 ) by embedding the blow-up bl v d (P 2 ) |O P 2 (d)| into a suitable moduli space of Bridgeland semistable objects on P 2 . cohomological strata and the Bridgeland walls. It was shown in [12] that for the case of N (6, 1), a cohomological strata may be object of several contractions when running the MMP, giving rise to several Bridgeland walls.Nevertheless, when χ = 0 we can identify all rank-1 walls even when Maican-type stratifications are unknown. In this case, by restricting the Bridgeland wall-crossing on a suitable subvariety of a model of N (d, 0) (d odd), and following the spirit of [1], we construct a sequence of flips for the blow-up of the linear series |O(d − 3)| along the Veronese surface, with the first of these flips coinciding with the one constructed by Vermeire in [30].Theorem 33. Let d ≥ 5 be an integer and let ν d−3 : P 2 → P(H 0 (O(d − 3))) ∨ = P N be (d − 3)-uple embedding. There exists a sequence of flips

Let $X$ be a smooth projective threefold of Picard number one for which the generalized Bogomolov–Gieseker inequality holds. We characterize the limit Bridgeland semistable objects at large volume in the vertical region of the geometric stability conditions associated to $X$ in complete generality and provide examples of asymptotically semistable objects. In the case of the projective space and $\operatorname {ch}^\beta (E)=(-R,0,D,0)$, we prove that there are only a finite number of nested walls in the $(\alpha ,s)$-plane. Moreover, when $R=0$ the only semistable objects in the outermost chamber are the 1-dimensional Gieseker semistable sheaves, and when $\beta =0$ there are no semistable objects in the innermost chamber. In both cases, the only limit semistable objects of the form $E$ or $E[1]$ (where $E$ is a sheaf) that do not get destabilized until the innermost wall are precisely the (shifts of) instanton sheaves.