ABSTRACT. We study the Hilbert scheme of twisted cubics in the three-dimensional projective space by using Bridgeland stability conditions. We use wall-crossing techniques to describe its geometric structure and singularities, which reproves the classical result of Piene and Schlessinger.

We study the Hilbert scheme of twisted cubics in the three-dimensional projective space by using Bridgeland stability conditions. We use wall-crossing techniques to describe its geometric structure and singularities, which reproves the classical result of Piene and Schlessinger.

Let X be a smooth complex projective variety. In 2002, Bridgeland [6] defined a notion of stability for the objects in 𝔇 b (X), the bounded derived category of coherent sheaves on X, which generalised the notion of slope stability for vector bundles on curves. There are many nice connections between stability conditions on X and the geometry of the variety. We construct new stability conditions for surfaces containing a curve C whose self-intersection is negative. We show that these stability conditions lie on a wall of the geometric chamber of Stab(X), the stability manifold of X.We then construct the moduli space Mσ (ℴ X ) of σ-semistable objects of class [ℴ X ] in K 0(X) after wall-crossing.

We identify limit stable pairs and stable framed sheaves as epimorphisms and monomorphisms, respectively, in tilts of the standard heart, under suitable conditions. We then identify the moduli spaces with the corresponding Quot spaces, obtaining the projectivity of the Quot spaces in these cases. We also prove a formula in a motivic Hall algebra relating the Quot spaces under a tilt.