Let be a smooth complex projective variety. In 2002, [Bri07] defined a notion of stability for the objects in  ( ), the bounded derived category of coherent sheaves on , which generalized the notion of slope stability for vector bundles on curves. There are many nice connections between stability conditions on and the geometry of the variety.We construct new stability conditions for surfaces containing a curve whose self-intersection is negative. We show that these stability conditions lie on a wall of the geometric chamber of Stab( ), the stability manifold of . We then construct the moduli space ( ) of -semistable objects of class [ ] in 0 ( ) after wall-crossing.

We bound the genus of a projective curve lying on a complete intersection surface in terms of its degree and the degrees of the defining equations of the surface on which it lies.

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.

Abstract. We use a polyhedral criterion for the existence of diagonal splittings to investigate which toric varieties X are diagonally split. Our results are stated in terms of the vector configuration given by primitive generators of the 1-dimensional cones in the fan defining X. We show, in particular, that X is diagonally split at all q if and only if this configuration is unimodular, and X is not diagonally split at any q if this configuration is not 2-regular. We also study implications for the possibilities for the set of q at which a toric variety X is diagonally split.

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