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Let $X$ be a smooth complex projective variety. In 2002, Bridgeland defined a notion of stability for the objects in $D^b(X)$, the bounded derived category of coherent sheaves on $X$, which generalized 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 ${rm Stab}(X)$, the stability manifold of $X$. We then construct the moduli space $M_{sigma}(mathcal{O}_X)$ of $sigma$-semistable objects of class $[mathcal{O}_X]$ in $K_0(X)$ after wall-crossing.
In this short note, we describe a problem in algebraic geometry where the solution involves Catalan numbers. More specifically, we consider the derived category of coherent sheaves on an elliptic surface, and the action of its autoequivalence group o
Shimura curves on Shimura surfaces have been a candidate for counterexamples to the bounded negativity conjecture. We prove that they do not serve this purpose: there are only finitely many whose self-intersection number lies below a given bound. P
Inspired by mirror symmetry, we investigate some differential geometric aspects of the space of Bridgeland stability conditions on a Calabi-Yau triangulated category. The aim is to develop theory of Weil-Petersson geometry on the stringy Kahler modul
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.
We give further counterexamples to the conjectural construction of Bridgeland stability on threefolds due to Bayer, Macr`i, and Toda. This includes smooth projective threefolds containing a divisor that contracts to a point, and Weierstra{ss} ellipti