New stability conditions on surfaces and new Castelnuovo-type inequalities for curves on complete-intersection surfaces
Abstract
Let X be a smooth complex projective variety. In 2002, [Bri07] defined a notion of stability for the
objects in Db(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.
In 2012, [BMT14] gave a conjectural stability condition for threefolds. In the case that X is a
complete intersection threefold, the existence of this stability condition would imply a Castelnuovo-type
inequality for curves on X. I give a new Castelnuovo-type inequality for curves on complete
intersection surfaces of high degree. I then show how this bound would imply the bound conjectured
in [BMT14] if a weaker bound could be shown for curves of lower degree.
I also construct new stability conditions for surfaces containing a curve C whose self-intersection
is negative. I show that these stability conditions lie on a wall of the geometric chamber of Stab(X),
the stability manifold of X. I then construct the moduli space Mσ (OX) of σ-semistable objects of
class [OX] in K0(X) after wall-crossing.