Stability, Hilbert scheme and PT moduli of genus four curves and failure of the MMP/Wall-crossing correspondence
Inspired by concepts in string theory, the notion of stability conditions on triangulated categories was introduced by Bridgeland in 2002. Its impact across mathematics includes the solutions of classical problems in algebraic geometry, which were hard to tackle directly. This concept leads to a wall-crossing machinery: there is a manifold of stability conditions, with a wall-and-chamber decomposition, such that the moduli space of stable objects only changes as we cross a wall. This has many geometrical applications. In the first part, we show that wall-crossing transformations can be more involved than was previously known, by proving the existence of a wall-crossing with unexpected behaviour. In particular, it fails an expected correspondence between wall-crossing and birational transformations. This significantly complicates the overall picture in this fundamentally important correspondence to applications of stability conditions to algebraic geometry. In the second part, we apply the machinery to answer some basic questions about the classical Hilbert scheme of canonical genus four curves in P 3 via an effective control over its wall-crossing. The strategy uses the space of PT-stable pairs as an intermediate step.