Moduli of ℚ-Gorenstein pairs and applications

We develop a framework to construct moduli spaces of Q {\mathbb {Q}} -Gorenstein pairs. To do so, we fix certain invariants; these choices are encoded in the notion of Q {\mathbb {Q}} -stable pair. We show that these choices give a proper moduli space with projective coarse moduli space and they prevent some pathologies of the moduli space of stable pairs when the coefficients are smaller than 1 2 \frac {1}{2} . Lastly, we apply this machinery to provide an alternative proof of the projectivity of the moduli space of stable pairs.