A note on O6 intersections in AdS flux vacua

The DGKT-CFI construction of AdS flux vacua in type IIA string theory has interesting features such as classical moduli stabilization and a parametric scale separation between the Hubble scale and the Kaluza-Klein scale. A possible worry regarding the consistency of these vacua is that pathologies could arise due to intersections of the O6-planes, which are not well understood in the 10D solution. In this note, we show in explicit examples that such intersections are absent if one compactifies on smooth Calabi-Yau manifolds instead of toroidal orbifolds. In particular, we show that the blow-up of the $T^6/\mathbb{Z}_3$ orbifold yields a single O6-plane which wraps a smooth submanifold without any (self-)intersections. On the other hand, blowing up the $T^6/\mathbb{Z}_3^2$ orbifold seems to yield an O6-plane which self-intersects. However, we show that this is due to an inconsistent orientifold involution in the original DGKT model. Imposing a consistent involution again yields a single smooth O6-plane without self-intersections on the blown-up manifold.