Categorical cones and quadratic homological projective duality

We introduce the notion of a categorical cone, which provides a categorification of the classical cone over a projective variety, and use our work on categorical joins to describe its behavior under homological projective duality. In particular, our construction provides well-behaved categorical resolutions of singular quadrics, which we use to obtain an explicit quadratic version of the main theorem of homological projective duality. As applications, we prove the duality conjecture for Gushel-Mukai varieties, and produce interesting examples of conifold transitions between noncommutative and honest Calabi-Yau threefolds.