Given a variety embedded in projective space, its secant variety is the closure of the union of all lines secant to it. We explore the role that secant varieties can play in the construction of projective models of hyper-Kähler manifolds.

In joint work with Agostini, Giovenzana and Ríos-Ortiz, we describe the singular locus of secant varieties, using recent results of Choi, Lacini, Park and Sheridan that generalize the classical Terracini's lemma. Our description provides a tool for constructing projective models and leads to further examples, while also revealing geometric features of independent interest.