Cayley-Bacharach theorems and conjectures

A theorem of Pappus of Alexandria, proved in the fourth century A.D., began a long development in algebraic geometry. In its changing expressions one can see reflected the changing concerns of the field, from synthetic geometry to projective plane curves to Riemann surfaces to the modern development of schemes and duality. We survey this development historically and use it to motivate a brief treatment of a part of duality theory. We then explain one of the modern developments arising from it, a series of conjectures about the linear conditions imposed by a set of points in projective space on the forms that vanish on them. We give a proof of the conjectures in a new special case.