Intersections of symmetric determinantal varieties and theta characteristics of canonical curves

From a block-diagonal (n+1)×(m+1)×(m+1) tensor symmetric in the last two entries one obtains two varieties: an intersection of symmetric determinantal hypersurfaces X in n-dimensional projective space, and an intersection of quadrics C in m-dimensional projective space. Under mild technical assumptions, we characterize the accidental singularities of X in terms of C. We apply our methods to algebraic curves and show how to construct theta characteristics of certain canonical curves of genera 3, 4, and 5, generalizing a classical construction of Cayley.