  From a block-diagonal $(n+1) \times (m+1) \times (m+1)$ 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 $\mathfrak{C}$ in $m$-dimensional projective space.
Under mild technical assumptions, we characterize the accidental singularities
of $X$ in terms of $\mathfrak{C}$. 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.
