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From a block-diagonal $(n+1) \times (m+1) \times (m+1)$ tensor symmetric inthe last two entries one obtains two varieties: an intersection of symmetricdeterminantal hypersurfaces $X$ in $n$-dimensional projective space, and anintersection of quadrics $\mathfrak{C}$ in $m$-dimensional projective space.Under mild technical assumptions, we characterize the accidental singularitiesof $X$ in terms of $\mathfrak{C}$. We apply our methods to algebraic curves andshow how to construct theta characteristics of certain canonical curves ofgenera 3, 4, and 5, generalizing a classical construction of Cayley.

On intersections of symmetric determinantal varieties and theta characteristics of canonical curves