In my research I recently stumbled upon a problem which involves trying to identify whether a given projective variety is determinantal or, even stronger, determinantal of a particular form. For example, I am interested in questions like 'can $X$ be written as a determinantal hypersurface, i.e. the zero-set of the polynomial $$\det\sum_{i=0}^n x_iA_i, $$ where the matrices $A_i$ are of rank $k$ at most'.
To do that I would like to learn as much as possible about specific properties of determinantal varieties. What are the good references for this topic? I have read the relevant chapters of Harris' 'Algebraic Geometry: A First Course', but couldn't find more in-depth, but sufficiently general, resources. The papers I have found tend to work with some very specific cases.
So what are, in your opinion, the things an algebraic geometer needs to know about determinantal varieties? I am particularly interested in their automorphism groups (and how to work out the automorphism group of a projective variety in general) and linear sections. I think I have found that linear sections of a generic determinantal hypersurface are generally non-isomorphic (except possibly some low-dimensional cases), and this seems like a very basic fact, but I haven't found it anywhere on the web and would really love some geometric intuition for why this is the case.
