Linear symmetric determinantal hypersurfaces (Q874373)

The paper under review deals with linear symmetric determinantal hypersurfaces in \(\mathbb{P}_n(\mathbb{C})\), i.e. hypersurfaces whose equation can be expressed as the determinant of a matrix whose entries are linear forms. This is a classical problem in algebraic geometry, many authors worked and work on this subject. By a result of \textit{A. Beauville} [Mich. Math. J. 48, 39--64 (2000; Zbl 1076.14534)] any plane curve has a linear symmetric determinantal representation, but every linear symmetric determinantal surface is singular. In the paper the author studies which combinations of singularities can occur on a linear symmetric determinantal cubic or quartic surface. For the cubics he finds all their linear symmetric determinantal representation and in the case of isolated singularities their combination are given by certain subgraphs of \(\tilde{E}_6\). The description in the case of quartics is similar but more complicated.