Determinantal representations of smooth cubic surfaces

Abstract

For every smooth (irreducible) cubic surface S we give an explicit construction of a representative for each of the 72 equivalence classes of determinantal representations. Equivalence classes (under GL3 × GL3 action by left and right multiplication) of determinantal representations are in one to one correspondence with the sets of six mutually skew lines on S and with the 72 (two-dimensional) linear systems of twisted cubic curves on S. Moreover, if a determinantal representation M corresponds to lines (a 1,...,a 6) then its transpose M t corresponds to lines (b 1,...,b 6) which together form a Schläfli’s double-six \(a_1\ldots a_6 \choose b_1\ldots b_6\) . We also discuss the existence of self-adjoint and definite determinantal representation for smooth real cubic surfaces. The number of these representations depends on the Segre type F i . We show that a surface of type F i , i = 1,2,3,4 has exactly 2(i−1) nonequivalent self-adjoint determinantal representations none of which is definite, while a surface of type F 5 has 24 nonequivalent self-adjoint determinantal representations, 16 of which are definite.