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 $\GL_3\times \GL_3$ 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 $\left(\genfrac{}{}{0ex}{}{a_1...a_6}{b_1...b_6}\right)$. 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.

24 pages, 2 figures; added motivation and historical remarks

References in corpus (1)

• Luigi Cremona and cubic surfaces