Determinantal representations of smooth cubic surfaces

2006/06/05 by Buckley, Anita, Košir, Tomaž
#14J60 #14M12 #15P05 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.math/0606098

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 (a₁,...,a₆) then its transpose M^t corresponds to lines (b₁,...,b₆) which together form a Schläfli's double-six a₁... a₆ \choose b₁... b₆. 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 F₅ has 24 nonequivalent self-adjoint determinantal representations, 16 of which are definite.

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