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.
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Buckley, A., Košir, T. Determinantal representations of smooth cubic surfaces. Geom Dedicata 125, 115–140 (2007). https://doi.org/10.1007/s10711-007-9144-x
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DOI: https://doi.org/10.1007/s10711-007-9144-x
Keywords
- Smooth cubic surface
- Determinantal representations
- Schlaefli’s double-six
- Linear systems of twisted cubic curves
- Real smooth cubic surfaces
- Self-adjoint and definite determinantal representations