Determinantal representations of smooth cubic surfaces
| dc.creator | Buckley, Anita | |
| dc.creator | Košir, Tomaž | |
| dc.date | 2006-06-05 | |
| dc.date | 2007-03-22 | |
| dc.date.accessioned | 2026-07-07T07:53:04Z | |
| dc.date.available | 2026-07-07T07:53:04Z | |
| dc.description | 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 $a_1... a_6 \choose b_1... b_6$. 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. | |
| dc.description | 24 pages, 2 figures; added motivation and historical remarks | |
| dc.identifier | https://arxiv.org/abs/math/0606098 | |
| dc.identifier | http://arxiv.org/abs/math/0606098 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/126111 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 14M12, 14J60, 15P05 | |
| dc.title | Determinantal representations of smooth cubic surfaces | |
| dc.type | text |