Real quartic surfaces containing 16 skew lines
It is well known that there is an open three-dimensional subvariety Ms of the Grassmannian of lines in ℙ3 which parametrizes smooth irreducible complex surfaces of degree 4 which are Heisenberg invariant, and each quartic contains 32 lines but only 16 skew lines, being determined by its configuratio...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2004-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171204308112 |
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| Summary: | It is well known that there is an open three-dimensional subvariety Ms of the Grassmannian of lines in ℙ3 which parametrizes smooth irreducible complex surfaces of degree 4 which are Heisenberg invariant, and each quartic contains 32 lines but only 16 skew lines, being determined by its configuration of lines, are called a double 16. We consider here the problem of visualizing in a computer the real Heisenberg invariant quartic surface and the real double 16. We construct a family of points l∈Ms parametrized by a two-dimensional semialgebraic variety such that under a change of coordinates of l into its Plüecker, coordinates transform into the real coordinates for a line L in ℙ3, which is then used to construct a program in Maple 7. The program allows us to draw the quartic surface and the set of transversal lines to L. Additionally, we include a table of a group of examples. For each test example we specify a parameter, the viewing angle of the image, compilation time, and other visual properties of the real surface and its real double 16. We include at the end of the paper an example showing the surface containing the double 16. |
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| ISSN: | 0161-1712 1687-0425 |