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5. Maximally Inflected Quintics
Since 9=3(5-2), a rational plane quintic has 9 ramification points, counted with
multiplicity.
The table below gives names for the different types of ramification points
possible for quartic
curves, together with the lowest order terms (germ) in a parameterization of the
ramification.
View the pictures of the germs.
| Name | Index | Partition | Germ |
| Flex | (0,1,3) | 1 |
(s,s3) |
| Cusp | (0,2,3) | 11 |
(s2,s3) |
| Planar | (0,1,4) | 2 |
(s,s3) |
| Claw | (0,2,4) | 21 |
(s2,s4) |
| Box | (0,3,4) | 22 |
(s3,s4) |
| Flat Flex | (0,1,5) | 3 |
(s3,s5) |
| Beak | (0,2,5) | 31 |
(s3,s5) |
| | (0,3,5) | 32 |
(s3,s5) |
| | (0,4,5) | 33 |
(s4,s5) |
( The Term `Box' refers to its Young diagram, a better term should be found)
In the future, we hope to display here examples of quintics with all
possibilities of ramification type.
This means drawing quite a lot of curves, and supplying (marginally)
intelligent text, which is quite a big project.
At present, the links below are to pages that we have created with pictures.
The first few files show pictures of quintics with only flexes, cusps, or
planar points. Not all of these have been completed.
- Quintics with 9 flexes.
- Quintics with 7 flexes and one cusp.
- Quintics with 7 flexes and one
planar point.
- Quintics with 5 flexes and 2 cusps.
Not yet begun
- Quintics with 5 flexes, one cusp, and one
planar point.
Not yet begun
- Quintics with 5 flexes and 2 planar
points.
Not yet begun
- Quintics with 3 flexes and 3 cusps.
Three pictures
- Quintics with 3 flexes, 2 cusps, and a
planar point.
Not yet begun
- Quintics with 3 flexes, 2 planar
points, and a cusp.
Not yet begun
- Quintics with 3 flexes and 3 planar
points.
Not yet begun
- Quintics with 1 flex.
Not yet begun
These next files have pictures of quintics with a ramification point of
multiplicity greater than 2.
Each file contains pictures of curves with at least one ramification point of
the indicated type.
- Quintics with a flat flex.
Mostly completed, but has a cool movie!
- Quintics with a claw.
Half-completed
- Quintics with a box point.
Barely begun
- Quintics with a beak.
Almost completed
- Quintics with a ramification of order at
least 5.
Previous: 4. Maximally Inflected Quartics
Next: 6. Classical Constructions
Up: Table of Contents