Togliatti Quintic

A classical question in algebraic geometry is how many cone-singularities a surface of a given degree can have. In 1940 Togliatti proved that for degree 5 surfaces 31 cone-singularities are possible. It took 40 years until Beauville could prove that this is indeed the maximal number. The movie shows a quintic with 31 cone-singularities.

This movie was made using surfex. A higher resolution version can be found at

http://www.iag.uni-hannover.de/~bothmer/Goettingen/togliattiBig.gif

Category:

Howto & Style

Tags:

• geometry
• algebraic
• mathematics
• togliatti
• quintic
• singularity
• bothmer
• surfex
• cg

[kalduglun]

i meant the vertices of the cones and the points where those "hats" meet pairwise ?(i forgot these last in my previous message) And can all these 31 points be located arbitrarily? It does not seem to me, the picture looks so symmetric.

[bothmer]

The 31 points can not be located arbitraily. If you move one, you have to move the others too. Still there is a 9-dimensional Family of Togliatti-Quintics, not all of them as symmetric as the one depicted here. (This equation for this beautyful symmetric Togliatti-Quintic was found by Barth in the 1990's)

[kalduglun]

Hello Hans-Christian, it is difficult to ask this question without pointing details on the picture (for true), but: are all these singularities located at the "summits" of the hats? in other words the vertices of the cones? Or are there some left "behind" the "dress"?

[bothmer]

The 31 singularities are the points where two vertices meet. There are 6 on the cones that stick out and 25 where the cones meet the "hat" and below them where we hat 5 teraedons. As you say, its difficult to explain without pointing...

[Meb8Rappa]

@Jhammo52 You just saw a rotating Togliatti Quintic. What else?

[dappledjoy]

what?

[Jhammo52]

Can somebody please tell me what I just saw?

[ambiamusic]

...but what Togliatti didn't know is that he'd actually designed a hat for a jester.