What is... a deformation?

This page host information on Nathan Ilten's talk "What is... a deformation?" at the WhatIsSeminar.

Video

A video recording with integrated Screenshots is available at http://www.scivee.tv/node/8161 .

Abstract

I plan to give a concise introduction to deformations of singularities. After showing some very pretty pictures, I will define what a deformation is. Additionally, I hope to make remarks concerning concepts such as flatness, induced deformations, and versality. Time permitting, I will present Pinkham's famous example of the cone over the rational normal curve of degree 4.

Pinkham's Example

I unfortunately didn't make it to Pinkham's example in the talk. Let $Y$ be the vanishing set of the 6 2x2 minors of the matrix

$A=\left(\begin{array}{c c c c}x_1& x_2& x_3& x_4\\ x_2& x_3& x_4& x_5\end{array}\right)$

for example $x_1 x_3- x_2^2$ etc. This is what is called the cone over the rational normal curve of degree four. What Pinkham did was calculate the base space of a versal deformation of this singularity; it turned out to have a three-dimensional component and a one-dimensional component. This was quite interesting because it was the first example of a singularity with multiple components in the versal base space.

The equations for the total space over the three-dimensional component are given by the 2x2 minors of the matrix

$B=\left(\begin{array}{c c c c}x_1& x_2& x_3& x_4\\ x_2+t_2& x_3+t_3& x_4+t_4& x_5\end{array}\right)$

whereas the equations for the total space over the one-dimensional component are given by the 2x2 minors of the matrix

$C=\left(\begin{array}{c c c }x_1& x_2& x_3\\ x_2&x_3+s& x_4\\ x_3& x_4& x_5\end{array}\right)$.

Notice that after setting $s=0$ in this second matrix, the 2x2 minors will give 6 polynomials which describe the same vanishing set as the original matrix $A$. The fact that deformations from the two components can't somehow be "combined" has to do with the fact that the map $\pi:X\to S$ is required to be "nice", which in this case is the condition of flatness.

Comments

Great talk! I really enjoyed it. The vague definition worried me first, but since you mostly talked about singularities, it went really well -- and pretty pictures are always nice.

Comparing it to your abstract now, it might have been good to focus either on more examples (to show the generality of the approach) or focus more on singularities (to get more applications). But I think the questions and discussions showed how good it was. Oh, and was Pinkham's name actually mentioned? Just wondering. Peter.

-- PeterKrautzberger - 10 Nov 2008
 

Topic revision: r5 - 10 Nov 2008, DE
 
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