Nathan Ilten (FU Berlin)

2008/11/07

at FU Berlin

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.

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.