= Nilpotent cone for conjugation of matrices
{title2=$\mathcal N=\{A:A^n=0\}$}
For conjugation of $n$ by $n$ complex matrices by the <special linear group>, the <affine null cone> consists exactly of the <nilpotent matrices>. If a conjugate tends to zero along an <algebraic one-parameter subgroup>, its constant <characteristic polynomial> must be $T^n$. Conversely, put a nilpotent matrix in strictly upper triangular form and conjugate it by $\operatorname{diag}(t^{n-1},t^{n-3},\ldots,t^{1-n})$. Every active matrix entry has positive weight and tends to zero. The <Hilbert-Mumford criterion for projective stability> therefore identifies the semistable locus of the projective conjugation representation as the complement of the projectivized nilpotent cone.
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