# Is $\Delta[\det,m]$ smooth?

In Geometric complexity theory the following variety $\Sigma[\text{det},m]$ is crucial.

Let $X=(x_1,\ldots,x_r)$ be a tuple of $r=m^2$ variables, so that $X$ can be thought of as an $m\times m$ variable matrix, identifying $x_i$'s with the entries of $X$ in any way. By a homogeneous symbolic determinant of size $m$ over $X=(x_1,\ldots,x_r)$, we mean the determinant of a symbolic $m \times m$ matrix, whose each entry is a homogeneous linear function over $K$ of $x_1,\ldots,x_r$. Let ${\cal X}$ be the vector space over $K$ of homogeneous polynomials of degree $m$ in the variables $x_1,\ldots,x_r$, and $P({\cal X})$ the projective space associated with ${\cal X}$. Let $\Sigma[\det,m] \subseteq P({\cal X})$ be the set of all points in $P({\cal X})$ that correspond to nonzero homogeneous polynomials in ${\cal X}$ that can be expressed as homogeneous symbolic determinants of size $m$ over $X$. Then $\Delta[\det,m] \subseteq P({\cal X})$ is the Zariski-closure $\overline{\Sigma[\det,m]}$ of $\Sigma[\det,m]$. Its dimension is $\le m^4$.

My question: is $\Delta[\det,m]$ smooth?

## 1 Answer

No, it is not even normal by the article of Shrawan Kumar.