Determinantal Varieties

Let {M=\mathbb{P}^{nm-1}} be the projectivization of {\mathrm{Mat}_{n\times m}(\mathbb{C})}. Then for all {k}, we have a variety {M_k} given as the set of matrices of rank at most {k}, which is given by the vanishing of the determinents of {(k+1)\times (k+1)} minors. We call these the generic determinantal varieties.

We can actually compute what it is for {k=1}. An {m\times n} matrix {Z} has rank 1 if and only if we can write it as {Z=U^tV} where {U=(u_1,\ldots,u_m)} and {W=(w_1,\ldots,w_n)} are vectors. Working out the equations then gives us {z_{i,j}z_{k,\ell}=z_{i,\ell}z_{k,j}}, which should be familiar, this is the image of the Segre map on {\mathbb{P}^{n-1}\times \mathbb{P}^{m-1}}.

The story of determinantal varieties seems to be the story of replacing the entries of a matrix of variables with polynomials. The next thing to try is to let {\Omega=(L_{i,j})} be a matrix of linear forms on {\mathbb{P}^\ell} which don’t all vanish simultaneously. Now, we set {\Sigma_k(\Omega)} to be the set where {\Omega(z)} has rank at most {k}. We call this a linear determinantal variety. One way to interpret this more geometrically, and using the earlier definitions, is that {\Omega} gives a map {\mathbb{P}^\ell\rightarrow M}, and then the linear determinantal varieties are the pullbacks along {\Omega} of the {M_k}. This suggests, to me, that the name “generic determinantal variety” isn’t quite right (though it’s what Harris uses in his “Algebraic Geometry: A First Course”), I think that perhaps universal is a better modifier, as at least when all the entries are of the same degree, we can get the varieties in this way.

Rational normal curves are of the above type, though. Given {\Omega=\left(\begin{array}{cccc}z_0 & z_1 & \ldots & z_{n-1}\\ z_1 & z_2 & \ldots & z_n\end{array}\right)} then {\Sigma_1(\Omega)} is the rational normal curve of degree {n}.

However, not all determinantal varieties are made from matrices where the degrees are constant. In fact, if {\Omega=(F_{i,j})} is a matrix of forms on {X\subset \mathbb{P}^k}, then we get a determinantal variety so long as {\deg F_{i,j}=d_{i,j}} can be written as {e_i-f_j} for some vectors {(e_1,\ldots,e_m)} and {(f_1,\ldots,f_n)}. Then the minors will all be homogeneous, and so their zero locus {Y\subseteq X} is a determinantal subvariety of {X}.

So, now let’s look at one type of variety that ISN’T a linear determinantal variety, so that this concept it actually useful: the generic surface of degree d. These are given by the determinants of d\times d matrices of linear forms that don’t identically vanish. Call the space of such matrices U_d. Then we can left or right multiply by scalar matrices and it doesn’t change the determinant, except a scalar. So we can quotient and get D_d=U_d/PGL_d\times PGL_d, which has dimension 4d^2-1-2(d^2-1)=2d^2+1.

However, the dimension of the space of degree d surfaces is \binom{d+3}{3}-1.  So then for d\geq 4 already, the generic surface is not determinantal! Perhaps more interestingly, cubic surfaces ARE, generically.  There’s a lot more to say, but honestly, I only have bits and pieces.

About Charles Siegel

Charles Siegel is currently a postdoc at Kavli IPMU in Japan. He works on the geometry of the moduli space of curves.
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