Let be the projectivization of . Then for all , we have a variety given as the set of matrices of rank at most , which is given by the vanishing of the determinents of minors. We call these the generic determinantal varieties.
We can actually compute what it is for . An matrix has rank 1 if and only if we can write it as where and are vectors. Working out the equations then gives us , which should be familiar, this is the image of the Segre map on .
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 be a matrix of linear forms on which don’t all vanish simultaneously. Now, we set to be the set where has rank at most . We call this a linear determinantal variety. One way to interpret this more geometrically, and using the earlier definitions, is that gives a map , and then the linear determinantal varieties are the pullbacks along of the . 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 then is the rational normal curve of degree .
However, not all determinantal varieties are made from matrices where the degrees are constant. In fact, if is a matrix of forms on , then we get a determinantal variety so long as can be written as for some vectors and . Then the minors will all be homogeneous, and so their zero locus is a determinantal subvariety of .
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 . These are given by the determinants of matrices of linear forms that don’t identically vanish. Call the space of such matrices . 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 , which has dimension .
However, the dimension of the space of degree surfaces is . So then for 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.