If we fill up a 3 x 3 matrix with polynomials in variables of some ambient space, then the locus where the matrix drops rank (from 3 to 2, or even down to 1 as I do here) defines some locus. Remarkably, this simple-minded trick seems to have a place in the classification of complex 3-folds.
The classfication of complex (projective) curves and surfaces is ancient (1850s, 1905, 1960s). In 3 dimensions we can say an enormous amount, but nothing like an explicit classification yet. The general shape of classification is familiar: just as with compact real orientable surfaces there are three broad classes: Fano 3-folds, Calabi-Yau 3-folds and 3-folds of general type (the analogues of positively curved spheres, flat tori and hyperbolic g-holed tori for g at least 2).
Fano 3-folds can be embedded in weighted projective space (the orbifold quotient of usual projective space by a finite abelian group), and we can get a concrete grip on them by writing down the equations in these embeddings. We know the Hilbert series of all possible such embeddings (including, sadly, many that don’t exist - if only we knew which). In low codimension (<= 3 or 4ish) we know a few hundred deformation families of Fano 3-folds that realise all the Hilbert series in those cases. But it can happen that more than one family realises a given Hilbert series. I will describe some families of Fano 3-folds whose equations look like those of the Segre embedding of P2 x P2 in P8 (so lie in codimension 4) and show how they fit in to the picture we know so far.
To put it in words: these varieties are the loci where general 3 x 3 matrices drop rank, and much of the birational geometry I would like to explain can be described in terms of hands-on linear algebra. The key is to calculate their Euler characteristic, and the matrix plays a role in that too, as I will explain. Thus the title could have been: New Fano 3-folds in P2 x P2 format.
(This is joint with Al Kasprzyk, Imran Qureshi and Enrico Fatighenti.)