Complex 3-folds as degeneracy loci of 3 x 3 matrices

Gavin Brown (Warwick)

Monday 9th October, 2017 16:00-17:00 Maths 311 B

Abstract

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.

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 P^2 x P^2 in P^8 (so lie in
codimension 4) and show how they fit in to the picture we know so far.

(This is joint with Al Kasprzyk, Imran Qureshi and Enrico Fatighenti.)

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