COW seminar

The organisers are Miles Reid at Warwick, and Gregory Sankaran at Bath.

Next meeting

The talks will be in room 1004 Blackett, on the 10th floor of the Blackett lab (physics department). This is attached to mathematics but is best reached from Prince Consort Road: see the map.

Tea after the first lecture will be in the 8th floor common room, directly under 1004, two floors down.

Social programme Usually a pub in South Kensington. The standard one is the The Queen's Arms, 30 Queen's Gate Mews, London SW7 5QL.

Abstracts

• Tyurin: I present a certain generalization of toric structures called pseudotoric structure. A pseudotoric structure on a symplectic manifold defines a Lagrangian fibrations with Lagrangian tori as smooth generic fibers. The singular fibers occurring in this construction have mild singularities, allowing us to extend notions such as Bohr-Sommerfeld cycles etc. to them. This allows us to exploit pseudotoric structures in Geometric Quantization and Mirror Symmetry problems, provided that the methods applicable in the toric case remain useable in this wider setting. I show that each smooth toric variety admits a pseudotoric structure, constructing exotic Chekanov tori in the projective plane and toric del Pezzo surfaces. On the other hand, there are nontoric varieties that are nevertheless pseudotoric. As an example I describe special Lagrangian fibration (in the sense of Dennis Auroux) on the 3-fold flag variety F₃ of P².
• Reid: By analogy with the famous theorem of Buchsbaum and Eisenbud in codimension 3, I construct varieties V_k of matrices, and prove that Gorenstein codimension 4 ideals in a graded polynomial ring S with (k+1) x 2k resolution are parametrised by an open subscheme of the morphism space Mor(Spec S,V_k). The idea is that the first syzygy module P₂ = 2kS has a nondegenerate symmetric quadratic form given by the Buchsbaum-Eisenbud symmetriser. V_k is the Spin-Hom variety of linear maps from a (k+1) dimensional vector space to an isotropic subspace of this quadratic form; it is an almost homogeneous space under GL(k+1) x Spin(2k), and has Cramer-spinor coordinates. This gives a satisfactory generalisation of some aspects of the Buchsbaum-Eisenbud theorem, although not yet in a form suitable for convenient application.