New constructions of Fano 3-folds from mirror symmetry

Liana Heuberger, Loughborough University. Part of the algebra seminar series.

Mirror symmetry conjecturally associates to a Fano orbifold a (very special type of) Laurent polynomial.  Laurent inversion is a method for reversing this process, obtaining a Fano variety from a candidate Laurent polynomial.  We apply this to construct previously unknown Fano 3-folds with terminal Gorenstein quotient singularities.  

A Laurent polynomial f determines, through its Newton polytope P, a toric variety X_P, which is in general highly singular. Laurent inversion constructs, from f and some auxiliary data, an embedding of X_P into an ambient toric variety F.  In many cases this embeds X_P as a complete intersection of line bundles on F, and the general section of these line bundles is the Q-Fano 3-fold that we seek.

This is joint work with Tom Coates, Al Kasprzyk and Giuseppe Pitton.