Birational mirrors to pairs of Laurent polynomials

Mirror symmetry gives a correspondence between Fano varieties and certain Laurent polynomials, translating the classification of Fano varieties up to deformation into a combinatorial problem. To understand the classification using this correspondence, we ask the question: ‘Which pairs of Laurent polynomials (f,g) are mirrors to pairs of Fano varieties that are related by a blow-up X_g --> X_f?’ The combinatorial criteria on (f,g) we are looking for will generalise the relationship between the fans of two toric varieties related by a toric blow-up.

I will present a new method of constructing Fano mirrors to Laurent polynomials, which uses mirror constructions from the Gross-Siebert program. This will include the construction of a birational morphism between the mirrors to a pair of Laurent polynomials in two variables that satisfies certain combinatorial conditions.