Maximally mutable Laurent polynomials.

Area of Science:

• Algebraic Geometry
• Mirror Symmetry
• Combinatorial Geometry

Background:

• Mirror symmetry posits dual pairs of Calabi-Yau manifolds.
• Fano varieties are key objects in algebraic geometry with rich structures.
• Developing a combinatorial approach to mirror symmetry for Fano varieties is an ongoing challenge.

Purpose of the Study:

• Introduce maximally mutable Laurent polynomials (MMLPs) as candidates for mirrors to Fano varieties.
• Classify rigid MMLPs in two and three variables.
• Establish a concrete dictionary between MMLPs and Fano varieties.

Main Methods:

• Definition and analysis of maximally mutable Laurent polynomials (MMLPs).
• Combinatorial classification of rigid MMLPs based on mutation classes.
• Computer-assisted enumeration for MMLPs in three variables with reflexive Newton polytopes.

Main Results:

• Identified 10 mutation classes of rigid MMLPs in two variables, corresponding to 10 deformation classes of smooth del Pezzo surfaces.
• Classified rigid MMLPs in three variables with reflexive Newton polytopes, corresponding to 98 deformation classes of 3D Fano manifolds.
• Demonstrated that all known mirrors to Fano manifolds are rigid MMLPs.

Conclusions:

• Maximally mutable Laurent polynomials provide a promising combinatorial framework for mirror symmetry to Fano varieties.
• The correspondence holds for low-dimensional Fano varieties and suggests a general principle for higher dimensions.
• This work offers a new perspective on constructing and understanding mirrors to Fano manifolds.