### References & Citations

# Mathematics > Algebraic Geometry

# Title: Equations of mirrors to log Calabi--Yau pairs via the heart of canonical wall structures

(Submitted on 17 Sep 2021)

Abstract: Gross and Siebert developed a program to construct a mirror family to a log Calabi--Yau pair $(X,D)$, consisting of a smooth projective variety $X$ with a normal-crossing anti-canonical divisor $D\subset X$, in arbitrary dimension. This construction uses a canonical wall structure, encoding tropical analogues of all $\mathbb{A}^1$-curves in $(X,D)$, which are rational stable maps to $(X,D)$, whose images touch $D$ at a single point. In this paper, we provide tools to practically compute explicit equations of the mirror family in the case when $X$ is obtained as a blow-up of a toric variety along hypersurfaces in its toric boundary, and $D$ is the strict transform of the toric boundary. Particularly, we introduce the "heart of the canonical wall structure" associated to such pairs $(X,D)$, which is constructed purely combinatorially following our previous work with Mark Gross, providing a combinatorial algorithm for computing $\mathbb{A}^1$-curves. As an application, using the heart of the canonical wall structure we compute mirror families to blow-ups of $\mathbb{P}^3$ along disjoint unions of hypersurfaces. This provides the first nontrivial concrete examples of equations of mirror families to log Calabi--Yau pairs in dimensions bigger than two.

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