[Paper Review] Explicit equations for mirror families to log Calabi-Yau surfaces

Lawrence Jack Barrott|arXiv (Cornell University)|Oct 19, 2018

Algebraic Geometry and Number Theory15 references6 citations

TL;DR

This paper constructs explicit algebraic equations for mirror Landau-Ginzburg models to log Calabi-Yau surfaces, specifically smooth del Pezzo surfaces of degree at least two with a cycle of rational curves as the anticanonical divisor. Using the Gross-Siebert program and tropical geometry, it lifts formal mirror families to algebraic families over a known base, confirming their correctness via Jacobian ring dimension checks and enumerative consistency.

ABSTRACT

Mirror symmetry for del Pezzo surfaces was studied by Auroux, Katzarkov and Orlov who suggested that the mirror should take the form of a Landau-Ginzburg model with a particular type of elliptic fibration. This problem was then considered again but from an algebro-geometric perspective by Gross, Hacking and Keel. Their construction allows one to construct a formal mirror family to a pair $(S,D)$ where $S$ is a smooth rational projective surface and $D$ a certain type of Weil divisor supporting an ample or anti-ample class. In the case of $S$ a Fano surface they proved that this family may be lifted to an algebraic family over an affine base. In this paper we perform this construction for all smooth del Pezzo surfaces of degree at least two and obtain explicit equations for the mirror families and explain some of the motivation for their construction. We also provide an implementation of the Kontsevich Soibelman lemma in Sage.

Motivation & Objective

• To construct explicit algebraic equations for mirror Landau-Ginzburg models of log Calabi-Yau surfaces (S,D), where S is a smooth del Pezzo surface of degree ≥2 and D is a cycle of rational curves.
• To extend the formal mirror family construction from Gross, Hacking, and Keel to an algebraic family over Spec k[NE(S)] when the self-intersection matrix of D is not negative semi-definite.
• To provide explicit equations for the mirror family in all cases of del Pezzo surfaces of degree ≥2, using the Gross-Siebert program and tropicalization.
• To verify the correctness of the constructed mirror families through enumerative checks, particularly the dimension of the Jacobian ring of the superpotential.

Proposed method

• Utilizes the Gross-Siebert program, which tropicalizes the SYZ conjecture and constructs mirrors via canonical scattering diagrams on a base B with piecewise linear functions.
• Applies the formal smoothing of the n-vertex (a cycle of planes) over Spec k[[NE(S)]) and lifts it to an algebraic family over Spec k[NE(S)] when the self-intersection matrix of D is not negative semi-definite.
• Constructs the mirror as a family of varieties defined by explicit polynomial equations in affine space, with the superpotential given as a sum of coordinates corresponding to toric divisors.
• Employs tropical curves and broken lines to model Maslov index zero disks and control gluing data in the scattering diagram.
• Uses Sage to compute the Jacobian ring of the superpotential by solving for critical points via vanishing minors of a 3×4 matrix of partial derivatives.
• Verifies mirror correctness by checking that the dimension of the Jacobian ring matches the expected rank of the quantum cohomology (12−d for degree d del Pezzo surfaces).

Experimental results

Research questions

• RQ1 Can the formal mirror family constructed by Gross, Hacking, and Keel for Looijenga pairs (S,D) be lifted to an algebraic family over Spec k[NE(S)] when the self-intersection matrix of D is not negative semi-definite?
• RQ2 What explicit algebraic equations define the mirror Landau-Ginzburg model for each smooth del Pezzo surface of degree at least two with a cycle of rational curves as the anticanonical divisor?
• RQ3 Does the dimension of the Jacobian ring of the superpotential on the mirror family match the rank of the quantum cohomology of the original del Pezzo surface?
• RQ4 How can the mirror construction be systematically verified using enumerative data such as Jacobian ring dimensions?

Key findings

• The mirror family for the degree 2 del Pezzo surface (dP₂) is explicitly constructed with defining equations in A⁴, and the Jacobian ring of the superpotential is computed to be 10-dimensional, matching the expected rank of 12−2=10.
• The construction confirms that the mirror family is algebraic over Spec k[NE(S)] for all del Pezzo surfaces of degree ≥2, under the non-negative semi-definite condition on the self-intersection matrix of D.
• For dP₂, the system of equations defining the critical locus of the superpotential yields a quotient ring of dimension 10, consistent with the quantum cohomology rank.
• The method successfully lifts formal mirrors to algebraic families, providing a concrete realization of the Gross-Siebert mirror symmetry program for log Calabi-Yau surfaces.
• The Jacobian ring computation via Sage confirms the mirror family's correctness for dP₂, supporting the broader claim that the construction yields the desired mirror families.