Abstract
The construction introduced by Gross, Hacking and Keel 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 that paper they proved two convergence results. Firstly that if the intersection matrix of $D$ is not negative semi-definite then the family they construct lifts to an algebraic family. Secondly they prove that if the intersection matrix is negative definite then their construction lifts along certain analytic strata on the base, and then over a formal neighbourhood of this. In the original version of that paper they claimed that if the intersection matrix were negative semi-definite then family in fact extends over an analytic neighbourhood of the origin but gave an incorrect proof. In this paper we correct this error. We explain how the general Gross-Siebert program can be used to reduce construction of the mirror to such a surface to calculating certain relative Gromov-Witten invariants. We then relate these invariants to the invariants of a new space where we can find explicit formulae for the invariants. From this we deduce analytic convergence of the mirror family, at least when the original surface has an $I_4$ fibre.
Highlights
The geometry of surfaces often is a rich playground in which to test new conjectures and constructions
In [14] the authors introduced a construction of a mirror family to certain pairs (S, D), called Looijenga pairs
This built upon work of [4] who had made predictions of the mirror families to various del Pezzo surfaces
Summary
The geometry of surfaces often is a rich playground in which to test new conjectures and constructions. In the second case one can perform a birational modification of the family to produce a type II degeneration where the central fibre is the union of only two components In this case the class S1 ∩ S2 is an anti-canonical curve on each component. We explore how the combinatorial nature of these objects limits the data we must calculate In our case this reduces the problem to calculating the number of rational curves meeting the fixed fibre F with tangency orders one and two. We extend their results to our case, where we are interested in relative invariants rather than absolute invariants With this warm up done we move to the main challenge of this paper, counting the curves tangent to the boundary at a single point with tangency order two.
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