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Toric systems and mirror symmetry. (English) Zbl 1396.14045
A (smooth, projective) toric surface \(X\) is a compactification of the torus \((\mathbb C^*)^2\) by torus invariant boundary divisors \(D_1,\ldots,D_m\). Actually, \(X\) is completely determined by the tuple of self-intersection numbers \(D_i^2 = \chi(\mathcal O(D_i))-2\). L. Hille and M. Perling in [Compos. Math. 147, No. 4, 1230–1280 (2011; Zbl 1237.14043)] showed that for any full exceptional sequence of line bundles \((\mathcal L_1,\ldots,\mathcal L_m)\) on a rational surface \(Y\), there is a toric surface \(X\) whose boundary divisors have self-intersection \(\chi(\mathcal L_{i+1} \otimes \mathcal L_i^{-1})-2\), where \(\mathcal L_{m+1} := \mathcal L_1 \otimes \omega_Y\).
Let \(Y\) be a weak toric del Pezzo surface, i.e. \(-K_Y\) is nef (there are \(16\) of those). On \(Y\), full cyclic strongly exceptional sequences of line bundles exist, and the toric surface \(X\) associated to such a sequence is again weak del Pezzo; see [loc. cit.]
This article gives an explanation of this last observation using dimer models. For a background on dimer models see [Mem. Am. Math. Soc. 1011, iii–vii, 86 p. (2012; Zbl 1237.14002)] by N. Broomhead. To a full cyclic strongly exceptional sequence of line bundles on \(Y\), the author constructs a dimer model \(Q\). The dual dimer model \(\check Q\) in turn, determines the weak del Pezzo \(X\).
There is a mirror symmetric aspect attached to this. To a dimer model \(Q\), one can associate a category of matrix factorisations \(\mathrm{Hmf}(Q)\) and a wrapped Fukaya category \(\mathrm{fuk}(Q)\). By R. Bocklandt (with an appendix by M. Abouzaid) [“Noncommutative mirror symmetry for punctured surfaces”, Preprint, arXiv:1111.3392], \(\mathrm{Hmf}(Q)\) and \(\mathrm{fuk}(\check Q)\) are equivalent as \(A_\infty\)-categories.

14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14J33 Mirror symmetry (algebro-geometric aspects)
14J45 Fano varieties
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
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