Bocklandt, Raf Toric systems and mirror symmetry. (English) Zbl 1396.14045 Compos. Math. 149, No. 11, 1839-1855 (2013). 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. Reviewer: Andreas Hochenegger (Köln) Cited in 2 Documents MSC: 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 14J33 Mirror symmetry (algebro-geometric aspects) 14J45 Fano varieties 14J32 Calabi-Yau manifolds (algebro-geometric aspects) Keywords:dimers; helices; del Pezzo surfaces; toric systems; homological mirror symmetry PDF BibTeX XML Cite \textit{R. Bocklandt}, Compos. Math. 149, No. 11, 1839--1855 (2013; Zbl 1396.14045) Full Text: DOI arXiv References: [1] doi:10.1016/j.jalgebra.2012.03.040 · Zbl 1263.14006 · doi:10.1016/j.jalgebra.2012.03.040 [3] doi:10.1007/s00029-009-0492-2 · Zbl 1204.14019 · doi:10.1007/s00029-009-0492-2 [4] doi:10.1093/qmath/45.4.515 · Zbl 0837.16005 · doi:10.1093/qmath/45.4.515 [5] doi:10.1088/1126-6708/2006/07/001 · doi:10.1088/1126-6708/2006/07/001 [8] doi:10.1070/SM2006v197n12ABEH003824 · Zbl 1161.14301 · doi:10.1070/SM2006v197n12ABEH003824 [10] doi:10.1016/j.aim.2009.10.001 · Zbl 1191.14008 · doi:10.1016/j.aim.2009.10.001 [12] doi:10.1088/1126-6708/2007/10/029 · doi:10.1088/1126-6708/2007/10/029 [13] doi:10.4310/ATMP.2008.v12.n3.a2 · Zbl 1144.81501 · doi:10.4310/ATMP.2008.v12.n3.a2 [14] doi:10.1016/j.jalgebra.2011.05.005 · Zbl 1250.14028 · doi:10.1016/j.jalgebra.2011.05.005 [15] doi:10.1215/S0012-7094-04-12422-4 · Zbl 1082.14009 · doi:10.1215/S0012-7094-04-12422-4 [16] doi:10.1016/j.jpaa.2007.03.009 · Zbl 1132.16017 · doi:10.1016/j.jpaa.2007.03.009 [18] doi:10.1016/j.jalgebra.2005.03.016 · Zbl 1069.14044 · doi:10.1016/j.jalgebra.2005.03.016 [19] doi:10.4007/annals.2008.167.867 · Zbl 1175.14030 · doi:10.4007/annals.2008.167.867 [20] doi:10.1017/S0017089512000080 · Zbl 1244.14042 · doi:10.1017/S0017089512000080 [21] doi:10.1007/s00222-006-0003-4 · Zbl 1110.14033 · doi:10.1007/s00222-006-0003-4 [22] doi:10.1112/S0010437X10005208 · Zbl 1237.14043 · doi:10.1112/S0010437X10005208 [27] doi:10.2140/gt.2010.14.627 · Zbl 1195.53106 · doi:10.2140/gt.2010.14.627 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.