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Mirror symmetry for del Pezzo surfaces was studied in~\cite{MirrorSymmetryForDelPezzoSurfacesVanishingCyclesAndCoherentSheaves} where they suggested that the mirror should take the form of a Landau-Ginzburg model with a particular type of elliptic fibration. This argument came from symplectic considerations of the derived categories involved. This problem was then considered again but from an algebro-geometric perspective by Gross, Hacking and Keel in~\cite{MirrorSymmetryforLogCY1}. 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 where the self intersection matrix for $D$ is not negative semi-definite it was shown in~\cite{MirrorSymmetryforLogCY1} 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 present the mirror to $dP_2$ as a double cover of $\PP^2$.
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참고문헌 (16)
참고문헌 신청
N. H. Argz / 2016 / Log geometric techniques for open invariants in mirror symmetry
D. Auroux / 2007 / Mirror symmetry and T-duality in the complement of an anticanonical divisor / J. Gokova Geom. Topol. GGT 1:51~91
Denis Auroux / 2006 / Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves / Inventiones mathematicae 166(3):537~582
M. Carl / 2018 / A tropical view of Landau-Ginzburg models
Man Wai Cheung / 2017 / The greedy basis equals the theta basis: A rank two haiku / Journal of Combinatorial Theory, Series A 145:150~171
D. Auroux / 2007 / Mirror symmetry and T-duality in the complement of an anticanonical divisor / J. Gokova Geom. Topol. GGT 1:51~91
Denis Auroux / 2006 / Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves / Inventiones mathematicae 166(3):537~582
M. Carl / 2018 / A tropical view of Landau-Ginzburg models
Man Wai Cheung / 2017 / The greedy basis equals the theta basis: A rank two haiku / Journal of Combinatorial Theory, Series A 145:150~171
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