Book contents
- Frontmatter
- Dedicaton
- Contents
- Contributors
- Happy Birthday
- On Stable Cohomology of Central Extensions of Elementary Abelian Groups
- On Projective 3-Folds of General Type with pg = 2
- 15-Nodal Quartic Surfaces. Part I: Quintic del Pezzo Surfaces and Congruences of Lines in P3
- Mori Flips, Cluster Algebras and Diptych Varieties Without Unprojection
- The Mirror of the Cubic Surface
- Semi-Orthogonal Decomposition of a Derived Category of a 3-Fold With an Ordinary Double Point
- Duality and Normalization, Variations on a Theme of Serre and Reid
- Rationality of Q-Fano Threefolds of Large Fano Index
- An Exceptional Locus in the Perfect Compactification of Ag
- Variation of Stable Birational Types of Hypersurfaces
- Triangle Varieties and Surface Decomposition of Hyper-Käahler Manifolds
- References
The Mirror of the Cubic Surface
Published online by Cambridge University Press: 25 October 2022
Book contents
- Frontmatter
- Dedicaton
- Contents
- Contributors
- Happy Birthday
- On Stable Cohomology of Central Extensions of Elementary Abelian Groups
- On Projective 3-Folds of General Type with pg = 2
- 15-Nodal Quartic Surfaces. Part I: Quintic del Pezzo Surfaces and Congruences of Lines in P3
- Mori Flips, Cluster Algebras and Diptych Varieties Without Unprojection
- The Mirror of the Cubic Surface
- Semi-Orthogonal Decomposition of a Derived Category of a 3-Fold With an Ordinary Double Point
- Duality and Normalization, Variations on a Theme of Serre and Reid
- Rationality of Q-Fano Threefolds of Large Fano Index
- An Exceptional Locus in the Perfect Compactification of Ag
- Variation of Stable Birational Types of Hypersurfaces
- Triangle Varieties and Surface Decomposition of Hyper-Käahler Manifolds
- References
Summary
A number of years ago, one of us (Mark Gross) was giving a lecture at the University of Warwick on the material on scattering diagrams from [GPS10]. Of course, Miles was in the audience, and he asked (paraphrasing as this was many years ago) whether, at some point, the lecturer would come back down to earth. The goal of this note is to show, in fact, we have not left the planet by considering a particularly beautiful example of the mirror symmetry construction of [GHK15b], namely the mirror to a cubic surface.
- Type
- Chapter
- Information
- Recent Developments in Algebraic GeometryTo Miles Reid for his 70th Birthday, pp. 150 - 182Publisher: Cambridge University PressPrint publication year: 2022
References
Abramovich, Dan and Qile Chen. Stable logarithmic maps to Deligne-Faltings pairs II. Asian J. Math., 18(3):465–488, 2014.Google Scholar
Abramovich, Dan, Qile Chen, Mark Gross and Bernd Siebert. Punctured logarithmic maps. arXiv:2009.07720, 2020.Google Scholar
Abramovich, Dan, Steffen Marcus and Jonathan Wise. Comparison theorems for Gromov-Witten invariants of smooth pairs and of degenerations. Ann. Inst. Fourier (Grenoble), 64(4):1611–1667, 2014.CrossRefGoogle Scholar
ArgUz, Hulya. Canonical scattering for log Calabi-Yau surfaces. In preparation.Google Scholar
Abramovich, Dan and Jonathan Wise. Birational invariance in logarithmic Gromov-Witten theory. Compos. Math., 154(3):595–620, 2018.CrossRefGoogle Scholar
Jack Barrott, Lawrence. Explicit equations for mirror families to log Calabi-Yau surfaces. Bull. Korean Math. Soc., 57(1):139–165, 2020.Google Scholar
Chen, Qile. Stable logarithmic maps to Deligne-Faltings pairs I. Ann. of Math. (2), 180(2):455–521, 2014.CrossRefGoogle Scholar
Cantat, Serge and Frank Loray. Dynamics on character varieties and Malgrange irreducibility of Painleve VI equation. Ann. Inst. Fourier (Grenoble), 59(7):2927–2978, 2009.CrossRefGoogle Scholar
Carl, Michael, Max Pumperla and Bernd Siebert. A tropical view of Landau-Ginzburg models. Available at http://www.math.uni-hamburg.de/home/siebert/preprints/LGtrop.pdf.Google Scholar
V. Dolgachev. Classical algebraic geometry. Cambridge University Press, Igor, Cambridge, 2012. A modern view.CrossRefGoogle Scholar
Gross, Mark, Paul Hacking and Sean Keel. Mirror symmetry for log Calabi-Yau surfaces II. Draft.Google Scholar
Gross, Mark, Paul Hacking and Sean Keel. Birational geometry of cluster algebras. Algebr. Geom., 2(2):137–175, 2015.Google Scholar
Gross, Mark, Paul Hacking and Sean Keel. Mirror symmetry for log Calabi-Yau surfaces I. Publ. Math. Inst. Hautes Etudes Sci., 122:65–168, 2015.CrossRefGoogle Scholar
Gross, Mark, Paul Hacking, Sean Keel and Maxim Kontsevich. Canonical bases for cluster algebras. J. Amer. Math. Soc., 31(2):497–608, 2018.CrossRefGoogle Scholar
Gross and Rahul Pandharipande. Quivers, Mark, curves, and the tropical vertex. Port. Math., 67(2):211–259, 2010.Google Scholar
Gross, Mark, Rahul Pandharipande, and Bernd Siebert. The tropical vertex. Duke Math. J., 153(2):297–362, 2010.CrossRefGoogle Scholar
Gross, Mark and Bernd Siebert. From real affine geometry to complex geometry. Ann. of Math. (2), 174(3):1301–1428, 2011.CrossRefGoogle Scholar
Gross, Mark and Bernd Siebert. Logarithmic Gromov-Witten invariants. J. Amer. Math. Sac., 26(2):451–510, 2013.Google Scholar
Gross, Mark and Bernd Siebert. Intrinsic mirror symmetry and punctured Gromov-Witten invariants. In Algebraic geometry: Salt Lake City 2015, volume 97 of Prac. Sympas. Pure Math., pages 199–230. Amer. Math. Soc., Providence, RI, 2018.CrossRefGoogle Scholar
Keel and Tony Yue Yu. The Frobenius structure theorem for affine log Calabi-Yau varieties containing a torus. arXiv:1908.09861, Sean, 2019.Google Scholar
Mandel. Theta bases and log Gromov-Witten invariants of cluster varieties. arXiv:1903.03042, Travis, 2019.Google Scholar
Oblomkov, Alexei. Double affine Hecke algebras of rank 1 and affine cubic surfaces. Int. Math. Res. Nat., (18):877–912, 2004.Google Scholar
Reid. Undergraduate algebraic geometry, volume 12 of London Mathematical Society Student Texts. Cambridge University Press, Miles, Cambridge, 1988.Google Scholar