Abstract
In this note we illustrate the Fanosearch programme of Coates, Corti, Galkin, Golyshev, and Kasprzyk in the example of the anticanonical cone over the smooth del Pezzo surface of degree 6.
Keywords
2010 Mathematics Subject Classification
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Akhtar, M., Coates, T., Galkin, S., Kasprzyk, A.M.: Minkowski polynomials and mutations. SIGMA Symmetry Integrability Geom. Methods Appl. 8, Paper 094, 17 (2012)
Akhtar, M., Coates, T., Corti, A., Heuberger, L., Kasprzyk, A., Oneto, A., Petracci, A., Prince, T., Tveiten, K.: Mirror symmetry and the classification of orbifold del Pezzo surfaces. Proc. Am. Math. Soc. 144(2), 513–527 (2016)
Altmann, K.: Minkowski sums and homogeneous deformations of toric varieties. Tohoku Math. J. 47(2), 151–184 (1995)
Altmann, K.: The versal deformation of an isolated toric Gorenstein singularity. Invent. Math. 128(3), 443–479 (1997)
Brown, G., Kerber, M., Reid, M.: Fano 3-folds in codimension 4, Tom and Jerry. Part I. Compos. Math. 148(4), 1171–1194 (2012)
Buch, A., Thomsen, J.F., Lauritzen, N., Mehta, V.: The Frobenius morphism on a toric variety. Tohoku Math. J. (2) 49(3), 355–366 (1997)
Coates, T., Givental, A.: Quantum Riemann-Roch, Lefschetz and Serre. Ann. Math. (2) 165(1), 15–53 (2007)
Coates, T., Corti, A., Galkin, S., Golyshev, V., Kasprzyk, A.: Mirror symmetry and Fano manifolds. In: European Congress of Mathematics, pp. 285–300. European Mathematical Society, Zürich (2013)
Coates, T., Corti, A., Galkin, S., Kasprzyk, A.: Quantum periods for 3–dimensional Fano manifolds. Geom. Topol. 20(1), 103–256 (2016)
Coates, T., Kasprzyk, A., Prince, T.: Laurent inversion. Preprint arXiv:1707.05842. https://arxiv.org/1707.05842
Corti, A., Hacking, P., Petracci, A.: Smoothing toric Fano threefolds (in preparation)
Cox, D.A., Little, J.B., Schenck, H.K.: Toric Varieties. Graduate Studies in Mathematics, vol. 124. American Mathematical Society, Providence (2011)
Cruz Morales, J.A., Galkin, S.: Upper bounds for mutations of potentials. SIGMA Symmetry Integrability Geom. Methods Appl. 9, Paper 005, 13 (2013)
Fujino, O.: Multiplication maps and vanishing theorems for toric varieties. Math. Z. 257(3), 631–641 (2007)
Fulton, W.: Introduction to Toric Varieties. Annals of Mathematics Studies, vol. 131. Princeton University Press, Princeton (1993). The William H. Roever Lectures in Geometry
Galkin, S., Usnich, A.: Mutations of Potentials. IPMU, Kashiwa, pp. 10–0100 (2010). Preprint
Givental, A.B.: Equivariant Gromov-Witten invariants. Int. Math. Res. Notices 13, 613–663 (1996)
Givental, A.: A mirror theorem for toric complete intersections. In: Topological field theory, primitive forms and related topics (Kyoto, 1996). Progress in Mathematics, vol. 160, pp. 141–175. Birkhäuser, Boston (1998)
Golyshev, V.V.: Classification problems and mirror duality. In: Surveys in Geometry and Number Theory: Reports on Contemporary Russian Mathematics. London Mathematical Society Lecture Note Series, vol. 338, pp. 88–121. Cambridge University, Cambridge (2007)
Jahnke, P., Radloff, I.: Terminal Fano threefolds and their smoothings. Math. Z. 269(3–4), 1129–1136 (2011)
Kasprzyk, A., Tveiten, K.: Maximally mutable Laurent polynomials (in preparation)
Kasprzyk, A.M., Nill, B.: Fano polytopes. In: Strings, Gauge Fields, and the Geometry Behind, pp. 349–364. World Science Publication, Hackensack (2013)
Katzarkov, L., Przyjalkowski, V.: Landau-Ginzburg models—old and new. In: Proceedings of the Gökova Geometry-Topology Conference 2011, pp. 97–124. International Press, Somerville (2012)
Manetti, M.: Differential graded Lie algebras and formal deformation theory. In: Algebraic geometry—Seattle 2005. Part 2. Proceedings of Symposia in Pure Mathematics, vol. 80, pp. 785–810. American Mathematical Society, Providence (2009)
Mavlyutov, A.R.: Deformations of toric varieties via Minkowski sum decompositions of polyhedral complexes. Preprint arXiv:0902.0967. https://arxiv.org/abs/0902.0967
Mustaţă, M.: Vanishing theorems on toric varieties. Tohoku Math. J. (2) 54(3), 451–470 (2002)
Oneto, A., Petracci, A.: On the quantum periods of del Pezzo surfaces with \(\frac 13 (1, 1)\) singularities. Adv. Geom. 18(3), 303–336 (2018)
Petracci, A.: Homogeneous deformations of toric pairs. Preprint arXiv:1801.05732. https://arxiv.org/1801.05732
Petracci, A.: Some examples of non-smoothable Gorenstein Fano toric threefolds. Math. Z. https://doi.org/10.1007/s00209-019-02369-8
Prince, T.: Cracked polytopes and Fano toric complete intersections. Manuscripta Math. https://doi.org/10.1007/s00229-019-01149-2
Przyjalkowski, V.: On Landau-Ginzburg models for Fano varieties. Commun. Number Theory Phys. 1(4), 713–728 (2007)
Przyjalkowski, V.V.: Weak Landau-Ginzburg models of smooth Fano threefolds. Izv. Math. 77(4), 772–794 (2013)
Totaro, B.: Jumping of the nef cone for Fano varieties. J. Algebraic Geom. 21(2), 375–396 (2012)
Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer, New York (1995)
Acknowledgements
I am indebted to Tom Coates, Alessio Corti, Paul Hacking, Alexander Kasprzyk, Thomas Prince, and the other members of the Fanosearch group for countless fruitful conversations about the topics of this note. I wish to thank the organisers of the conference “Birational geometry and moduli spaces”, held in Rome in June 2018, for giving me the opportunity to present a poster about this subject. Finally, I would like to thank Enrica Floris, Luigi Lunardon, and Diletta Martinelli for useful comments on a preliminary version of this note.
The author was funded by Kasprzyk’s EPSRC Fellowship EP/N022513/1.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Petracci, A. (2020). An Example of Mirror Symmetry for Fano Threefolds. In: Colombo, E., Fantechi, B., Frediani, P., Iacono, D., Pardini, R. (eds) Birational Geometry and Moduli Spaces. Springer INdAM Series, vol 39. Springer, Cham. https://doi.org/10.1007/978-3-030-37114-2_10
Download citation
DOI: https://doi.org/10.1007/978-3-030-37114-2_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-37113-5
Online ISBN: 978-3-030-37114-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)