American Institute of Mathematical Sciences

Advanced
  2016, 23: 52-68. doi: 10.3934/era.2016.23.006

Arithmetically Gorenstein Calabi--Yau threefolds in $\mathbb{P}^7$

Stephen Coughlan 1, , Łukasz Gołębiowski 2, , Grzegorz Kapustka 3, and  Michał Kapustka 4,

1. 

Mathematisches Institut, Lehrstuhl Mathematik VIII, Universität Bayreuth, Universitätsstrasse 30, D-95447 Bayreuth, Germany
2. 

Jagiellonian University Cracow, Poland
3. 

Jagiellonian University Cracow, University of Zürich, Poland
4. 

University of Stavanger, Department of Mathematics and Natural Sciences, NO-4036 Stavanger, Norway

Received  September 2016 Revised  October 2016 Published  December 2016

We present a list of arithmetically Gorenstein Calabi--Yau threefolds in $\mathbb{P}^7$ and give evidence that this is a complete list. In particular we construct three new families of arithmetically Gorenstein Calabi--Yau threefolds in $\mathbb{P}^7$ for which no mirror construction is known.
Keywords: Gorenstein structure., Calabi-Yau threefolds.
Mathematics Subject Classification: 14J3.
Citation: Stephen Coughlan, Łukasz Gołębiowski, Grzegorz Kapustka, Michał Kapustka. Arithmetically Gorenstein Calabi--Yau threefolds in $\mathbb{P}^7$. Electronic Research Announcements, 2016, 23: 52-68. doi: 10.3934/era.2016.23.006
References:
[1]

H. Ahmadinezhad, On pliability of del Pezzo fibrations and Cox rings,, J. Reine Angew. Math., ().   Google Scholar

[2]

A. Bertin, Examples of Calabi-Yau 3-folds in $\mathbbP^7$ with $\rho = 1$,, Canad. J. Math., 61 (2009), 1050.  doi: 10.4153/CJM-2009-050-2.  Google Scholar

[3]

G. Bini and P. Penegini, New fourfolds from F-theory,, Math. Nachrichten., ().   Google Scholar

[4]

W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I. The user language,, J. Symbolic Comput., 24 (1997), 235.  doi: 10.1006/jsco.1996.0125.  Google Scholar

[5]

G. Brown, A. Corti and F. Zucconi, Birational geometry of 3-fold Mori fibre spaces,, Proc. of the, (2002).   Google Scholar

[6]

G. Brown and K. Georgiadis, Polarised Calabi-Yau threefold in codimension $4$,, , (2015).   Google Scholar

[7]

G. Brown, A. Kasprzyk and L. Zhu, Gorenstein formats, canonical and Calabi-Yau threefolds,, , ().   Google Scholar

[8]

G. Brown and F. Zucconi, Graded rings of rank 2 Sarkisov links,, Nagoya Math. J., 197 (2010), 1.  doi: 10.1017/S0027763000009855.  Google Scholar

[9]

P. Candelas, A. Constantin and C. Mishra, Calabi-yau threefolds with small hodge numbers,, preprint, ().   Google Scholar

[10]

S. Coughlan, Ł. Gołebiowski, G. Kapustka and M. Kapustka, Calabi-Yau threefolds in codimension $4$,, in preparation., ().   Google Scholar

[11]

S. Cynk, Hodge numbers of double octic with non-isolated singularities,, Ann. Pol. Math., 73 (2000), 221.   Google Scholar

[12]

C. Di Natale, E. Fatighenti and D. Fiorenza, Hodge theory and deformations of affine cones of subcanonical projective varieties,, , ().   Google Scholar

[13]

D. Eisenbud, Commutative Algebra: With a View Toward Algebraic Geometry,, Springer, (1995).  doi: 10.1007/978-1-4612-5350-1.  Google Scholar

[14]

D. Eisenbud and J. Harris, On varieties of minimal degree (a centennial account),, in Algebraic Geometry, (1985), 3.  doi: 10.1090/pspum/046.1/927946.  Google Scholar

[15]

I. Fausk, Pfaffian Calabi-Yau Threefolds, Stanley-Reisner Schemes and Mirror Symmetry,, PhD thesis, ().   Google Scholar

[16]

S. Galkin, A. Kuznetsov and M. Movshev, An explicit construction of Miura's varieties,, in prepration., ().   Google Scholar

[17]

B. Grünbaum and V. P. Sreedharan, An enumerative of simplicial 4-polytopes with 8 vertices,, Journal of Combinatorial Theory, 2 (1967), 437.  doi: 10.1016/S0021-9800(67)80055-3.  Google Scholar

[18]

D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in algebraic geometry,, available from: , ().   Google Scholar

[19]

H. Jockers, V. Kumar, J. M. Lapan and D. R. Morrison, Two-sphere partition functions and Gromov-Witten invariants,, Commun. Math. Phys., 325 (2014), 1139.  doi: 10.1007/s00220-013-1874-z.  Google Scholar

[20]

R. Hartshorne, Generalized divisors and biliaison,, Illinois J. Math., 51 (2007), 83.   Google Scholar

[21]

D. Inoue, A. Ito and M. Miura, Complete intersection Calabi-Yau manifolds with respect to homogeneous vector bundles on Grassmannians,, , ().   Google Scholar

[22]

G. Kapustka, Primitive contractions of Calabi-Yau threefolds II,, Journal of LMS, 79 (2009), 259.  doi: 10.1112/jlms/jdn069.  Google Scholar

[23]

G. Kapustka, Projections of del Pezzo surfaces and Calabi-Yau threefolds,, Adv. Geom., 15 (2015), 143.  doi: 10.1515/advgeom-2015-0002.  Google Scholar

[24]

G. Kapustka and M. Kapustka, A cascade of Calabi-Yau threefolds,, Math. Nachr., 283 (2010), 1795.  doi: 10.1002/mana.200910057.  Google Scholar

[25]

G. Kapustka and M. Kapustka, Tonoli Calabi-Yau revisited,, , ().   Google Scholar

[26]

G. Kapustka and M. Kapustka, Calabi-Yau threefolds in $\mathbbP^6$,, Ann. Math. Pur. Appl., 195 (2016), 529.  doi: 10.1007/s10231-015-0476-0.  Google Scholar

[27]

G. Kapustka and M. Kapustka, Bilinkage in codimension $3$ and canonical surfaces of degree 18 in $\mathbbP^5$,, Ann. Scuola Norm-Sci., (2016).   Google Scholar

[28]

M. Kapustka, Geometric transitions between Calabi-Yau threefolds related to Kustin-Miller unprojections,, J. Geom. Phys., 61 (2011), 1309.  doi: 10.1016/j.geomphys.2011.02.022.  Google Scholar

[29]

Y. Kawamata, Flops connect minimal models,, Publ. Res. Inst. Math. Sci., 44 (2008), 419.  doi: 10.2977/prims/1210167332.  Google Scholar

[30]

M. Kühnel, Calabi-Yau-threefolds with Picard number $\rho(X) = 2$ and their Kähler cone I,, Math. Z., 245 (2003), 233.  doi: 10.1007/s00209-003-0540-0.  Google Scholar

[31]

M. Kühnel, Calabi-Yau-threefolds with Picard number $\rho(X) = 2$ and their Kähler cone II,, Pacific. J. Math., 217 (2004), 115.  doi: 10.2140/pjm.2004.217.115.  Google Scholar

[32]

Y. Namikawa, Deformation theory of Calabi-Yau threefolds and certain invariants of singularities,, Journal of Algebraic Geometry, 6 (1997), 753.   Google Scholar

[33]

S. Nollet, Even linkage classes,, Trans. Amer. Math. Soc., 348 (1996), 1137.  doi: 10.1090/S0002-9947-96-01552-8.  Google Scholar

[34]

Ch. Okonek, Note on varieties of codimension $3$ in $\mathbbP^N$,, Manuscripta Math., 84 (1994), 421.  doi: 10.1007/BF02567467.  Google Scholar

[35]

S. A. Papadakis and M. Reid, Kustin-Miller unprojection without complexes,, J. Algebraic Geom., 13 (2004), 563.  doi: 10.1090/S1056-3911-04-00343-1.  Google Scholar

[36]

C. Peskine and L. Szpiro, Liaison des variétés algébriques. I,, Invent. Math., 26 (1974), 271.  doi: 10.1007/BF01425554.  Google Scholar

[37]

M. I. Qureshi and B. Szendroi, Calabi-Yau threefolds in weighted flag varieties,, Adv. High Energy Physics, 2012 (2012).   Google Scholar

[38]

G. V. Ravindra and V. Srinivas, The Grothendieck-Lefschetz theorem for normal projective varieties,, J. Alg. Geom., 15 (2006), 563.  doi: 10.1090/S1056-3911-05-00421-2.  Google Scholar

[39]

M. Reid, Graded rings and birational geometry,, in Proc. of Algebraic Geometry Symposium (Kinosaki, (2000), 1.   Google Scholar

[40]

M. Reid, Gorenstein in codimension 4 - the general structure theory,, in Algebraic Geometry in East Asia (Taipei Nov 2011), (2011).   Google Scholar

[41]

F. Tonoli, Construction of Calabi-Yau 3-folds in $\mathbbP^6$,, J. Alg. Geom., 13 (2004), 209.  doi: 10.1090/S1056-3911-03-00371-0.  Google Scholar

[42]

Ch. Walter, Pfaffian subschemes,, J. Alg. Geom., 5 (1996), 671.   Google Scholar

show all references

References:
[1]

H. Ahmadinezhad, On pliability of del Pezzo fibrations and Cox rings,, J. Reine Angew. Math., ().   Google Scholar

[2]

A. Bertin, Examples of Calabi-Yau 3-folds in $\mathbbP^7$ with $\rho = 1$,, Canad. J. Math., 61 (2009), 1050.  doi: 10.4153/CJM-2009-050-2.  Google Scholar

[3]

G. Bini and P. Penegini, New fourfolds from F-theory,, Math. Nachrichten., ().   Google Scholar

[4]

W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I. The user language,, J. Symbolic Comput., 24 (1997), 235.  doi: 10.1006/jsco.1996.0125.  Google Scholar

[5]

G. Brown, A. Corti and F. Zucconi, Birational geometry of 3-fold Mori fibre spaces,, Proc. of the, (2002).   Google Scholar

[6]

G. Brown and K. Georgiadis, Polarised Calabi-Yau threefold in codimension $4$,, , (2015).   Google Scholar

[7]

G. Brown, A. Kasprzyk and L. Zhu, Gorenstein formats, canonical and Calabi-Yau threefolds,, , ().   Google Scholar

[8]

G. Brown and F. Zucconi, Graded rings of rank 2 Sarkisov links,, Nagoya Math. J., 197 (2010), 1.  doi: 10.1017/S0027763000009855.  Google Scholar

[9]

P. Candelas, A. Constantin and C. Mishra, Calabi-yau threefolds with small hodge numbers,, preprint, ().   Google Scholar

[10]

S. Coughlan, Ł. Gołebiowski, G. Kapustka and M. Kapustka, Calabi-Yau threefolds in codimension $4$,, in preparation., ().   Google Scholar

[11]

S. Cynk, Hodge numbers of double octic with non-isolated singularities,, Ann. Pol. Math., 73 (2000), 221.   Google Scholar

[12]

C. Di Natale, E. Fatighenti and D. Fiorenza, Hodge theory and deformations of affine cones of subcanonical projective varieties,, , ().   Google Scholar

[13]

D. Eisenbud, Commutative Algebra: With a View Toward Algebraic Geometry,, Springer, (1995).  doi: 10.1007/978-1-4612-5350-1.  Google Scholar

[14]

D. Eisenbud and J. Harris, On varieties of minimal degree (a centennial account),, in Algebraic Geometry, (1985), 3.  doi: 10.1090/pspum/046.1/927946.  Google Scholar

[15]

I. Fausk, Pfaffian Calabi-Yau Threefolds, Stanley-Reisner Schemes and Mirror Symmetry,, PhD thesis, ().   Google Scholar

[16]

S. Galkin, A. Kuznetsov and M. Movshev, An explicit construction of Miura's varieties,, in prepration., ().   Google Scholar

[17]

B. Grünbaum and V. P. Sreedharan, An enumerative of simplicial 4-polytopes with 8 vertices,, Journal of Combinatorial Theory, 2 (1967), 437.  doi: 10.1016/S0021-9800(67)80055-3.  Google Scholar

[18]

D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in algebraic geometry,, available from: , ().   Google Scholar

[19]

H. Jockers, V. Kumar, J. M. Lapan and D. R. Morrison, Two-sphere partition functions and Gromov-Witten invariants,, Commun. Math. Phys., 325 (2014), 1139.  doi: 10.1007/s00220-013-1874-z.  Google Scholar

[20]

R. Hartshorne, Generalized divisors and biliaison,, Illinois J. Math., 51 (2007), 83.   Google Scholar

[21]

D. Inoue, A. Ito and M. Miura, Complete intersection Calabi-Yau manifolds with respect to homogeneous vector bundles on Grassmannians,, , ().   Google Scholar

[22]

G. Kapustka, Primitive contractions of Calabi-Yau threefolds II,, Journal of LMS, 79 (2009), 259.  doi: 10.1112/jlms/jdn069.  Google Scholar

[23]

G. Kapustka, Projections of del Pezzo surfaces and Calabi-Yau threefolds,, Adv. Geom., 15 (2015), 143.  doi: 10.1515/advgeom-2015-0002.  Google Scholar

[24]

G. Kapustka and M. Kapustka, A cascade of Calabi-Yau threefolds,, Math. Nachr., 283 (2010), 1795.  doi: 10.1002/mana.200910057.  Google Scholar

[25]

G. Kapustka and M. Kapustka, Tonoli Calabi-Yau revisited,, , ().   Google Scholar

[26]

G. Kapustka and M. Kapustka, Calabi-Yau threefolds in $\mathbbP^6$,, Ann. Math. Pur. Appl., 195 (2016), 529.  doi: 10.1007/s10231-015-0476-0.  Google Scholar

[27]

G. Kapustka and M. Kapustka, Bilinkage in codimension $3$ and canonical surfaces of degree 18 in $\mathbbP^5$,, Ann. Scuola Norm-Sci., (2016).   Google Scholar

[28]

M. Kapustka, Geometric transitions between Calabi-Yau threefolds related to Kustin-Miller unprojections,, J. Geom. Phys., 61 (2011), 1309.  doi: 10.1016/j.geomphys.2011.02.022.  Google Scholar

[29]

Y. Kawamata, Flops connect minimal models,, Publ. Res. Inst. Math. Sci., 44 (2008), 419.  doi: 10.2977/prims/1210167332.  Google Scholar

[30]

M. Kühnel, Calabi-Yau-threefolds with Picard number $\rho(X) = 2$ and their Kähler cone I,, Math. Z., 245 (2003), 233.  doi: 10.1007/s00209-003-0540-0.  Google Scholar

[31]

M. Kühnel, Calabi-Yau-threefolds with Picard number $\rho(X) = 2$ and their Kähler cone II,, Pacific. J. Math., 217 (2004), 115.  doi: 10.2140/pjm.2004.217.115.  Google Scholar

[32]

Y. Namikawa, Deformation theory of Calabi-Yau threefolds and certain invariants of singularities,, Journal of Algebraic Geometry, 6 (1997), 753.   Google Scholar

[33]

S. Nollet, Even linkage classes,, Trans. Amer. Math. Soc., 348 (1996), 1137.  doi: 10.1090/S0002-9947-96-01552-8.  Google Scholar

[34]

Ch. Okonek, Note on varieties of codimension $3$ in $\mathbbP^N$,, Manuscripta Math., 84 (1994), 421.  doi: 10.1007/BF02567467.  Google Scholar

[35]

S. A. Papadakis and M. Reid, Kustin-Miller unprojection without complexes,, J. Algebraic Geom., 13 (2004), 563.  doi: 10.1090/S1056-3911-04-00343-1.  Google Scholar

[36]

C. Peskine and L. Szpiro, Liaison des variétés algébriques. I,, Invent. Math., 26 (1974), 271.  doi: 10.1007/BF01425554.  Google Scholar

[37]

M. I. Qureshi and B. Szendroi, Calabi-Yau threefolds in weighted flag varieties,, Adv. High Energy Physics, 2012 (2012).   Google Scholar

[38]

G. V. Ravindra and V. Srinivas, The Grothendieck-Lefschetz theorem for normal projective varieties,, J. Alg. Geom., 15 (2006), 563.  doi: 10.1090/S1056-3911-05-00421-2.  Google Scholar

[39]

M. Reid, Graded rings and birational geometry,, in Proc. of Algebraic Geometry Symposium (Kinosaki, (2000), 1.   Google Scholar

[40]

M. Reid, Gorenstein in codimension 4 - the general structure theory,, in Algebraic Geometry in East Asia (Taipei Nov 2011), (2011).   Google Scholar

[41]

F. Tonoli, Construction of Calabi-Yau 3-folds in $\mathbbP^6$,, J. Alg. Geom., 13 (2004), 209.  doi: 10.1090/S1056-3911-03-00371-0.  Google Scholar

[42]

Ch. Walter, Pfaffian subschemes,, J. Alg. Geom., 5 (1996), 671.   Google Scholar

[1]

Janos Kollar. The Nash conjecture for threefolds. Electronic Research Announcements, 1998, 4: 63-73.

[2]

Jungkai A. Chen and Meng Chen. On projective threefolds of general type. Electronic Research Announcements, 2007, 14: 69-73. doi: 10.3934/era.2007.14.69
[3]

Michael Hutchings. Mean action and the Calabi invariant. Journal of Modern Dynamics, 2016, 10: 511-539. doi: 10.3934/jmd.2016.10.511
[4]

Hongnian Huang. On the extension and smoothing of the Calabi flow on complex tori. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6153-6164. doi: 10.3934/dcds.2017265
[5]

Piotr Pokora. The orbifold Langer-Miyaoka-Yau Inequality and Hirzebruch-type inequalities. Electronic Research Announcements, 2017, 24: 21-27. doi: 10.3934/era.2017.24.003
[6]

Asaf Kislev. Compactly supported Hamiltonian loops with a non-zero Calabi invariant. Electronic Research Announcements, 2014, 21: 80-88. doi: 10.3934/era.2014.21.80
[7]

Jianjun Tian, Bai-Lian Li. Coalgebraic Structure of Genetic Inheritance. Mathematical Biosciences & Engineering, 2004, 1 (2) : 243-266. doi: 10.3934/mbe.2004.1.243
[8]

V. Balaji, I. Biswas and D. S. Nagaraj. Principal bundles with parabolic structure. Electronic Research Announcements, 2001, 7: 37-44.

[9]

Vincent Giovangigli, Lionel Matuszewski. Structure of entropies in dissipative multicomponent fluids. Kinetic & Related Models, 2013, 6 (2) : 373-406. doi: 10.3934/krm.2013.6.373
[10]

Kurusch Ebrahimi-Fard, Dominique Manchon. The tridendriform structure of a discrete Magnus expansion. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1021-1040. doi: 10.3934/dcds.2014.34.1021
[11]

Assane Lo, Nasser-eddine Tatar. Exponential stabilization of a structure with interfacial slip. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6285-6306. doi: 10.3934/dcds.2016073
[12]

Alberto A. Pinto, João P. Almeida, Telmo Parreira. Local market structure in a Hotelling town. Journal of Dynamics & Games, 2016, 3 (1) : 75-100. doi: 10.3934/jdg.2016004
[13]

Philipp Fuchs, Ansgar Jüngel, Max von Renesse. On the Lagrangian structure of quantum fluid models. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1375-1396. doi: 10.3934/dcds.2014.34.1375
[14]

Boris Hasselblatt and Jorg Schmeling. Dimension product structure of hyperbolic sets. Electronic Research Announcements, 2004, 10: 88-96.

[15]

Claudio Qureshi, Daniel Panario, Rodrigo Martins. Cycle structure of iterating Redei functions. Advances in Mathematics of Communications, 2017, 11 (2) : 397-407. doi: 10.3934/amc.2017034
[16]

Fabio Augusto Milner, Ruijun Zhao. A deterministic model of schistosomiasis with spatial structure. Mathematical Biosciences & Engineering, 2008, 5 (3) : 505-522. doi: 10.3934/mbe.2008.5.505
[17]

Manfred Deistler. Singular arma systems: A structure theory. Numerical Algebra, Control & Optimization, 2019, 9 (3) : 383-391. doi: 10.3934/naco.2019025
[18]

Yakov Krasnov, Alexander Kononovich, Grigory Osharovich. On a structure of the fixed point set of homogeneous maps. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1017-1027. doi: 10.3934/dcdss.2013.6.1017
[19]

Ling Jia. The structure theorems for Yetter-Drinfeld comodule algebras. Electronic Research Announcements, 2013, 20: 31-42. doi: 10.3934/era.2013.20.31
[20]

Marco A. Fontelos, Lucía B. Gamboa. On the structure of double layers in Poisson-Boltzmann equation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1939-1967. doi: 10.3934/dcdsb.2012.17.1939

2018 Impact Factor: 0.263

Tools

Metrics

  • PDF downloads (37)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences