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), 1050-1072. doi: 10.4153/CJM-2009-050-2.  Google Scholar

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[5]

G. Brown, A. Corti and F. Zucconi, Birational geometry of 3-fold Mori fibre spaces, Proc. of the "Fano conference'' held in Toronto, 2002. Google Scholar

[6]

G. Brown and K. Georgiadis, Polarised Calabi-Yau threefold in codimension $4$, arXiv:1508.05130v1, Math. Nachrichten., to appear, (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), 1-44. doi: 10.1017/S0027763000009855.  Google Scholar

[9]

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

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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), 221-226.  Google Scholar

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

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

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D. Eisenbud and J. Harris, On varieties of minimal degree (a centennial account), in Algebraic Geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proceedings of Symposia in Pure Mathematics, 46, Part 1, Amer. Math. Soc., Providence, RI, 1987, 3-13. 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

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

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B. Grünbaum and V. P. Sreedharan, An enumerative of simplicial 4-polytopes with 8 vertices, Journal of Combinatorial Theory, 2 (1967), 437-465. doi: 10.1016/S0021-9800(67)80055-3.  Google Scholar

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

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H. Jockers, V. Kumar, J. M. Lapan and D. R. Morrison, Two-sphere partition functions and Gromov-Witten invariants, Commun. Math. Phys., 325 (2014), 1139-1170. doi: 10.1007/s00220-013-1874-z.  Google Scholar

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R. Hartshorne, Generalized divisors and biliaison, Illinois J. Math., 51 (2007), 83-98.  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), 259-271. doi: 10.1112/jlms/jdn069.  Google Scholar

[23]

G. Kapustka, Projections of del Pezzo surfaces and Calabi-Yau threefolds, Adv. Geom., 15 (2015), 143-158. doi: 10.1515/advgeom-2015-0002.  Google Scholar

[24]

G. Kapustka and M. Kapustka, A cascade of Calabi-Yau threefolds, Math. Nachr., 283 (2010), 1795-1809. doi: 10.1002/mana.200910057.  Google Scholar

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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), 529-556. 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), 1309-1318. doi: 10.1016/j.geomphys.2011.02.022.  Google Scholar

[29]

Y. Kawamata, Flops connect minimal models, Publ. Res. Inst. Math. Sci., 44 (2008), 419-423. 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), 233-254. 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), 115-137. 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), 753-776.  Google Scholar

[33]

S. Nollet, Even linkage classes, Trans. Amer. Math. Soc., 348 (1996), 1137-1162. 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), 421-442. doi: 10.1007/BF02567467.  Google Scholar

[35]

S. A. Papadakis and M. Reid, Kustin-Miller unprojection without complexes, J. Algebraic Geom., 13 (2004), 563-577. 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), 271-302. 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), 547317, 14pp.  Google Scholar

[38]

G. V. Ravindra and V. Srinivas, The Grothendieck-Lefschetz theorem for normal projective varieties, J. Alg. Geom., 15 (2006), 563-590. 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, Oct 2000) (ed. K. Ohno), 1-72. Google Scholar

[40]

M. Reid, Gorenstein in codimension 4 - the general structure theory, in Algebraic Geometry in East Asia (Taipei Nov 2011), Adv. Stud. in Pure Math., 65, 2013. Google Scholar

[41]

F. Tonoli, Construction of Calabi-Yau 3-folds in $\mathbbP^6$, J. Alg. Geom., 13 (2004), 209-232. doi: 10.1090/S1056-3911-03-00371-0.  Google Scholar

[42]

Ch. Walter, Pfaffian subschemes, J. Alg. Geom., 5 (1996), 671-704.  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), 1050-1072. 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), 235-265. 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 "Fano conference'' held in Toronto, 2002. Google Scholar

[6]

G. Brown and K. Georgiadis, Polarised Calabi-Yau threefold in codimension $4$, arXiv:1508.05130v1, Math. Nachrichten., to appear, (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), 1-44. 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), 221-226.  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, Bowdoin, 1985 (Brunswick, Maine, 1985), Proceedings of Symposia in Pure Mathematics, 46, Part 1, Amer. Math. Soc., Providence, RI, 1987, 3-13. 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), 437-465. 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), 1139-1170. doi: 10.1007/s00220-013-1874-z.  Google Scholar

[20]

R. Hartshorne, Generalized divisors and biliaison, Illinois J. Math., 51 (2007), 83-98.  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), 259-271. doi: 10.1112/jlms/jdn069.  Google Scholar

[23]

G. Kapustka, Projections of del Pezzo surfaces and Calabi-Yau threefolds, Adv. Geom., 15 (2015), 143-158. doi: 10.1515/advgeom-2015-0002.  Google Scholar

[24]

G. Kapustka and M. Kapustka, A cascade of Calabi-Yau threefolds, Math. Nachr., 283 (2010), 1795-1809. 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), 529-556. 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), 1309-1318. doi: 10.1016/j.geomphys.2011.02.022.  Google Scholar

[29]

Y. Kawamata, Flops connect minimal models, Publ. Res. Inst. Math. Sci., 44 (2008), 419-423. 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), 233-254. 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), 115-137. 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), 753-776.  Google Scholar

[33]

S. Nollet, Even linkage classes, Trans. Amer. Math. Soc., 348 (1996), 1137-1162. 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), 421-442. doi: 10.1007/BF02567467.  Google Scholar

[35]

S. A. Papadakis and M. Reid, Kustin-Miller unprojection without complexes, J. Algebraic Geom., 13 (2004), 563-577. 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), 271-302. 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), 547317, 14pp.  Google Scholar

[38]

G. V. Ravindra and V. Srinivas, The Grothendieck-Lefschetz theorem for normal projective varieties, J. Alg. Geom., 15 (2006), 563-590. 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, Oct 2000) (ed. K. Ohno), 1-72. Google Scholar

[40]

M. Reid, Gorenstein in codimension 4 - the general structure theory, in Algebraic Geometry in East Asia (Taipei Nov 2011), Adv. Stud. in Pure Math., 65, 2013. Google Scholar

[41]

F. Tonoli, Construction of Calabi-Yau 3-folds in $\mathbbP^6$, J. Alg. Geom., 13 (2004), 209-232. doi: 10.1090/S1056-3911-03-00371-0.  Google Scholar

[42]

Ch. Walter, Pfaffian subschemes, J. Alg. Geom., 5 (1996), 671-704.  Google Scholar

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