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January  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

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