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

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

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.
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., (). 

[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.

[3]

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

[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.

[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.

[6]

G. Brown and K. Georgiadis, Polarised Calabi-Yau threefold in codimension $4$, arXiv:1508.05130v1, Math. Nachrichten., to appear, (2015).

[7]

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

[8]

G. Brown and F. Zucconi, Graded rings of rank 2 Sarkisov links, Nagoya Math. J., 197 (2010), 1-44. doi: 10.1017/S0027763000009855.

[9]

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

[10]

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

[11]

S. Cynk, Hodge numbers of double octic with non-isolated singularities, Ann. Pol. Math., 73 (2000), 221-226.

[12]

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

[13]

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

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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.

[15]

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

[16]

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

[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.

[18]

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

[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.

[20]

R. Hartshorne, Generalized divisors and biliaison, Illinois J. Math., 51 (2007), 83-98.

[21]

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

[22]

G. Kapustka, Primitive contractions of Calabi-Yau threefolds II, Journal of LMS, 79 (2009), 259-271. doi: 10.1112/jlms/jdn069.

[23]

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

[24]

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

[25]

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

[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.

[27]

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

[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.

[29]

Y. Kawamata, Flops connect minimal models, Publ. Res. Inst. Math. Sci., 44 (2008), 419-423. doi: 10.2977/prims/1210167332.

[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.

[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.

[32]

Y. Namikawa, Deformation theory of Calabi-Yau threefolds and certain invariants of singularities, Journal of Algebraic Geometry, 6 (1997), 753-776.

[33]

S. Nollet, Even linkage classes, Trans. Amer. Math. Soc., 348 (1996), 1137-1162. doi: 10.1090/S0002-9947-96-01552-8.

[34]

Ch. Okonek, Note on varieties of codimension $3$ in $\mathbbP^N$, Manuscripta Math., 84 (1994), 421-442. doi: 10.1007/BF02567467.

[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.

[36]

C. Peskine and L. Szpiro, Liaison des variétés algébriques. I, Invent. Math., 26 (1974), 271-302. doi: 10.1007/BF01425554.

[37]

M. I. Qureshi and B. Szendroi, Calabi-Yau threefolds in weighted flag varieties, Adv. High Energy Physics, 2012 (2012), 547317, 14pp.

[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.

[39]

M. Reid, Graded rings and birational geometry, in Proc. of Algebraic Geometry Symposium (Kinosaki, Oct 2000) (ed. K. Ohno), 1-72.

[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.

[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.

[42]

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

show all references

References:
[1]

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

[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.

[3]

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

[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.

[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.

[6]

G. Brown and K. Georgiadis, Polarised Calabi-Yau threefold in codimension $4$, arXiv:1508.05130v1, Math. Nachrichten., to appear, (2015).

[7]

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

[8]

G. Brown and F. Zucconi, Graded rings of rank 2 Sarkisov links, Nagoya Math. J., 197 (2010), 1-44. doi: 10.1017/S0027763000009855.

[9]

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

[10]

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

[11]

S. Cynk, Hodge numbers of double octic with non-isolated singularities, Ann. Pol. Math., 73 (2000), 221-226.

[12]

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

[13]

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

[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.

[15]

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

[16]

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

[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.

[18]

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

[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.

[20]

R. Hartshorne, Generalized divisors and biliaison, Illinois J. Math., 51 (2007), 83-98.

[21]

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

[22]

G. Kapustka, Primitive contractions of Calabi-Yau threefolds II, Journal of LMS, 79 (2009), 259-271. doi: 10.1112/jlms/jdn069.

[23]

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

[24]

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

[25]

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

[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.

[27]

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

[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.

[29]

Y. Kawamata, Flops connect minimal models, Publ. Res. Inst. Math. Sci., 44 (2008), 419-423. doi: 10.2977/prims/1210167332.

[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.

[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.

[32]

Y. Namikawa, Deformation theory of Calabi-Yau threefolds and certain invariants of singularities, Journal of Algebraic Geometry, 6 (1997), 753-776.

[33]

S. Nollet, Even linkage classes, Trans. Amer. Math. Soc., 348 (1996), 1137-1162. doi: 10.1090/S0002-9947-96-01552-8.

[34]

Ch. Okonek, Note on varieties of codimension $3$ in $\mathbbP^N$, Manuscripta Math., 84 (1994), 421-442. doi: 10.1007/BF02567467.

[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.

[36]

C. Peskine and L. Szpiro, Liaison des variétés algébriques. I, Invent. Math., 26 (1974), 271-302. doi: 10.1007/BF01425554.

[37]

M. I. Qureshi and B. Szendroi, Calabi-Yau threefolds in weighted flag varieties, Adv. High Energy Physics, 2012 (2012), 547317, 14pp.

[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.

[39]

M. Reid, Graded rings and birational geometry, in Proc. of Algebraic Geometry Symposium (Kinosaki, Oct 2000) (ed. K. Ohno), 1-72.

[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.

[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.

[42]

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

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