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Banach limit in convexity and geometric means for convex bodies
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 |
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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[20] |
R. Hartshorne, Generalized divisors and biliaison,, Illinois J. Math., 51 (2007), 83.
|
[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. |
[23] |
G. Kapustka, Projections of del Pezzo surfaces and Calabi-Yau threefolds,, Adv. Geom., 15 (2015), 143.
doi: 10.1515/advgeom-2015-0002. |
[24] |
G. Kapustka and M. Kapustka, A cascade of Calabi-Yau threefolds,, Math. Nachr., 283 (2010), 1795.
doi: 10.1002/mana.200910057. |
[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. |
[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. |
[29] |
Y. Kawamata, Flops connect minimal models,, Publ. Res. Inst. Math. Sci., 44 (2008), 419.
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.
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.
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.
|
[33] |
S. Nollet, Even linkage classes,, Trans. Amer. Math. Soc., 348 (1996), 1137.
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.
doi: 10.1007/BF02567467. |
[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. |
[36] |
C. Peskine and L. Szpiro, Liaison des variétés algébriques. I,, Invent. Math., 26 (1974), 271.
doi: 10.1007/BF01425554. |
[37] |
M. I. Qureshi and B. Szendroi, Calabi-Yau threefolds in weighted flag varieties,, Adv. High Energy Physics, 2012 (2012).
|
[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. |
[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. |
[42] |
Ch. Walter, Pfaffian subschemes,, J. Alg. Geom., 5 (1996), 671.
|
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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[20] |
R. Hartshorne, Generalized divisors and biliaison,, Illinois J. Math., 51 (2007), 83.
|
[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. |
[23] |
G. Kapustka, Projections of del Pezzo surfaces and Calabi-Yau threefolds,, Adv. Geom., 15 (2015), 143.
doi: 10.1515/advgeom-2015-0002. |
[24] |
G. Kapustka and M. Kapustka, A cascade of Calabi-Yau threefolds,, Math. Nachr., 283 (2010), 1795.
doi: 10.1002/mana.200910057. |
[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. |
[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. |
[29] |
Y. Kawamata, Flops connect minimal models,, Publ. Res. Inst. Math. Sci., 44 (2008), 419.
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.
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.
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.
|
[33] |
S. Nollet, Even linkage classes,, Trans. Amer. Math. Soc., 348 (1996), 1137.
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.
doi: 10.1007/BF02567467. |
[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. |
[36] |
C. Peskine and L. Szpiro, Liaison des variétés algébriques. I,, Invent. Math., 26 (1974), 271.
doi: 10.1007/BF01425554. |
[37] |
M. I. Qureshi and B. Szendroi, Calabi-Yau threefolds in weighted flag varieties,, Adv. High Energy Physics, 2012 (2012).
|
[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. |
[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. |
[42] |
Ch. Walter, Pfaffian subschemes,, J. Alg. Geom., 5 (1996), 671.
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