Construction of Calabi-Yau $3$-folds in $\mathbb P^6$
Author:
Fabio Tonoli
Journal:
J. Algebraic Geom. 13 (2004), 209-232
DOI:
https://doi.org/10.1090/S1056-3911-03-00371-0
Published electronically:
October 15, 2003
MathSciNet review:
2047696
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Abstract |
References |
Additional Information
Abstract: We give examples of smooth Calabi-Yau $3$-folds in $\mathbb P^6$ of low degree, up to the first difficult case, which occurs in degree 17. In this case we show the existence of three unirational components of their Hilbert scheme, all having the same dimension $23+48=71$. The constructions are based on the Pfaffian complex, choosing an appropriate vector bundle starting from their cohomology table. This translates into studying the possible structures of their Hartshorne-Rao modules. We also give a criterium to check the smoothness of $3$-folds in $\mathbb P^6$.
[ACGH85]ACGH E. Arbarello, M. Cornalba, P. Griffiths, and J. Harris, Geometry of Algebraic Curves, vol. I, Springer Grundlehren, 267:xvi+386, 1985.
[Bat94]Ba V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in algebraic tori, J. Alg. Geom., 3:493–535, 1994.
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[Bog78]BOG F.A. Bogomolov, Hamiltonian Kähler manifolds, Sov. Math. Doklady, 19:1462–1465, 1978.
[Cat97]CAT97 F. Catanese, Homological algebra and algebraic surfaces, A.M.S. Proc. Symp. Pure Math., 62:3–56, 1997.
[CdlOK95]COK P. Candelas, X. de la Ossa, and S. Katz, Mirror symmetry for Calabi-Yau hypersurfaces in weighted ${\mathbb {P}^{4}}$ and extensions of Landau-Ginzburg theory, Nucl. Phys., B 450:267–290, 1995.
[Cyn99]CYN2 S. Cynk, Double coverings of octic arrangements with isolated singularities, Adv. Theor. Math. Phys., 3:217–225, 1999.
[CK99]CK D. Cox and S. Katz, Mirror symmetry and algebraic geometry, AMS, Mathematical Surveys and Monographs, 68:xxii+469, 1999.
[CS98]CYN1 S. Cynk and T. Szemberg, Double covers and Calabi-Yau varieties, In Singularities Symposium–Lojasiewicz 70, B. Jakubczyk et al. editors, Banach Canter Publ., 44:93–101, 1998.
[DES93]DES W. Decker, L. Ein, and F.-O. Schreyer, Construction of surfaces in $\mathbb P^4$. J. Alg. Geom., 2:185–237, 1993.
[DS00]DS W. Decker and F.-O. Schreyer, Non-general type surfaces in $\mathbb P^4$: Some remarks on bounds and constructions, J. Symbolic Comput., 29:545–585, 2000.
[ELMS89]ELMS D. Eisenbud, H. Lange, G. Martens, and F.-O. Schreyer, The Clifford dimension of a projective curve, Compos. Math., 72:173–204, 1989.
[EPW01]EPW99 D. Eisenbud, S. Popescu, and C. Walter, Lagrangian subbundles and codimension 3 subcanonical subschemes, Duke Math. J., 107:427–467, 2001.
[Gre99]GR M.L. Green, The Eisenbud-Koh-Stillman conjecture on linear syzygies, Invent. Math., 136:411–418, 1999.
[GS99]MAC D. Grayson and M. Stillman, Macaulay 2 – a software system for algebraic geometry and commutative algebra, Available at http://www.math.uiuc.edu/Macaulay2, 1999.
[IP99]IP V.A. Iskovskikh and Y.G. Prokhorov, Fano Varieties, Algebraic Geometry V, A.N. Parshin and I.R. Shafarevich eds., 183–190, 1992; Encyclopaedia of Mathematical Sciences, 47, 1999.
[Kaw92]KAW Y. Kawamata, Unobstructed deformations – a remark on a paper of Z. Ran, J. Alg. Geom, 1:183–190, 1992; erratum ibid. 6:803–804, 1997.
[LT82]LT A. Lanteri and C. Turrini, Some formulas concerning nonsingular algebraic varieties embedded in some ambient variety, Atti Accad. Sc. Torino, 116:463–474, 1982.
[Ran92]RAN Z. Ran, Deformations of manifolds with torsion or negative canonical bundle, J. Alg. Geom, 1:279–291, 1992.
[ST02]ST F.-O. Schreyer and F. Tonoli, Needles in a haystack: special varieties via small fields, In Computations in algebraic geometry with Macaulay2, D. Eisenbud et al. editors, Springer, Algorithms Comput. Math., 8:251–279 (2002).
[Tia87]TIAN G. Tian, Smoothness of the univeral deformation space of compact Calabi-Yau manifolds and its Peterson-Weil metric, In Mathematical aspects of string theory, S. T. Yau editor, World Scientific Press, Singapore, 629–646, 1987.
[Wal96]WAL C. Walter, Pfaffian subschemes, J. Alg. Geom., 5:671–704, 1996.
[ACGH85]ACGH E. Arbarello, M. Cornalba, P. Griffiths, and J. Harris, Geometry of Algebraic Curves, vol. I, Springer Grundlehren, 267:xvi+386, 1985.
[Bat94]Ba V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in algebraic tori, J. Alg. Geom., 3:493–535, 1994.
[BE77]BE77 D. A. Buchsbaum and D. Eisenbud, Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3, Am. J. Math., 99:447–485, 1977.
[Bog78]BOG F.A. Bogomolov, Hamiltonian Kähler manifolds, Sov. Math. Doklady, 19:1462–1465, 1978.
[Cat97]CAT97 F. Catanese, Homological algebra and algebraic surfaces, A.M.S. Proc. Symp. Pure Math., 62:3–56, 1997.
[CdlOK95]COK P. Candelas, X. de la Ossa, and S. Katz, Mirror symmetry for Calabi-Yau hypersurfaces in weighted ${\mathbb {P}^{4}}$ and extensions of Landau-Ginzburg theory, Nucl. Phys., B 450:267–290, 1995.
[Cyn99]CYN2 S. Cynk, Double coverings of octic arrangements with isolated singularities, Adv. Theor. Math. Phys., 3:217–225, 1999.
[CK99]CK D. Cox and S. Katz, Mirror symmetry and algebraic geometry, AMS, Mathematical Surveys and Monographs, 68:xxii+469, 1999.
[CS98]CYN1 S. Cynk and T. Szemberg, Double covers and Calabi-Yau varieties, In Singularities Symposium–Lojasiewicz 70, B. Jakubczyk et al. editors, Banach Canter Publ., 44:93–101, 1998.
[DES93]DES W. Decker, L. Ein, and F.-O. Schreyer, Construction of surfaces in $\mathbb P^4$. J. Alg. Geom., 2:185–237, 1993.
[DS00]DS W. Decker and F.-O. Schreyer, Non-general type surfaces in $\mathbb P^4$: Some remarks on bounds and constructions, J. Symbolic Comput., 29:545–585, 2000.
[ELMS89]ELMS D. Eisenbud, H. Lange, G. Martens, and F.-O. Schreyer, The Clifford dimension of a projective curve, Compos. Math., 72:173–204, 1989.
[EPW01]EPW99 D. Eisenbud, S. Popescu, and C. Walter, Lagrangian subbundles and codimension 3 subcanonical subschemes, Duke Math. J., 107:427–467, 2001.
[Gre99]GR M.L. Green, The Eisenbud-Koh-Stillman conjecture on linear syzygies, Invent. Math., 136:411–418, 1999.
[GS99]MAC D. Grayson and M. Stillman, Macaulay 2 – a software system for algebraic geometry and commutative algebra, Available at http://www.math.uiuc.edu/Macaulay2, 1999.
[IP99]IP V.A. Iskovskikh and Y.G. Prokhorov, Fano Varieties, Algebraic Geometry V, A.N. Parshin and I.R. Shafarevich eds., 183–190, 1992; Encyclopaedia of Mathematical Sciences, 47, 1999.
[Kaw92]KAW Y. Kawamata, Unobstructed deformations – a remark on a paper of Z. Ran, J. Alg. Geom, 1:183–190, 1992; erratum ibid. 6:803–804, 1997.
[LT82]LT A. Lanteri and C. Turrini, Some formulas concerning nonsingular algebraic varieties embedded in some ambient variety, Atti Accad. Sc. Torino, 116:463–474, 1982.
[Ran92]RAN Z. Ran, Deformations of manifolds with torsion or negative canonical bundle, J. Alg. Geom, 1:279–291, 1992.
[ST02]ST F.-O. Schreyer and F. Tonoli, Needles in a haystack: special varieties via small fields, In Computations in algebraic geometry with Macaulay2, D. Eisenbud et al. editors, Springer, Algorithms Comput. Math., 8:251–279 (2002).
[Tia87]TIAN G. Tian, Smoothness of the univeral deformation space of compact Calabi-Yau manifolds and its Peterson-Weil metric, In Mathematical aspects of string theory, S. T. Yau editor, World Scientific Press, Singapore, 629–646, 1987.
[Wal96]WAL C. Walter, Pfaffian subschemes, J. Alg. Geom., 5:671–704, 1996.
Additional Information
Fabio Tonoli
Affiliation:
Mathematisches Institut der Universität Bayreuth, Universitätstrasse, D-95440 Bayreuth Deutschland
Email:
fabio.tonoli@uni-bayreuth.de
Received by editor(s):
May 31, 2001
Published electronically:
October 15, 2003
