×

zbMATH — the first resource for mathematics

Hilbert schemes and toric degenerations for low degree Fano threefolds. (English) Zbl 1445.14006
Smooth Fano threefolds \(V\) with very ample anticanonical divisor \(-K_V\) lead to points \([V]\in\mathcal H_d\) in the Hilbert scheme of degree \(d:=-K_V^3\) subvarieties of \(\mathbb{P}(|-K_V|)\). These point lie on unique irreducible components. Maximal families of those Fanos, i.e. the set of those components of \(\mathcal H_d\) are completely classified. For small degree \(d\), these families can be distinguished by \((d,b_2,b_3)\) where \(b_i\) denote the \(i\)-th Betti number of \(V\). The dimensions of the corresponding components, i.e. \(h^0(\mathcal N_{V})\), are known, too.
On the other hand, toric Fano threefolds \(X\) of degree \(d\) with Gorenstein singularities correspond to three-dimensional reflexive polytopes. They are classified, too. They do also provide points \([X]\in\mathcal H_d\) which might, however, sit in several components. Presenting this incidence for \(d\leq 12\) is the main point of the paper. Thus, it provides a complete overview about toric degenerations of the above \(V\) which is important from the view point of mirror symmetry.
One of the tools is the identification of certain Stanley-Reisner schemes \(S\) wich provide points \([S]\in\mathcal H_d\), too. These are the most degenerate versions. Degenerations from \(X\) to \(S\) are understood by unimodular triangulations of the reflexive polytopes. Moreover, deformation theory of \(X\) is used because the components of \(\mathcal H_d\) containing \([X]\) are identified by the components of the tangent cone of \(\mathcal H_d\) in \([X]\). The equations of the latter are obtained via explicit computeralgebra computations.

MSC:
14C05 Parametrization (Chow and Hilbert schemes)
14J10 Families, moduli, classification: algebraic theory
14J45 Fano varieties
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Altmann K. and Christophersen J. A., Cotangent cohomology of Stanley-Reisner rings, Manuscripta Math. 115 (2004), no. 3, 361-378. · Zbl 1071.13008
[2] Altmann K. and Christophersen J. A., Deforming Stanley-Reisner schemes, Math. Ann. 348 (2010), 513-537. · Zbl 1203.13022
[3] Brown G. and Kasprzyk A., The graded ring database, .
[4] Christophersen J. A. and Ilten N. O., Degenerations to unobstructed Stanley-Reisner schemes, Math. Z., to appear. · Zbl 1341.14015
[5] Christophersen J. A. and Ilten N. O., Hilbert schemes and toric degenerations of low degree Fano threefolds, supplementary material 2014, .
[6] Coates T., Cort A., Galkin S., Golyshev V. and Kasprzyk A., Mirror symmetry and Fano manifolds, preprint 2012, .
[7] Coates T., Cort A., Galkin S. and Kasprzyk A., Quantum periods for 3-dimensional Fano manifolds, preprint 2013, . · Zbl 1348.14105
[8] Cox D. A., Little J. B. and Schenck H. K., Toric varieties, Grad. Stud. Math. 124, American Mathematical Society, Providence 2011. · Zbl 1223.14001
[9] Dicks D., Surfaces with pg=3, k²=4 and extension-deformation theory, PhD thesis, Warwick 1988.
[10] Galkin S., Small toric degenerations of Fano 3-folds, preprint 2007, .
[11] Grayson D. R. and Stillman M. E., Macaulay2. A software system for research in algebraic geometry, .
[12] Ilten N. O., Versal deformations and local Hilbert schemes, J. Software Algebra Geom. 3 (2012), 12-16. · Zbl 1311.14011
[13] Ilten N. O., Lewis J. and Przyjalkowski V., Toric degenerations of Fano threefolds giving weak Landau-Ginzburg models, J. Algebra 374 (2013), 104-121. · Zbl 1270.14020
[14] Ishida M.-N. and Oda T., Torus embeddings and tangent complexes, Tôhoku Math. J. (2) 33 (1981), no. 3, 337-381. · Zbl 0456.14005
[15] Iskovskih V. A., Fano threefolds. II, Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), no. 3, 506-549. · Zbl 0407.14016
[16] Iskovskih V. A., Anticanonical models of three-dimensional algebraic varieties (in Russian), Current problems in mathematics. Vol. 12, VINITI, Moscow (1979), 59-157, 239.
[17] Kleppe J. O., Deformations of graded algebras, Math. Scand. 45 (1979), no. 2, 205-231. · Zbl 0436.14004
[18] Kollár J., Complex algebraic geometry, IAS/Park City Math. Ser. 3, American Mathematical Society, Providence 1997.
[19] Kreuzer M. and Skarke H., Classification of reflexive polyhedra in three dimensions, Adv. Theor. Math. Phys. 2 (1998), no. 4, 853-871. · Zbl 0934.52006
[20] Mori S. and Mukai S., Classification of Fano 3-folds with \({B_{2}≥ 2}\), Manuscripta Math. 36 (1981/82), no. 2, 147-162. · Zbl 0478.14033
[21] Mori S. and Mukai S., Classification of Fano 3-folds with \({B_{2}≥ 2}\). I, Algebraic and topological theories (Kinosaki 1984), Kinokuniya, Tokyo (1986), 496-545. · Zbl 0800.14021
[22] Mukai S., Plane quartics and Fano threefolds of genus twelve, The Fano conference (Torino 2002), Università di Torino, Turin (2004), 563-572. · Zbl 1068.14050
[23] Przhiyalkovskiĭ V. V., Chel’tsov I. A. and Shramov K. A., Hyperelliptic and trigonal Fano threefolds, Izv. Ross. Akad. Nauk Ser. Mat. 69 (2005), no. 2, 145-204. · Zbl 1081.14059
[24] Przyjalkowski V., Weak Landau-Ginzburg models for smooth Fano threefolds, preprint 2009, . · Zbl 1281.14033
[25] Rambau J., TOPCOM: Triangulations of point configurations and oriented matroids, Mathematical software—ICMS 2002 (Beijing 2002), World Scientific, Singapore (2002), 330-340. · Zbl 1057.68150
[26] Sernesi E., Deformations of algebraic schemes, Grundlehren Math. Wiss. 334, Springer-Verlag, Berlin 2006.
[27] Stanley R. P., Combinatorics and commutative algebra, Progr. Math. 41, Birkhäuser-Verlag, Boston 1983. · Zbl 0537.13009
[28] Stevens J., Rolling factors deformations and extensions of canonical curves, Doc. Math. 6 (2001), 185-226, electronic. · Zbl 0989.14008
[29] Sturmfels B., Gröbner bases and convex polytopes, Univ. Lecture Ser. 8, American Mathematical Society, Providence 1996.
[30] The 4ti2 Team , 4ti2—a software package for algebraic, geometric and combinatorial problems on linear spaces, .
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.