On the Cynk-Hulek criterion for crepant resolutions of double covers. (English) Zbl 1509.14033
Summary: A collection \(A=\{D_1,\dots,D_n\}\) of divisors on a smooth variety \(X\) is an arrangement if the intersection of every subset of \(A\) is smooth. We show that, if \(X\) is defined over a field of characteristic not equal to 2, a double cover of \(X\) ramified on an arrangement has a crepant resolution under additional hypotheses. Namely, we assume that all intersection components that change the canonical divisor when blown up are splayed, a property of the tangent spaces of the components first studied by E. Faber [Publ. Res. Inst. Math. Sci. 49, No. 3, 393–412 (2013; Zbl 1277.32032)]. This strengthens a result of S. Cynk and K. Hulek [Can. Math. Bull. 50, No. 4, 486–503 (2007; Zbl 1141.14009)], which requires a stronger hypothesis on the intersection components. Further, we study the singular subscheme of the union of the divisors in \(A\) and prove that it has a primary decomposition where the primary components are supported on exactly the subvarieties which are blown up in the course of constructing the crepant resolution of the double cover.
MSC:
| 14E20 | Coverings in algebraic geometry |
| 14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |
| 14B05 | Singularities in algebraic geometry |
| 13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |
References:
| [1] | Atiyah, M. F.; Macdonald, I. G., Introduction to commutative algebra, Addison-Wesley Series in Mathematics (2016), Westview Press: Westview Press Boulder, CO · Zbl 1351.13002 |
| [2] | Batyrev, Victor, Birational Calabi-Yau n-folds have equal Betti numbers, (Hulek, K.; Reid, M.; Peters, C.; Catanese, F., New Trends in Algebraic Geometry, vol. 264 (1999), Cambridge University Press), 1-12 · Zbl 0955.14028 |
| [3] | Bourbaki, Nicolas, Commutative Algebra. Chapters 1-7, Elements of Mathematics (Berlin) (1998), Springer-Verlag: Springer-Verlag Berlin, Translated from the French, Reprint of the 1989 English translation · Zbl 0902.13001 |
| [4] | Bridgeland, T.; King, A.; Reid, M., The McKay correspondence as an equivalence of derived categories, J. Am. Math. Soc., 14, 535-554 (2001) · Zbl 0966.14028 |
| [5] | Cynk, S.; Hulek, K., Higher-dimensional modular Calabi-Yau manifolds, Can. Math. Bull., 50, 4, 486-503 (2007) · Zbl 1141.14009 |
| [6] | Cynk, S.; Schütt, M.; van Straten, D., Hilbert modularity of some double octic Calabi-Yau threefolds, J. Number Theory, 210, 313-332 (2020) · Zbl 1445.14059 |
| [7] | Cynk, S.; Szemberg, T., Double covers of \(\mathbb{P}^3\) and Calabi-Yau varieties, Banach Cent. Publ., 44, 93-101 (1998) · Zbl 0915.14025 |
| [8] | Faber, Eleonore, Towards transversality of singular varieties: splayed divisors, Publ. Res. Inst. Math. Sci., 49, 3, 393-412 (2013) · Zbl 1277.32032 |
| [9] | Grothendieck, Alexander, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Publ. Math. Inst. Hautes Études Sci., 32, 361 (1967) · Zbl 0153.22301 |
| [10] | Jacobson, Nathan, Lectures in Abstract Algebra, III. Theory of Fields and Galois Theory, Graduate Texts in Mathematics, vol. 32 (1975), Springer-Verlag: Springer-Verlag New York-Heidelberg, Second corrected printing · Zbl 0124.27002 |
| [11] | Kawamata, Yujiro, d-equivalence and k-equivalence, J. Differ. Geom., 61, 1, 147-171 (2002) · Zbl 1056.14021 |
| [12] | Logan, Adam, Modularity of two double covers of \(\mathbb{P}^5\) branched along 12 hyperplanes (2018) |
| [13] | Matsuki, Kenji, Introduction to the Mori Program, Universitext (2002), Springer-Verlag: Springer-Verlag Berlin · Zbl 0988.14007 |
| [14] | Meyer, Christian, Modular Calabi-Yau Threefolds, Fields Institute Monographs, vol. 22 (2005), American Mathematical Society · Zbl 1096.14032 |
| [15] | Mumford, David, The Red Book of Varieties and Schemes, Lecture Notes in Mathematics, vol. 1358 (1999), Springer-Verlag: Springer-Verlag Berlin, Includes the Michigan Lectures (1974) on curves and their Jacobians, with contributions by Enrico Arbarello · Zbl 0945.14001 |
| [16] | Orlik, P.; Terao, H., Arrangements of Hyperplanes, Grundlehren der Mathematischen Wissenschaften, vol. 300 (1992), Springer-Verlag: Springer-Verlag Berlin · Zbl 0757.55001 |
| [17] | Raynaud, Michel, Anneaux Locaux Henséliens, Lecture Notes in Mathematics, vol. 169 (1970), Springer-Verlag: Springer-Verlag Berlin-New York · Zbl 0203.05102 |
| [18] | Zariski, O.; Samuel, P., Commutative Algebra. Vol. II, The University Series in Higher Mathematics (1960), D. Van Nostrand Co., Inc.: D. Van Nostrand Co., Inc. Princeton, N. J.-Toronto-London-New York · Zbl 0121.27801 |
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