Double covers of quadratic degeneracy and Lagrangian intersection loci
- PDF / 494,816 Bytes
- 35 Pages / 439.37 x 666.142 pts Page_size
- 10 Downloads / 199 Views
Mathematische Annalen
Double covers of quadratic degeneracy and Lagrangian intersection loci Olivier Debarre1 · Alexander Kuznetsov2,3,4 Received: 16 May 2018 / Revised: 9 May 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract We explain a general construction of double covers of quadratic degeneracy loci and Lagrangian intersection loci based on reflexive sheaves. We relate the double covers of quadratic degeneracy loci to the Stein factorizations of the relative Hilbert schemes of linear spaces of the corresponding quadric fibrations. We give a criterion for these double covers to be nonsingular. These results are an extension of O’Grady’s construction of double covers of EPW sextics and provide an alternate construction of Iliev–Kapustka–Kapustka–Ranestad’s EPW cubes. Mathematics Subject Classification 14E20 · 14C20 · 14J35 · 14J40 · 14J70 · 14M99
Communicated by Ngaiming Mok. Alexander Kuznetsov was partially supported by the HSE University Basic Research Program, Russian Academic Excellence Project “5–100” and by the Program of the Presidium of the Russian Academy of Sciences 01 “Fundamental Mathematics and its Applications” under Grant PRAS-18-01.
B
Alexander Kuznetsov [email protected] Olivier Debarre [email protected]
1
Département de Mathématiques et Applications, Université Paris-Diderot, PSL Research University, CNRS, École normale supérieure, 45 rue d’Ulm, 75230 Paris Cedex 05, France
2
Algebraic Geometry Section, Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkin Str., Moscow 119991, Russia
3
Interdisciplinary Scientific Center J.-V. Poncelet, Independent University of Moscow, Moscow, Russia
4
Laboratory of Algebraic Geometry, HSE University, Moscow, Russia
123
O. Debarre, A. Kuznetsov
1 Introduction When double coverings are mentioned, one usually thinks of double coverings branched over divisors. These are very classical objects in algebraic geometry. Let D be an effective Cartier divisor on a scheme S such that the line bundle O S (D) is a square in the Picard group of S, that is, O S (−D) M ⊗2 for some line bundle M . The double covering S of S branched over D is defined as the relative spectrum S := Spec S (O S ⊕ M ), where the algebra structure on O S ⊕ M is such that the multiplication on M is given by the composition ·s D M ⊗ M O S (−D) −−→ O S (here s D is a section of O S (D) with divisor D). This construction depends on the choice of the line bundle M (there may be several choices if the Picard group has nontrivial 2-torsion) and of the section s D (the various choices form a torsor over the group H 0 (S, O S× ) of invertible functions on S, and two choices provide isomorphic double coverings if and only if their ratio is the square of an invertible function). The construction also works for the zero divisor D and produces an étale double covering of S (which is nontrivial if the square root M of O S (−D) = O S is nontrivial, and with the same ambiguity for the choice). There is an extension of this construction which
Data Loading...
