Varieties of planes on intersections of three quadrics
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Varieties of planes on intersections of three quadrics Brendan Hassett1 · Yuri Tschinkel2
Received: 28 December 2019 / Accepted: 8 July 2020 © Springer Nature Switzerland AG 2020
Abstract We study the geometry of spaces of planes on smooth complete intersections of three quadrics, with a view toward rationality questions. Keywords Intersections of three quadrics · Varieties of planes · Cohomology Mathematics Subject Classification 14E08 · 14M10
1 Introduction This note studies the geometry of smooth complete intersections of three quadrics X ⊂ Pn, with a view toward rationality questions over nonclosed fields k. We review what is known over C: • X is irrational for n 6 [1]; • X may be either rational or irrational for n = 7, and the rational ones are dense in moduli [11]; • X is always rational for n 8 [20, Corollary 5.1]. The analysis in higher dimensions relies on the geometry of planes in X . Indeed, when X contains a plane P defined over k then projection from P gives a birational map ∼
π P : X P n−3.
Brendan Hassett was partially supported by NSF Grants 1551514 and 1701659, and the Simons Foundation; Yuri Tschinkel was partially supported by NSF Grant 1601912.
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Yuri Tschinkel [email protected] Brendan Hassett [email protected]
1
Department of Mathematics, Brown University, 151 Thayer Street, Providence, RI 02912, USA
2
Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY 10012, USA
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B. Hassett, Yu. Tschinkel
This leads us to study the variety of planes F2 (X ) ⊂ Gr(3, n + 1). When n 12, the geometry of these varieties gives a quick and uniform proof of rationality over finite fields and function fields of complex curves (see Theorem 2.1). We are therefore interested in the intermediate cases n = 8, 9, 10, 11, and especially in n = 8 and 9. For generic X ⊂ P8, the variety F2 (X ) is finite of degree 1024 (see Proposition 4.1); we explore the geometry of the associated configurations of planes in P8. We then focus most on the case n = 9. Here the variety F2 (X ) is a threefold of general type with complicated geometry—we analyze its numerical invariants. Our original motivation was to understand certain singular complete intersections of three quadrics in P9 associated with universal torsors over degree 4 del Pezzo surfaces fibered in conics over P1 [3]. Conjectures of Colliot-Thélène and Sansuc predict that such torsors are rational when they admit a point, over number fields. Rationality of torsors has significant arithmetic applications, e.g., to proving the uniqueness of the Brauer–Manin obstruction to the Hasse principle and weak approximation. It has geometric consequences as well, e.g., the construction of new examples of nonrational but stably rational threefolds over C. The geometry of smooth intersections, presented here, turned out to be quite rich and interesting on its own. Here is a road map of the paper: Sect. 2 presents uniform proofs of rationality for high-dimensional cases. Section 3 is devoted to determinantal
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