Hypersurfaces that are not stably rational

By B. Totaro. 2015

135 citations recorded by the source corpus.

A fundamental problem of algebraic geometry is to determine which varieties are rational, that is, isomorphic to projective space after removing lower-dimensional subvarieties from both sides. In particular, we want to know which smooth hypersurfaces in projective space are rational. An easy case is that smooth complex hypersurfaces of degree at least n + 2 in P are not covered by rational curves and hence are not rational. By far the most general result on rationality of hypersurfaces is Kollar’s theorem that for d at least 2d(n + 3)/3e, a very general complex hypersurface of degree d in P is not ruled and therefore not rational [13, Theorem V.5.14]. Very little is known about rationality in lower degrees, except for cubic 3-folds and quintic 4-folds [4], [19, Chapter 3]. A rational variety is also stably rational, meaning that some product of the variety with projective space is rational. Many techniques for proving non-rationality give no information about stable rationality. Voisin made a breakthrough in 2013 by showing that a very general quartic double solid (a double cover of P ramified over a quartic surface) is not stably rational [23]. These Fano 3-folds were known to be non-rational over the complex numbers, but stable rationality was an open question. Voisin’s method was to show that the Chow group of zero-cycles is not universally trivial (that is, the Chow group becomes nontrivial over some extension of the base field), by degenerating the variety to a nodal 3-fold which has a resolution of singularities X with nonzero torsion in H3(X,Z). Colliot-Thelene and Pirutka simplified and generalized Voisin’s degeneration method. They deduced that very general quartic 3-folds are not stably rational [6]. This was striking, in that non-rationality of smooth quartic 3-folds was the original triumph of Iskovskikh-Manin’s work on birational rigidity, while stable rationality of these varieties was unknown [11]. Beauville applied the method to prove that very general sextic double solids, quartic double 4-folds, and quartic double 5-folds are not stably rational [1, 2]. In this paper, we show that a wide class of hypersurfaces in all dimensions are not stably rational. Namely, for all d ≥ 2d(n + 2)/3e and n ≥ 3, a very general complex hypersurface of degree d in P is not stably rational (Theorem 2.1). The theorem covers all the degrees in which Kollar proved non-rationality. In fact, we get a bit more, since Kollar assumed d ≥ 2d(n + 3)/3e. For example, very general quartic 4-folds are not stably rational, whereas it was not even known whether these varieties are rational. The method applies to some smooth hypersurfaces over Q in each even degree. Section 3 gives some examples over Q which are not stably rational over C. The idea is that the most powerful results are obtained by degenerating a smooth complex variety to a singular variety in positive characteristic, rather than to a sin-

Subfields: Applied Mathematics, Geometry and Topology

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Citations in the corpus

On the rationality problem for quadric bundles (2017, methodology): “t resolution ofthose can be avoided. The ﬁrst step is the following small but crucial improvement of the original technique in [Voi2] and [CTP1]; the proof is inspired by [Voi2, CTP1], Totaro’s paper [T2] and the original arguments of Bloch and Srinivas. Proposition 26. Let R be a discrete valuation”
The motivic nearby fiber and degeneration of stable rationality (2017, background): “The latter specialization result has been used to solve some long-standing questions about stable irrationality of certain very general cyclic coverings, high degree hypersurfaces in projective spaces and conic bundles [5,10,15,37,40].”
Measures of irrationality for hypersurfaces of large degree (2015, background): “have been introduced to study questions of rationality, such as Koll´ar’s passage to characteristic p > 0 [18], the Chow-theoretic ideas used by Voisin [33], and the combination of these by Totaro [31]. There have also been ideas that rationality could be detected in the geometry of derived categ”
Stable rationality and conic bundles (2015, methodology): “lies of double covers by Beauville [11, 12]; using a mixed characteristic version, introduced in [17], and an idea of Koll´ar [23], Totaro proved that many hypersurfaces in Pn are not stably rational [40]. The problem of rationality of conic bundles has a long history; see, e.g., [9] [22, Conjecture”