Birational self-maps of threefolds of (un)-bounded genus or gonality
We study the complexity of birational self-maps of a projective threefold by looking at the birational type of surfaces contracted. These surfaces are birational to the product of the projective line with a smooth projective curve. We prove that the genus of the curves occuring is unbounded if and only if is birational to a conic bundle or a fibration into cubic surfaces. Similarly, we prove that the gonality of the curves is unbounded if and only if is birational to a conic bundle.
1991 Mathematics Subject Classification:14E07, 14J45
Let be a smooth projective complex algebraic variety. One way of studying the complexity of the geometry of elements of the group of birational self-maps of consists of studying the complexity of the irreducible hypersurfaces contracted by elements of . If is a curve, then , so there is nothing to be said. If is a surface, every irreducible curve contracted by an element of is rational. The case of threefolds is then the first interesting to study in this context.
If , then every irreducible surface contracted by a birational transformation is birational to for some smooth projective curve . There are then two natural integers that one can associate to in this case, namely its genus and its gonality (the minimal degree of a dominant morphism ). We then define the genus (respectively the gonality ) of to be the maximum of the genera (respectively of the gonalities ) of the smooth projective curves such that a hypersurface of contracted by is birational to .
This notion of genus of elements of was already defined in [Frumkin] with another definition, which is in fact equivalent to ours by [Lamy2014]. Moreover, for each , the set of elements of of genus form a subgroup, so we get a natural filtration on , studied in [Frumkin, Lamy2014]. This naturally raises the question of finding the threefolds for which this filtration is infinite, namely the threefolds for which the genus of is unbounded (see [Lamy2014, Question 11]). Analogously, we get a filtration given by the gonality. Of course, the gonality being bounded if the genus is bounded, the unboundedness of the gonality is stronger than the unboundedness of the genus.
Note that the boundedness of the genus (respectively of the gonality) of elements of is a birational invariant. Our main result (Theorem 1.1) describes the threefolds having this property.
Recall that a variety is a conic bundle (respectively a del Pezzo fibration of degree ) if admits a morphism such that the generic fibre is a conic (respectively a del Pezzo surface of degree ) over the field of rational functions of . If the conic has a rational point (or equivalently the conic bundle has a rational section), then it is isomorphic to and in this case we say that the conic bundle is trivial.
Let be a smooth projective complex algebraic threefold.
If is birational to a conic bundle, the gonality and the genus of the elements are both unbounded.
If is birational to a del Pezzo fibration of degree , the genus of the elements of is unbounded.
If is not birational to a conic bundle, the gonality of the elements of is bounded.
If is not birational to a conic bundle and to a del Pezzo fibration of degree , then both the genus and the gonality of elements of are bounded.
We can generalise the above notions to higher dimensions. If is a smooth projective variety of dimension , every irreducible hypersurface contracted by an element of is birational to for some variety of dimension . When , and there are then many ways to study the complexity of this variety. One possibility is the covering gonality of , namely the smallest integer such that through a general point of there is an irreducible curve birational to a smooth curve of gonality . As before, we say that the covering gonality of is bounded if the covering gonality of the irreducible varieties such that a hypersurface contracted by an element is birational to is bounded. Since the covering gonality of a smooth curve is its gonality, this notion is the same as the gonality defined before, in the case of threefolds. As in dimension , this is again a birational invariant.
In Corollary LABEL:corollary:solid-any-dimensions, we prove that that if is a solid Fano variety (see [AhmadinezhadOkada, Definition 1.4]), then the covering gonality of elements of are bounded by a constant that depends only on . In particular, the covering gonality of birational selfmaps of birationally rigid Fano varieties of dimension (see [CheltsovPark, Definition 1.1.2]) are bounded by a constant that depends only on .
In Proposition 2.4, we prove that if is a trivial conic bundle of any dimension , then the covering gonality of the elements of
is unbounded. This raises the following two questions:
Let be a projective variety of dimension and let be a non-trivial conic bundle. Is the covering gonality of elements of unbounded?
Let be a projective variety of dimension that is not birational to a conic bundle. Is the covering gonality of elements of bounded?
A rough idea of the proof of Theorem 1.1 is as follows. Since the boundedness of the genus and gonality is a birational invariant, we can run the MMP and replace with a birational model (with terminal singularities) such that either is nef or has a Mori fibre space structure . In the former case any birational self-map is a pseudo-automorphism [Hanamura1987, Lemma 3.4] and so the genus and gonality are bounded in this case. If is a Mori fibre space, then any birational map is a composition of Sarkisov links (see Sect. 3). If a link involves a Mori fibre space which is (generically) a conic bundle, then we apply an explicit construction of Sect. 2 to get unboundedness (and thus obtain Theorem 1.1(i)). If a link involves Mori fibre spaces and , then we use the boundedness result for Fano threefolds [KMMT-2000] (see also [BirkarS]). This result is also used to prove the assertions 1.1(iii)-(iv) (see Lemma LABEL:Lemm:g0c0moregeneral). The unboundeness of the genus for del Pezzo fibrations of degree (Theorem 1.1(ii)) is obtained by finding -sections of large genus and applying Bertini involutions associated to these curves, see Section LABEL:Sec:DP3 for the detailed construction.
This research was supported through the programme “Research in Pairs” by the Mathematisches Forschungsinstitut Oberwolfach in 2018. Jérémy Blanc acknowledges support by the Swiss National Science Foundation Grant “Birational transformations of threefolds” 200020_178807. Ivan Cheltsov and Yuri Prokhorov were partially supported by the Royal Society grant No. IESR1180205, and the Russian Academic Excellence Project 5-100. Alexander Duncan was partially supported by National Security Agency grant H98230-16-1-0309. We thank Serge Cantat, Stéphane Lamy and Egor Yasinsky for interesting discussions during the preparation of this text.
2. The case of conic bundles
Every conic bundle is square birational equivalent to a conic bundle that can be seen as a conic in a (Zariski locally trivial) -bundle. Using a rational section of the -bundle, one can do the following construction:
Let be a conic bundle over an irreducible normal variety and let be its generic fibre. Then is a conic over the function field . The anicanonical linear system defines an embedding . Fix a -point . The projection from is a double cover. Let be the corresponding Galois involution. It induces a fibrewise birational involution .
Suppose now that our conic bundle is embedded into a -bundle and suppose that we are given a section whose image is not contained in . This section defines a point and therefore defines an involution as above.
[BlancLamy15, Lemma 15] If, in the above notation, is an irreducible hypersurface that is not contained in the discriminant locus of and such that , the hypersurface of is contracted by onto the codimension subset .
Let be a conic bundle over an irreducible normal variety , given by the restriction of a -bundle . Let be an irreducible hypersurface such that the restriction of gives a trivial conic bundle . Then, there exists an involution
that contracts the hypersurface onto the image of a rational section of .
Since the restriction of gives a trivial conic bundle , there is a rational section . We then extend this section to a rational section whose image is not contained in . This can be done locally, on a open subset where is a trivial -bundle. Lemma 2.2 provides an involution that contracts onto the image of . ∎
Let be a projective variety of dimension , let be a conic bundle and let us assume that either is trivial (admits a rational section) or that . Then, the covering gonality (and the genus if of elements of is unbounded.
We can assume that is the restriction of a -bundle .