FANO THREEFOLDS WITH LARGE AUTOMORPHISM GROUPS
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1 FANO THREEFOLDS WITH LARGE AUTOMORPHISM GROUPS CONSTANTIN SHRAMOV Let G be a finite group and k a field. We can consider the notion of G-rationality, G- nonrationality, etc., by considering G-equivariant birational maps X Y. This is the analogue, working over an algebraically closed field, of having the action of a Galois group around. 1. Motivating Problem Question 1.1. How can we describe Bir(P n ), the birational automorphism group of P n? Note that we always have PGL n+1 (k) Bir(P n ). For n = 1, one actually has Bir(P 1 ) = PGL 2 (k). For n = 2, we have that Bir(P 2 ) is generated by PGL 3 (k) and a single quadratic transformation ( standard quadratic transformation, which can be chosen to be a birational involution). But starting from n 3, there is no nice set of generators. Namely, any generating set contains elements defined by polynomials of arbitrary large degree. Question 1.2. What can we say about the finite subgroups of Bir(P n )? As a generalization of these questions, we can consider: Question 1.3. What can we say about Bir(X) for a rationally connected variety X? Here is one construction. If G acts birationally on X, then there exists a resolution X X and a (regular) action of G on X so that X X is a G-equivariant birational morphism. Also, one can run a G-Minimal Model Program (G-MMP), i.e., there exists a birational model X of X and a structure of G-Mori fiber space X S. We won t exactly define this notion. In dimension 3, the output X of the G-MMP that starts from a rationally connected variety is one of: A G-Fano variety, i.e. X has terminal Q-factorial singularities, the anticanonical class K X is ample, and rk Cl(X ) G = 1. A G-conic bundle over a rational surface with rk Cl(X /S) G = 1. A G-del Pezzo fibration X P 1 with rk Cl(X /S) G = 1. The G-Mori fiber spaces in dimension 3 are the G-conic bundles and G-del Pezzo fibrations as above. In these cases, there is a reduction to lower dimension using the G-equivariant fibration structure. Namely, to study a group action on X one can use information about possible group actions the base and on a generic fiber, which both have lower dimension. 1
2 There is an expectation that the G-Fano varieties should be bounded. This means that they should be parameterized by finitely many algebraic families. Conjecture 1.4 (Borisov Alexeev Borisov). Fano varieties of a given dimension with ε-log terminal singularities are bounded for any fixed n and ε. In dimension 3 and when the singularities are at most canonical, this conjecture is proved by J. Kollár, Y. Miyaoka, S. Mori, and H. Takagi. 2. Finite subgroups of Bir(P n ) Finite subgroups of Bir(P 1 ) = PGL 2 (k) are classically known. A (mostly complete) classification of finite subgroups of Bir(P 2 C ) was obtained by I. Dolgachev and V. Iskovskikh. There are some partial results in dimension 3. Theorem 2.1 (Yu. Prokhorov). The finite nonabelian simple subgroups of Bir(P 3 ) are: A 5, A 6, PSL 2 (F 7 ), A 7, SL 2 (F 8 ), PSp 4 (F 3 ) Note that A 5 acts on P 1, that A 6 and PSL 2 (F 7 ) do not act on P 1 but act on P 2, while the rest do not act (even by birational transformations) on P 2. This is one of the first illustrations of the above classification principle: the groups A 5, A 6, and PSL 2 (F 7 ) are the groups coming from lower dimensions, while the remaining three groups are new three-dimensional objects related to the class of G-Fano threefolds. Question 2.2. What general properties do finite subgroups of Bir(P n ) have? What about finite subgroups of Bir(X) for X a rationally connected variety? Note that over a given field k, we have GL n (k) Bir(P n ), which certainly contributes finite subgroups. In particular, we cannot hope to have a more explicit answer to Question 2.2 than in the case of the group GL n (k). There are certain kinds of boundedness results for finite subgroups of GL n (k). Theorem 2.3 (H. Minkowski). If k is a number field then there exists a constant B(n, k) 1 such that for every finite subgroup G GL n (k), we have that G B(n, k). Say that GL n (k) has the Minkowski property. Theorem 2.4 (C. Jordan). There exists J(n) 1 such that for every finite subgroup G GL n (C), there exists a normal abelian subgroup A G with [G : A] J(n). Say that GL n (C) has the Jordan property. It is easy to derive from Theorem 2.4 that Jordan property holds for the group GL n (k) for any field k of zero characteristic. Note also that it fails in finite characteristic. Indeed, if k = F p, then Aut(P 1 k ) contains subgroups isomorphic to PSL 2(F q ) for any q = p k. J.-P. Serre was the first to observe that some birational automorphism groups enjoy the above properties. Theorem 2.5 (J.-P. Serre). If k is a number field then the group Bir(P 2 ) has the Minkowski property. If k = C, then Bir(P 2 ) has the Jordan property. 2
3 In fact, C can be replaced by any field of characteristic zero in the statement. On the other hand, after replacing P 2 by an arbitrary surface, the analog of Theorem 2.5 may fail. Example 2.6 (Yu. Zarhin). If S = E P 1 with E an elliptic curve then Bir(S) does not have the Jordan property. Still in dimension 2, the Jordan property is completely understood. Theorem 2.7 (V. Popov). Let S be a surface that is not birational to E P 1 with E an elliptic curve. Then the group Bir(S) has the Jordan property. There is a partial generalization of Theorems 2.5 and 2.7 to higher dimensions. Theorem 2.8 (Yu. Prokhorov C. Shramov, assuming the BAB Conjecture). We have the following: (1) If k = C, then the group Bir(P n ) (and also Bir(X) for X rationally connected) has the Jordan property. In fact, can get a bound for J(3) around 10 9 that works for every rationally connected threefold X. (2) If k is a number field, then Bir(P n ) has the Minkowski property. 3. Conclusion There is a correspondence between G-actions on rational varieties of dimension n and embeddings G Bir(P n ). Two embeddings G 1 Bir(P n ) and G 2 Bir(P n ) are conjugate if and only if there is a G-equivariant birational map X 1 X 2 between the corresponding rational varieties with G-action. Slogan 3.1. Varieties with an action of a large group lead to interesting geometry. 4. Examples of interesting geometry 4.1. Considering the 1-parameter family of S 6 -invariant quartic threefolds (from Cheltsov s talk), we can give a proof of the irrationality of the generic element in the family by looking at the S 6 action on the intermediate Jacobian The natural actions of S 5 and A 5 on P 4 leave the hyperplane P 3 defined by i x i = 0 invariant. Consider the family of invariant quartic surfaces in this P 3 : i x4 i t ( i x2 i ) 2 = 0 for t C. Consider the double cover X P 3 branched over a quartic in this family. C. Voisin proved that X, as long as it is smooth, is a nonrational Fano threefold. But we are mostly interested in rational varieties with G-action! So we should consider singular members of the family. One can check that the values of t corresponding to singular quartics are 13, 1, 7, and 1 (except for the non-reduced case t = that we ignore) The resulting threefolds are X 5, X 10, X 10, and X 15, having 5, 10, 10 and 15 singular points, 3
4 respectively. In each of these four cases the singularities of X are nodes, and they form a single A 5 -orbit. There is a sequence of results about rationality of singular quartic double solids. Theorem 4.1. Concerning singular quartic double solids we have: (1) (A. Beauville, via intermediate Jacobian) A nodal quartic double solid is nonrational if it has a unique singular point. (2) (O. Debarre, via intermediate Jacobian) A nodal quartic double solid is nonrational if it has at most 5 singular points, subject to a genericity assumption. (3) (R. Varley, via intermediate Jacobian) There exist nonrational quartic double solids with 6 nodes. (4) (M. Artin, D. Mumford, via torsion in third cohomology) There exist quartic double solids with 10 nodes that are not stably rational. (5) (C. Voisin, via degeneration method and Artin Mumford example) The very general quartic double solid with at most 7 nodes is not stably rational. Theorem 4.2 (I. Cheltsov V. Przyjalkowski C. Shramov, via intermediate Jacobian). If a nodal quartic double solid has at most 6 singularities then it is nonrational. If a nodal quartic double solid has at least 11 singularities then it is rational. In conclusion: X 5 is nonrational and X 15 is rational. The quartic double solids X 10 and X 10 are particular cases of the Artin Mumford type examples whose rationality is already obstructed by H 3 ( X, Z) tors. (Here and below we denote by X the resolution of singularities of X obtained by blowing up its nodes.) 4.3. Consider the central Spin extension 2.A 5 of A 5 by {±1}. There exists a 4-dimensional representation U 4 of 2.A 5, so the group A 5 acts on P 3. Computing invariants, we see that there is a pencil P of A 5 -invariant quartics in P 3. One can check that there are exactly four singular quartics in this family: two with non-isolated singularities along twisted cubic curves, and two with 10 nodes. As before, construct double covers X P 3 branched over members of this family. Since we are mostly interested in rational threefolds that can arise as results of the (equivariant) Minimal Model Program, we ignore the smooth quartic double solids and those that have non-isolated singularities. Therefore, the most interesting cases here are the two 10-nodal varieties. One can also note that the action of the group A 5 on P 3 can be extended to an action of S 5. Unlike the previous example, the action of S 5 on P is non-trivial, and in particular the group S 5 /A 5 = Z/2 interchanges the two 10-nodal quartics in P. Therefore, up to isomorphism in our setup we have only one quartic double solid X 10 that is interesting from the birational point of view. For the threefold X = X 10 all currently known birational invariants vanish! The intermediate Jacobian J( X) is trivial, as well as the torsion H 3 ( X, Z) tors. Furthermore, X is not birationally rigid, and that there are no differential forms in reductions to characteristic p. It is highly possible that even the universal (non-)triviality of the Chow group cannot be applied to obstruct rationality of X. 4
5 Expectation 4.3 (S. Gorchinsky). The Chow motive of X is of Lefschetz type (over Z), and the group CH 0 ( X) is universally trivial. A possible approach to rationality of X is given by the following Conjecture 4.4 (V. Shokurov). If V T is a nice ( standard ) three-dimensional conic bundle with discriminant T such that 2K T + 0 then V is not rational. The quartic double solid X can be birationally transformed to a standard conic bundle over a blow up T of 9 points on P 2, with discriminant curve 2K T. Thus Conjecture 4.4 predicts that X is non-rational. 5
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