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AFFINE THREEFOLDS ADMITTING Ga-ACTIONS R.V. Gurjar, M. Koras

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AFFINE THREEFOLDS ADMITTING Ga-ACTIONS R.V. Gurjar, M. Koras AFFINE THREEFOLDS ADMITTING Ga-ACTIONS R.V. GURJAR, M. KORAS, K. MASUDA, M. MIYANISHI AND P. RUSSELL R.V. Gurjar, M. Koras, K. Masuda, M. Miyanishi, P. Russell had the wonderful opportunity to meet at BIRS during August 23-30, 2015, for the BIRS event Unipotent Geometry 15frg175. Some portion of our manuscript, including the Introduction of our manuscript below, describes some of the results we proved during our stay. Since we had only one week for discussions the Theorems 2.4 and 2.5 can be consid- ered as the most complete results we proved during our stay. These results were previously not known and we believe that they will be appreciated by other experts interested in Ga actions. We also con- structed several examples of Ga action on smooth affine 3-folds. These examples are significant since there is a general paucity of such exam- ples in the literature. Some of the results in the manuscript were proved during a Research In Pairs program at Oberwolfach, Germany, in August 2014. All the collaborators in this FRG, except Kayo Masuda, were also part of that program. We still need to work out some details of proofs of results in other parts of the manuscript. Date:September25,2015. 2000 Mathematics Subject Classification. Primary: 14R20; Secondary: 14R25. Key words and phrases. affine threefold, Ga-action, quotient surface, singularity. The second author is supported by Polish Grant MNiSW. The third author is supported by Grant-in-Aid for Scientific Research (B), No. 24340006, JSPS. The fourth author is supported by NSERC, Canada. 1 2R.V.GURJAR,M.KORAS,K.MASUDA,M.MIYANISHIANDP.RUSSELL Abstract. Affine varieties of dimension greater than two can be explored their structures with the help of fibrations by the affine line or plane and quotient morphisms by Ga-actions. We consider Ga-actions on affine threefolds and discuss the structure and the singularities of the quotient surface as well as the singular fibers 4 of the quotient morphism. Relative Ga and Gm-actions on A or on an affine 4-fold having A3-fibrations over A1 are discussed when the actions leave one variable invariant or the fibration invariant. Contents Introduction 2 1. The quotient space of smooth affine threefolds by Ga-actions 4 2. Singular fibers of the quotient morphism by a Ga-action 10 4 1 3. Relative fixpoint free Ga-actions on A over A 21 4 1 4. Relative effective Gm-actions on A over A 26 5. Danielewski surfaces of non-hypersurface type 31 6. Some results about A1-andP1-fibrations 39 7. Twisted additive groupschemes andtheiractions 42 References 48 Introduction Let X be a smooth affine threefold defined over an algebraically closed field k of characteristic zero. If X has a nontrivial Ga-action, there exists the quotient morphism q : X Y = X//G and Y has → a dimension two. Since the Ga-action is defined by a locally nilpotent k-derivation δ on the coordinate ring A of X, the quotient surface is defined by Y =SpecB with B =Kerδ.Thenthereexistsanopen 1 set U of Y such that U =SpecB[1/u]withu B and q− (U) = 1 ∈ ∼ U A . This implies that general fibers of q are Ga-orbits, but q has, in× general, singular fibers outside the open set U.Infact,topologicalor algebro-geometric properties of X are greatly affected by the singular fibers of q and the algebraic quotient surface Y ,andvice versa.In Theorem 1.1, we consider the case where X is, in addition, factorial and given two independent Ga-actions. Here we say that two Ga-actions on X are independent if the orbits by two Ga-actions have independent tangential directions at some point of X. Then it is shown that the quotient surface Y by one Ga-action is isomorphic to the affine plane A2 or an affine hypersurface x2 +y3 +z5 = 0 which is isomorphic to the quotient A2/Γ, where Γis the binary icosahedral subgroup of SL(2,k). AFFINE THREEFOLDS AND Ga-ACTIONS 3 Furthermore, Y is isomorphic to A2 if and only if X is simply connected. If one weakens the factoriality condition to the Q-factoriality, there are 2 examples for which Y ∼= A /Γ, where Γis now an arbitrary finite subgroup of SL(2,k). Some basic description of singular fibers of the quotient morphism q : X Y is given in [23]. Among other things, it is notable that if → asingularfiberF0 of q is one-dimensional, then every irreducible com- ponent of F0 is a contractible curve (k = C), and it is an interesting problem to ask whether it is always isomorphic to the affine line A1.In Theorem 2.1, it is shown that if an irreducible component has multi- plicity one, i.e., it is reduced, then the component is isomorphic to A1. If the Ga-action is given by a homogeneous locally nilpotent deriva- tion on the polynomial ring k[x, y, z] with a positive grading (quasi- homogeneous case), then the quotient morphism q : A3 A2,where → X = A3 and Y = A2 has only the affine line as the (unique) irreducible component of the (unique) singular fiber. This is shown by a detailed analysis of the singular fiber in Lemma 2.2 and Theorem 2.3. In the general case with only assumption that Y be smooth, every irreducible fiber component is isomorphic to A1 in Theorem 2.5. The singular lo- cus Sing(q)ofthequotientmorphismq is also observed. It is shown in Theorem 2.8 that if X and Y are smooth and the Ga-action is analyti- cally reduced everywhere on X then the union of multiple components of q coincide with the fixedpoint locus XGa . In the third section, we consider some of well-established results about Ga-actions acting on the affine space defined over k in the rel- ative setting where we add one free variable to the polynomial ring and the Ga-action keeps the additional variable invariant. We consider 3 Kaliman’s theorem on a free Ga-action on A and Rentschler’s theo- 2 rem on the normalization of a Ga-action on A in the relative setting (see Kaliman’s Theorem and Example 3.5). Kaliman [27] has recently 4 proved that a fixed point free, proper Ga-action on A is a translation after a suitable change of variables. We analyzed this result from our point of view. In the fourth section, we consider a Gm-action on an affine 4-fold 1 3 fibered over A whose every fiber is isomorphic to A so that Gm acts on each fiber and the parameter space invariant. Theorem 4.1 shows that the 4-fold is actually trivial, i.e., isomorphic to A3 Y and the × Gm-action is diagonalized. In the fifth section, we treat a smooth affine surface V (m, 1) with 1 1 1 1 m 1whichhasanA -fibration ρ : V (m, 1) A such that ρ− (A ) = 1≥ 1 1 → ∗ ∼ A A and ρ− (0) consists of two reduced irreducible components ∗ × isomorphic to A1.Ifm =1,V (1, 1) is isomorphic to the hypersurface 4R.V.GURJAR,M.KORAS,K.MASUDA,M.MIYANISHIANDP.RUSSELL xy = z2 1, which is one of the Danielewski surfaces. However, if m − ≥ 2, V (m, 1) is not a hypersurface. In fact V (m, 1) = V (m#, 1) if m = '∼ 1 ' m#. Meanwhile, we have an isomorphism V (m, 1) A = V (m#, 1) × ∼ × A1.Hencethesesurfacesprovideanewkindofcounterexamplesto the cancellation problem (see Theorem 5.3). The surface V (m, 1) is constructed as the quotient variety of a Gm-action with Gm acting on the hypersurface threefold X(m, 1) = xy zmu =1 . In the sixth section, various generalizations{ − of the} Nori exact se- quence of the fundamental groups of a fibration f : X Y with connected fibers such that each fiber contains at least one→reduced ir- reducible component (see [43]). This exact sequence is very effective when we use topological arguments for affine varieties. Hence its gener- alizations have been considered, and the section gathers together those generalizations. Applications of generalized results are also observed. In the seventh section, we defined an -twisted additive group scheme L Ga, over a scheme Y for an invertible sheaf on Y .Thisenablesus L L to treat A1-fibrations (even defined over a complete variety) from the viewpoint of the additive group scheme action. The idea is originally due to Dubouloz [11]. We denote by R[n] apolynomialringoveraringR in n variables whenever we do not specify variables of the polynomial ring. A locally nilpotent derivation is aften abbreviated as lnd. This article is an outcome of the joint research conducted as a RIP program of Mathematisches Forschungsinstitut Oberwolfach (MFO) during the period August 4 to 15, 2014 and as a workshop entitled unipotent geometry at the BanffInternational Research Station for Mathematical Innovation and Discovery (BIRS) during the period Au- gust 23 to 30, 2015. We are very grateful to MFO and BIRS for their splendid research environments and warm hospitality. 1. The quotient space of smooth affine threefolds by Ga-actions We are interested in a smooth, factorial, affine threefold X with Ga- actions and its quotient surface Y by one of the Ga-actions. Central problems are if Y is smooth or what kind of singularity it admits pro- vided Y is singular. Given two Ga-actions on X,wesaythatthetwo actions are independent if two respective Ga-orbits passing a point P of X have independent tangential directions. If X admits two indepen- dent Ga-actions, then we have the following result.
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