A Fulton-Hansen theorem for almost homogeneous spaces

We prove a generalization of the Fulton-Hansen connectedness theorem, where ${\mathbb P}^n$ is replaced by a normal variety on which an algebraic group acts with a dense orbit.

Our main result is Theorem 9, which generalizes the above results. We replace (PGL n+1 , P n ) by a pair (G, X), where X is a normal quasi-projective variety and G is an algebraic group acting on X with a fixed point and a dense open orbit. We obtain the Fulton-Hansen theorem by applying Theorem 9 to X = P n and G ⊂ PGL n+1 the stabilizer of a point; see Corollary 11. This also implies the original form of the Fulton-Hansen theorem, which begins with a map Z → P n × P n and shows that the preimage of the diagonal is connected, see Corollary 12.
The usual approaches to Fulton-Hansen-type theorems exploit ampleness properties of the tangent bundle T X . We get the strongest result when T X is ample, which holds only for X = P n [Mor79]. For other homogeneous spaces X = G/P , one can develop a theory that measures the failure of ampleness [Som78,Som79,FL81,Fal81,Gol82]. This leads to results that give π 1 -surjectivity if dim Y 1 +dim Y 2 is large enough. For example, let 0 ≤ r ≤ n/2 and let X = Gr(P r , P n ). By [Gol82, Connectedness Theorem, p. 361] the map π 1 (Y 1 ∩Y 2 ) → π 1 (Y 1 )×π 1 (Y 2 ) is surjective provided dim Y 1 + dim Y 2 > dim X + dim Gr(P r−1 , P n−2 ). ( * ) Here dim X is what we expect on the right hand side of ( * ) from the naive dimension count. For r = 0, dim Gr(P r−1 , P n−2 ) = 0, and we recover the Fulton-Hansen theorem. However, for r ≫ 0 the extra term gets quite large. Nonetheless, this bound is optimal, see Example 29.
Our aim is to prove that, for many group actions, the naive dimension estimate dim Y 1 + dim Y 2 > dim X is enough to give π 1 -surjectivity, unless the intersection Y 1 ∩ Y 2 is very degenerate.
Notation 1. We work over an algebraically closed field k unless otherwise specified. Let G be a connected algebraic group acting on a normal k-variety X. We usually assume that the action is faithful.
The action is denoted by (g, x) → gx. If p : Y → X is a morphism then it is convenient to use the shorthand gY → X for g • p : Y → X.
We consider the case when the G-action has a dense open orbit, denoted by X • , and also a fixed point x 0 ∈ X. For any n ≥ 0, G acts linearly on the nth order infinitesimal neighborhood O X,x0 /m n x0 , where (O X,x0 , m x0 ) is the local ring at x 0 and its maximal ideal. If we assume the action of G on X is faithful, this induced action is also faithful for sufficiently large n, and so, in this case, G is a linear algebraic group.
We use π 1 (Z, z 0 ) to denote the topological fundamental group of Z(C) if we are over C and theétale fundamental group of Z otherwise. We will usually include basepoints, but sometimes omit them when it is not important to keep track of them.
The main result is easier to state for normal subvarieties over C. We prove a more general version in Theorem 9, where the characteristic is arbitrary, the stabilizer may be disconnected and the Y i are not necessarily subvarieties.
Theorem 2. Let G be a connected algebraic group acting on a normal variety X defined over C. Assume that the action has a fixed point x 0 ∈ X and a dense open orbit X • ⊂ X such that the stabilizer H ⊂ G of any point in X • is connected. Let Y 1 , Y 2 be irreducible, normal, locally closed subvarieties of X containing x 0 . Assume that that contains x 0 and is not disjoint from X • . Then, Note that the conclusion of Theorem 2 is pretty much the strongest that one can hope for: If Y 1 ∩ Y 2 is disjoint from X • , then g 1 Y 1 ∩ g 2 Y 2 is never in 'general position,' and if an irreducible component does not contain the base point x 0 , then its contribution to the fundamental group is hard to control; see also Examples 28-29. For counterexamples with disconnected stabilizers see Examples 31-32.
We next state and prove Theorem 9, from which Theorem 2 follows. We then derive generalizations of the Fulton Proof of the main theorem.
We start with some preliminary lemmas.
Lemma 3. Let f : U → V be a dominant morphism of irreducible varieties and Proof. It is enough to show that the geometric generic fiber of f is connected. By irreducibility of U , the generic fiber of f is irreducible. The generic fiber may fail to be geometrically irreducible, but all geometric components must pass through the image of σ, and hence the geometric generic fiber is connected.
We also require the following slight variant of [Kle74, Theorem 2], which follows from [Kle74, Theorem 2] and upper semicontinuity of dimension of fibers.
Lemma 4. Let G be a connected algebraic group acting transitively on a normal k-variety X. Let Y 1 and Y 2 be irreducible, normal varieties and p i : Y i → X quasi-finite morphisms. Then there is a dense open subset U ⊂ G × G such for all Notation 5. Let X be a k-variety, x 0 ∈ X a closed point and X • ⊂ X a dense open subset. Let p : Z → X be a quasi-finite morphism. We let Z • denote p −1 (X • ). For z 0 ∈ p −1 (x 0 ), let Z(z 0 ) ⊂ Z denote the union of those irreducible components that contain z 0 and whose images are not disjoint from X • . We say that z 0 is a good limit point of Z if Z(z 0 ) = ∅.
Lemma 6. Let G be a connected algebraic group acting on a normal variety X with a fixed point x 0 ∈ X and a dense open orbit X • with stabilizer H ⊂ G. Let H 0 ⊂ H denote the identity component of H.
Let Y 1 and Y 2 be irreducible, normal varieties and p i : (6.1) (3) I • has at most |H/H 0 | irreducible components.
Let j index those irreducible componentsĪ • (z 0 , j) ⊂Ī • that contain the image of π z0 . Then the following hold.
(5) For any j as above, the image of If H is connected then π −1 (g 1 , g 2 ) is connected for general (g 1 , g 2 ) ∈ G × G.
Remark 6.8. It can happen that g 1 Y 1 × X g 2 Y 2 is always reducible, though this seems to be rare; see Example 25.
Proof. Claims (2) and (4) are clear and (2) implies (3). There is nothing further to prove if there are no good limit points.
The proof of Theorem 9 will follow from the above, together with the following important fact from topology.
Remark 7. Suppose we are given a map f : X → Y of varieties over C. We claim that there is a Zariski open subset U ⊂ Y such that X × Y U → U is locally a trivial bundle in the Euclidean topology. More precisely, for any point p ∈ U an , the analytic space associated to U , there is an analytic open set V ⊂ U an with p ∈ V and a homeomorphism of topological spaces spaces (f an The claim above follows from [GM88, Part I, Theorem 1.7, p. 43] and its proof, although it is not explicitly stated there, so we comment on the details. First, [Hir77, §4, Theorem 1] shows that the pieces in the stratification of [GM88, Part I, Theorem 1.7, p. 43] can be taken to be complex analytic. (Note that the pieces of the stratification are not necessarily real analytic when the map is real analytic, see [GM88,Caution,p. 43].) To prove the existence of the desired Zariski open set U ⊂ Y , one may work with proper compactifications X, Y of X and Y , which exist by Nagata's compactification theorem. We may then use the last sentence of [GM88, Part I, Theorem 1.7, p. 43] to ensure the boundaries of these closures are also unions of strata of the stratifications. Hence, we also obtain stratifications of the open parts X and Y . Then, [GM88, Part I, §1.6, p. 43] explains why the resulting map is topologically trivial over pieces of the stratification. Finally, to show that one of the pieces of the stratification of Y is Zariski open, we note that there is a piece whose complement is a complex analytic proper closed subvariety, hence algebraic by Chow's theorem.

(Fundamental groups of fibers). Let
Case 8.1. k = C. By Remark 7, there is an open, dense subset S • ⊂ S such that q is a topological fiber bundle over S • . Set W • := q −1 (S • ). Note that W • → S • has a section that gets contracted by p. Thus Case 8.2. char k = 0. By the Lefschetz principle we get that (8.1.a) holds for theétale fundamental group.
Remark 8.4 The complication in 8.3 is that S • H depends on H, and, if char k > 0,then ∩ H S • H may be empty. That is, there may not be any closed point s ∈ S for which π 1 W s , σ(s) → π 1 (Z, z 0 ) is surjective; see Example 30.
Combining Lemma 6 with Paragraph 8, can now prove our main theorem.
Theorem 9. Let G be a connected algebraic group acting on a normal k-variety X with a fixed point x 0 ∈ X and a dense open orbit X • with stabilizer H ⊂ G. Let (y 1 , Y 1 ), (y 2 , Y 2 ) be irreducible, normal, pointed varieties and p i : that the natural map, Proof. Fix an index j as in Lemma 6 and consider the diagram whose right hand side is defined in (6.9). Set Next apply the discussions in Paragraph 8 to conclude that is surjective if char k = 0, and surjective on finite quotients if char k > 0.

Fulton-Hansen-type theorems.
We get the following version of the Fulton-Hansen theorem.
Corollary 10 (Fulton-Hansen theorem I). Let G be a connected algebraic group acting 2-transitively on a quasi-projective variety X. Let Z be a normal, irreducible variety with dim Z > dim X and p : Z → X × X quasi-finite. Then, for general (g 1 , g 2 ) ∈ G × G, and ∆ X : X → X × X the diagonal map, (1) Z × (g1,g2)•p,X×X,∆X X is connected and (2) the natural map, is surjective if char k = 0 and surjective on finite quotients if char k > 0.
Proof (assuming Corollary 17). For general (g 1 , g 2 ) ∈ G × G choose z 0 = (z, x) ∈ Z × (g1,g2)•p,X×X,∆X X and define x 0 ∈ X as the point such that (x 0 , x 0 ) = ((g 1 , g 2 )• p)(z). Let G ⊂ G denote the identity component of the stabilizer of x 0 . Note that G acts transitively on X \ {x 0 }. (By Remark 13, the stabilizer in G of x 0 is already connected, though we will not need this fact.) We aim to apply Theorem 9 to G × G acting on X × X with fixed point (x 0 , x 0 ) and dense open orbit (X \ {x 0 }) × (X \ {x 0 }). We now check the hypotheses. As an itinitial reduction, the statement for general translations of both Z and X is equivalent to the analogous one for only translates of Z because We could have proven Corollary 10 using Remark 13 in place of Corollary 17. We opted to use the latter as it leads to a more self-contained proof.
An important special case of Corollary 10 is the the case where Z = Y 1 × Y 2 and p = p 1 × p 2 for p i : Y i → X quasi-finite morphisms. Because this is typically how Corollary 10 is applied, we now restate it in this case.
Corollary 11 (Fulton-Hansen theorem II). Let G be a connected algebraic group acting 2-transitively on a quasi-projective variety X. Let Y 1 , Y 2 be normal, irreducible varieties and p i : Y i → X quasi-finite morphisms. Assume that dim Y 1 + dim Y 2 > dim X. Then, for general (g 1 , g 2 ) ∈ G × G, (1) g 1 Y 1 × X g 2 Y 2 is connected and (2) the natural map, is surjective if char k = 0 and surjective on finite quotients if char k > 0.
If X and Z as in Corollary 10 are proper, then connectedness of Z× (g1,g2)•p,X×X,∆X X for general (g 1 , g 2 ) ∈ G × G implies connectedness for every (g 1 , g 2 ). This is sometimes called the Enriques-Severi-Zariski connectedness principle, proved by combining Stein factorization with Zariski's main theorem.
Thus we recover the original setting of the Fulton-Hansen therem [FH79].
Corollary 12 (Fulton-Hansen theorem III). Let G be a connected algebraic group acting 2-transitively on a projective variety X. Let Z be a normal irreducible proper variety with dim Z > dim X and p : Z → X × X a finite morphism. Then Z × p,X×X,∆X X is connected. In particular, if Z = Y 1 × Y 2 and p = p 1 × p 2 for p i : Y i → X finite morphisms, then Y 1 × p1,X,p2 Y 2 is connected.
Remark 13. The most important example of a 2-transitive action is (PGL n+1 , P n ), The 2-transitive case seems much more general, but in fact there are very few such pairs (G, X). By [Kno83] (PGL n+1 , P n ) is the only pair with X projective. The pairs with X quasi-projective are all of the form (G ⋉ G n a , A n ) where G ⊂ GL n is a product C · G, for C a subgroup of the central G m ⊂ GL n and G is one of the following: (1) n = 1, G = GL 1 , (2) n ≥ 2, G = SL n , (3) n = 2m is even, G = Sp m , (4) n = 6, the characteristic is 2, and G = G 2 .
(Note that G 2 does not have a nontrivial 6-dimensional representation in characteristics = 2.) There is, however, a very long list of pairs (G, X) such that G(R) acts 2transitively on X(R); see [Tit55,Kra03]. So the following variant applies in many more cases.
Corollary 14. Let X be a variety defined over R and G a connected algebraic group acting on it such that the G(R) action on X(R) is 2-transitive. Assume that for x 0 = x 1 ∈ X(R), the stabilizer of the ordered pair (x 0 , x 1 ) is connected (over C). Let Z 1 , Z 2 be irreducible, normal varieties and p i : Z i → X quasi-finite morphisms. Assume that dim Z 1 + dim Z 2 > dim X and the Z i have smooth real points. Then, There are also some non-transitive group actions for which we get a Fulton-Hansen-type result, with obvious exceptions.
Example 15 (Orthogonal group). Let GO q := G m · O q be the group of orthogonal similitudes acting on the n-dimensional vector space V n , where q is a nondegenerate quadratic form. There are 3 orbits, {0}, (q = 0) \ {0}, and the dense open orbit is V n \ (q = 0).
Claim 15.1. Let 0 ∈ Y i ⊂ A n be irreducible, normal, locally closed subvarieties. Assume that Y i ⊂ (q = 0) and dim Y 1 + dim Y 2 > n. Then Proof. Since A n is smooth, Thus we have a good limit point if Since Y i ⊂ (q = 0), we see that dim 0 g i Y i ∩(q = 0) ≤ dim Y i −1. Since (q = 0)\{0} is homogeneous, using Lemma 4 we see that The above arguments show that our approach gives the best results if the orbits of an action are fully understood. In the most extreme case, we have the following classification. The proof relies on some results of [Kno83], that we recall afterwards.
Proposition 16. Let X be an irreducible, normal variety of dimension ≥ 2 over a field k and G a connected linear algebraic group acting on X. Assume that all orbits have dimension either 0 or dim X.
(1) There is at most 1 orbit of dimension 0.
(2) If char k = 0 and there is a 0-dimensional orbit, then X is isomorphic to either an affine or a projective cone over a projective, homogeneous Gvariety Y .
Proof. For (1) we may assume that k is algebraically closed. We may then replace G by its reduction to assume G is smooth. Let P = {p i } be the union of the 0-dimensional orbits and assume P is nonempty. By Proposition 19 there is a projective G-variety Y and a G-equivariant, affine, surjective morphism f : X \ P → Y , whose general fiber is 1-dimensional by Lemma 18. By [Sum74, Theorem 3], there is a normal, G-equivariant compactificationX ⊃ X. Let Z be the normalization of the closure of the graph of f with projections π X and π Y .
Since G acts transitively on X \ P , it also acts transitively on Y , hence E := Z \(X \P ) is a union of G-orbits. Thus every fiber of π Y : Z → Y is a geometrically rational curve and E is a disjoint union of (possibly multiple) sections.
For any p i ∈ P there is an irreducible component E i ⊂ E that is contracted by π X to p i . Let C ⊂ Y be a general curve, X C the normalization of π −1 Y (C) and F i ⊂ X C the preimage of E i . Then X C → C is a P 1 -bundle. Note that a P 1 -bundle over a smooth, projective curve contains at most 1 curve with negative self-intersection, and this curve is a section. Thus P has at most 1 point.
In order to prove (2), we assume from now on that char k = 0. Then π Y : Z → Y is a P 1 -bundle and E consists of 1 or 2 sections.
If E consists of 2 sections, then Z = P Y (O Y + L) for some anti-ample line bundle L on Y . Thus X is the affine cone over (Y, L −1 ).
If E consists of 1 section, then Z = P Y (E) where E is obtained as an extension for some anti-ample line bundle L on Y . Now we again use that char k = 0, hence −K Y is ample and Kodaira's vanishing theorem implies that the extension splits. Thus X is the projective cone over (Y, L −1 ).
Remark 16.3. We believe that (2) is not true if char k = 0, but it should be possible to get a complete description of all cases.
From this we deduce a useful corollary, used in the proof of Corollary 11 above.
Corollary 17. Let X be an irreducible, smooth variety over a field k and G a connected linear algebraic group acting on X with 2 orbits, one of which is a point x ∈ X. Then X \ {x} ∼ = G/H where H is connected.
Proof. This is clear if dim X = 1, so assume that dim X ≥ 2. Let H 0 ⊂ H be the identity component. If H 0 = H then G/H 0 → G/H is ań etale cover, which extends to anétale cover π :X → X by purity. The G-action onX has an open orbitX \ π −1 (x) and π −1 (x) is a union of 0-dimensional orbits. Thus deg π = 1 by Propositon 16.
Proof. Assume that dim X ≥ dim Y + 1. Choose normal compactifications X ⊃ X,Ȳ ⊃ Y and let Z be the normalization of the closure of the graph of f with projections π X and π Y . Note that π X cannot contract a whole fiber of π Y . Thus there is pointȳ ∈Ȳ and an irreducible component Z z ⊂ π −1 Y (ȳ) such that Z z ∩ π −1 X (P ) and Z z \ π −1 X (P ) are both nonempty. Set W = π X (Z z ). Note that P ∩ W = ∅, so W • := W ∩ X is dense in W . Thus W • ∩ (X \ P ) is the fiber of f , hence affine. By Hartogs's theorem, this implies dim W = 1. So the general fiber dimension of f is ≤ 1.
The following group theoretic result is proved, but not stated, on [Kno83,p. 443]. It does not seem to be well known, so we now state it and give a proof in Paragraph 22, following suggestions of Brian Conrad and Zhiwei Yun.
Proposition 19. Suppose X = G/H is a homogeneous space for a smooth connected linear algebraic group G over an algebraically closed field k. Then, there exists a parabolic subgroup H ⊂ P ⊂ G with P/H affine. In particular, there exists a projective variety Y and a surjective and affine map X → Y .
Remark 20. Knop's proof of Proposition 19 in [Kno83] is slightly different from ours in that he produces a specific choice of P H associated to H, whereas our proof merely takes P to be an arbitrary minimal parabolic containing H. It is not clear to us what internal property distinguishes it from the other choices.
An interesting aspect is that P is usually not unique and the set of such parabolics has neither a smallest nor a largest element. For example, for In what follows, for G an algebraic group, we use R u (G) to denote its unipotent radical, the maximal smooth normal connected unipotent subgroup of G.
Lemma 21. Let H be a subgroup of a smooth connected reductive group G over a perfect field k. Then H is either reductive or contained in a k-parabolic subgroup of G.
Proof. The key input in this proof is the fact that any smooth connected unipotent subgroup U of a connected linear algebraic group G over a perfect field k is contained in the unipotent radical of a parabolic k-subgroup P ⊂ G. This follows from a theorem of Bruhat-Tits, see the "Refined Theorem" in [Mat12].
Applying this to our situation, suppose H is not reductive. We wish to show H is contained in a proper parabolic subgroup. By the above fact, there is some P G with R u (H) ⊂ R u (P ). It follows that R u (H) ⊂ H ⊂ N G (R u (H)) ⊂ N G (R u (P )) ⊂ N G (P ) = P . Therefore, H ⊂ P G for P parabolic. Knop's Proposition 19.). We can write X = G/H for H ⊂ G a subgroup. Let P denote a minimal parabolic containing H. We wish to show P/H is affine. Let L := P/R u (P ) and let K denote the image of H in L. Because P was chosen to be a minimal parabolic containing H, K is not contained in any proper parabolic subgroup of L. By Lemma 21, K is reductive. Let U := ker(H → K). Then, U ⊂ R u (P ) so U is unipotent. The quotient R u (P )/U is an affine group scheme acting on P/H with quotient L/K. It follows that P/H is a principal R u (P )/U -bundle over L/K, and so to show P/H is affine, it suffices to show L/K is. This follows from the general claim that a quotient of a connected reductive group by a connected reductive subgroup is affine, see [Bor85, Theorem 1.5].

Other applications.
Example 23 (Projective homogeneous spaces). These are of the form X = G/P where G is a semisimple algebraic group and P ⊂ G a parabolic subgroup. We get a Schubert cell decomposition with a single fixed point x 0 and an open cell X * ⊂ X. Note that X * is a homogeneous space under the unipotent radical U ⊂ P . The stabilizer of the U -action on X * is trivial, hence connected. Thus we get the following.
Claim 23.1. Let Y 1 , Y 2 be irreducible, normal varieties and p i : Y i → X quasifinite morphisms. Assume that there is an irreducible component where Z denotes the closure of Z * . Then, for general (g 1 , g 2 ) ∈ U × U , (hece also for general (g 1 , g 2 ) ∈ G × G), the natural map Note that we could consider instead the P -action, which has a usually larger open orbit X • ⊃ X * . This gives the following variant.
Proof. For dimension reasons there is an irreducible component Z • ⊂ Y 1 × X Y 2 × X X • that contains a good limit point z 0 . Next we use the P -action to see that z 0 is also a good limit point in gZ • × X X * for general g ∈ P . Now we can apply (1).
Results of this type have been considered in [Som79,Fal81,Gol82,Han83]. Our bound (23.2.a) is optimal in some cases, but weaker in several of them. Using the full G-action, as in the above articles, leads to further improvements, but we did not find a natural way to recover the bounds of [Fal81,Gol82] in all cases.
Example 24 (Prehomogeneous vector spaces). A prehomogeneous vector space is a pair (G, V n ) where V n is a k-vector space of dimension n and G ⊂ GL n is a connected subgroup that has a dense orbit W ⊂ V n . See [Kim03] for an introduction and detailed classification.
The infinite series of irreducible ones all have connected generic stabilizers. Using the original Sato-Kimura numbering as in [Kim03], the basic examples are built from (1) (SL n , V n ), (2) (SL n , Sym 2 V n ), (3) (SL n , ∧ 2 V n ), (13) (Sp n , V 2n ), (15) (O n , V n ), These lead to further examples by enlarging the group to contain the scalars or replacing (G, V n ) with (G × SL m , V n ⊗ V m ) for certain values of m.
Most of the sporadic examples either have disconnected generic stabilizer or the connectedness is not known. A nice example is (E 6 · G m , V 27 ), which is no. 27 1 on the list. The connected component of the generic stabilizer is F 4 . Since F 4 has no outer automorphisms, the stabilizer is F 4 , hence connected. See also [WY92,Yuk97,Spr06,KY18] for several other examples. Counterexamples.
Example 25. Start with (GL 3 , A 3 xyz ) and let Y 1 , Y 2 ⊂ A 3 be cones with vertex 0. Then g 1 Y 1 ∩ g 2 Y 2 conists of deg Y 1 · deg Y 2 lines for general g 1 , g 2 ∈ GL 3 . Thus, so long as Y 1 and Y 2 are not both planes, (g 1 Y 1 ∩ g 2 Y 2 )(0) = g 1 Y 1 ∩ g 2 Y 2 is reducible and the origin is a good limit point.
Example 26. Consider (G 3 m , A 3 xyz ) and let let Y 1 , Y 2 ⊂ A 3 be surfaces that contain the z-axis. Usually g 1 Y 1 ∩ g 2 Y 2 is reducible, having both moving and fixed irreducible components for g 1 , g 2 ∈ G 3 m . If Y 1 , Y 2 intersect transversally at the origin then the origin is not a good limit point, even though g 1 Y 1 ∩ g 2 Y 2 may have non-empty intersection with the dense open orbit. For example, this happens for Y 1 = (y = x 2 ), Y 2 = (x = y 2 ).
In any case, the origin is a good limit point if Y 1 ∩Y 2 has an irreducible component Z that passes through the origin but is not contained in a coordinate hyperplane.
Example 27. Suppose k has characteristic 0 (or at least has characteristic not equal to 2). Let X ⊂ P 4 be the projective cone over a smooth quadric surface with vertex x 0 . Let G ⊂ Aut(X) be the identity component. Then, G acts with 2 orbits: {x 0 } and X \ {x 0 }. Let Y 1 be a 2-plane contained in X and containing x 0 . We claim that there is a divisor Y 2 with x 0 ∈ Y 2 ⊂ X such that g 1 Y n 1 × X g 2 Y n 2 is the union of a curve and a point for general (g 1 , g 2 ) ∈ G × G. Thus one preimage of x 0 is a good limit point, the other is not.
The computation is local at x 0 , thus we choose affine coordinates such that X = (xy − uv = 0). We can then choose Y 1 = (x = u = 0). We choose Y 2 to be the complete intersection (xy − uv = 4u − (x − y) 2 = 0). We can eliminate u to get This is an irreducible hypersurface, but shows that it is non-normal along (x = y = 0) and Y n 2 has 2 points over the origin. We will next show that only one of them is a good limit point.
A typical translate of Y 1 by G is (x − c −1 v = y − cu = 0). Add these to the equations x and eliminate u and v to get Here (27.2) defines a curve whose tangent line at the origin is y = 0. With + sign, (27.1) defines a curve whose tangent line at the origin is x = 0. So we get an isolated intersection point at the origin. Finally, satisfies both (27.1) with a + sign and (27.2).
Example 28. Suppose k has characteristic 0. For X = P n × P n , consider G = PGL n+1 × PGL n+1 , Y 1 = P n × C, and Y 2 = P n × H, where C ⊂ P n is a smooth projective curve of positive genus and H ⊂ P n a hypersurface. Then dim Y 1 + dim Y 2 = dim X + n.
For general (g 1 , g 2 ) ∈ G × G, Thus, the intersection is disconnected and its conected components are simply connected. So, they do not contribute to the fundamental group of C and hence π 1 (g 1 Y 1 ∩g 2 Y 2 , z 0 ) → π 1 (Y 1 , y 1 ) has infinite index for any basepoint z 0 ∈ g 1 Y 1 ∩g 2 Y 2 .
Example 29 (Variant of [Han83, Example, p. 634]). Suppose k has characteristic 0. In X = Gr(P 1 , P n ), take G = PGL n+1 and consider Y 1 = (lines through a point) ∼ = P n−1 , Z L = (lines in a hyperplane L ⊂ P n ) ∼ = Gr(P 1 , P n−1 ), and where C ⊂P n is a smooth projective curve of positive genus parametrizing hyperplanes {L c : c ∈ C}. Then The key property is that g 1 Y 1 ∩g 2 Z L is either empty or is isomorphic to P n−2 . Thus, for general (g 1 , g 2 ) ∈ G × G, g 1 Y 1 ∩ g 2 Y 2 is reducible and its connected components are isomorphic to P n−2 . So again they do not contribute to the fundamental group of C.
Example 30. Let k be a field of positive characteristic, take X = P 2 with the standard action of G = PGL 3 . Let Y 1 ⊂ P 2 be a line and let Y 2 ⊂ P 2 be the complement of a line. A general translate of an intersection of Y 1 with Y 2 is isomorphic to A 1 , and the map Examples with disconnected stabilizers.
We conclude by giving examples showing that π 1 (g 1 Y 1 × X g 2 Y 2 , z 0 ) → π 1 (Y 1 , y 1 )× π 1 (Y 2 , y 2 ) may be non-surjective when H is disconnected in the setting of Theorem 9. For the remainder of the paper, we assume k has characteristic 0.
Example 31. If X as in Theorem 9 has nontrivial fundamental group, we can take Y 1 = Y 2 = X, and the resulting map π 1 (X, x 0 ) → π 1 (X, x 0 ) × π 1 (X, x 0 ) will fail to be surjective. By applying Theorem 9, we conclude that H must be disconnected, and have at least π 1 (X, x 0 ) components.
For a concrete example of such a variety, take X to be the moduli space parametrizing unordered pairs of distinct points in P 2 . Then X is homogeneous under PGL 3 . The open orbit in X × X is formed by those ({p 1 , p 2 }, {p ′ 1 , p ′ 2 }) for which no three points are on a line. The stabilizer is Z/2 × Z/2. The space parametrizing ordered pairs of distinct points in P 2 is the universal cover of X. Thus π 1 (X) ∼ = Z/2.
We conclude by giving a somewhat more involved example where H is disconnected, but nevertheless π 1 (X, x 0 ) = 1.
Example 32 (Constructing X and G). Let X be the moduli space of smooth plane conics in P 4 over C. The group GL 5 acts transitively on X via its action on P 4 .
Choose coordinates x 0 , . . . , x 4 and a reference conic C 0 = (x 2 0 + x 2 1 + x 2 3 = x 3 = x 4 = 0). The stablizer of C 0 is the set of matrices ThusG is connected and so isG ∩ SL 5 as shown by the retraction Thus X ∼ = SL 5 /(G ∩ SL 5 ) is simply connected. Since we prefer faithful actions, our group G ⊂ PGL 5 is the image ofG. For Z ⊂ P 4 , let Z denote the linear span of Z in P 4 . Observe that G has a dense orbit X • ⊂ X, consisting of those conics C 1 so that C 0 ∩ C 1 is a point p and neither of the C i contains p.
If C 1 = (x 0 = x 1 = x 2 2 + x 2 3 + x 2 4 = 0) then the stabilizer of the ordered pair (C 0 , C 1 ) is the set of matrices Thus H has 4 connected components, which can be geometrically described as follows. Set p := C 0 ∩ C 1 . From p one can draw two distinct tangent lines to each C i . Let these tangent lines be {T 0 0 , T 1 0 } and {T 0 1 , T 1 1 }. The H action permutes these lines, giving a surjection H ։ Z/2 × Z/2.
Example 33 (Constructing Y 1 and Y 2 ). Continuing Example 32, we next construct the subvarieties Y 1 and Y 2 ⊂ X.
Choose a point q ∈ C 0 \C 0 and let Y 1 := Y 1 (q) ⊂ X be the set of those conics C for which q ∈ C \ C. Then Y 1 is a smooth, locally closed subvariety of dimension 9 in X.
To construct Y 2 , fix 2 distinct points x 1 , x 2 ∈ C 0 and general 2-planes S i such that x i ∈ S i . Let Y 2 := Y 2 (S 1 , S 2 ) ⊂ X be the set of conics C with the following two properties.
(1) We have that c i := C ∩ S i is a single point and c 1 = c 2 .
(2) Let τ (C) ∈ C \ C denote the intersection point of the lines tangent to C at the c i . Then τ (C) ∈ C 0 \ C 0 .
Given distinct c i ∈ S i and p ∈ C 0 \ C 0 , the set of such conics for c i = C ∩ S i and τ (C) = p is a principal G m -bundle whenever c 1 , c 2 , and p are not collinear. Indeed, the conics in P 2 that pass through (1 : 0 : 0), (0 : 1 : 0) which have tangent lines at those points intersecting at (0 : 0 : 1) are precisely the hyperbolas x 1 x 2 = λx 2 0 for λ ∈ G m (C). Thus Y 2 is a smooth, locally closed subvariety of dimension 7 in X.
The intersection Z := Y 1 ∩ Y 2 consists of those conics in Y 2 for which τ (C) = q. Thus Z is a smooth, locally closed subvariety of dimension 5 in X.
Proposition 34. We use the notation of Examples 32-33. For general (g 1 , g 2 ) ∈ G × G, g 1 Y 1 ∩ g 2 Y 2 is irreducible and the map Proof. Note first that the group action sends Y 1 (q) to Y 1 (g 1 q) and Y 2 (S 1 , S 2 ) to Y 2 (g 2 S 1 , g 2 S 2 ). Thus, letting Z = Y 1 ∩ Y 2 , it is enough to show that The index is at most 4 by Theorem 9 because H has 4 connected components, as shown in Example 32. We now show the index is at least 4. Note that Y 1 has a connected degree 2 finiteétale cover Y 1 → Y 1 parametrizing pairs (C, c) where C ∈ Y 1 and c ∈ C is one of the 2 points of C whose tangent line passes through q. Similarly, Y 2 has a connected degree 2 finiteétale cover parametrizing pairs (C, d) where d is one of the two points of C 0 whose tangent line passes through τ (C). Let q 1 and q 2 denote the two points of C 0 whose tangent lines pass through q. Then, the restriction of the cover Y 1 × Y 2 to Z splits into the 4 connected components Z i,j := {(C, C ∩ S i , q j )} ⊂ Z × × Y1×Y2 ( Y 1 × Y 2 ) for 1 ≤ i, j ≤ 2.
Conflict of Interest Statement. On behalf of all authors, the corresponding author states that there is no conflict of interest.