aa r X i v : . [ m a t h . AG ] N ov RATIONAL MORPHISMS BETWEEN QUASILINEARHYPERSURFACES
STEPHEN SCULLY
Abstract.
We prove analogues of several well-known results concerning rational mor-phisms between quadrics for the class of so-called quasilinear p -hypersurfaces. Thesehypersurfaces are nowhere smooth over the base field, so many of the geometric methodswhich have been successfully applied to the study of projective homogeneous varietiesover fields cannot be used. We are therefore forced to take an alternative approach, whichis partly facilitated by the appearance of several non-traditional features in the studyof these objects from an algebraic perspective. Our main results were previously knownfor the class of quasilinear quadrics. We provide new proofs here, because the originalproofs do not immediately generalise for quasilinear hypersurfaces of higher degree. Introduction
Considerable progress has been made in recent years towards understanding the con-ditions under which a rational morphism X Y can exist between smooth projectivevarieties X and Y over a field. We can mention, for example, M. Rost’s degree formulasand their generalisations (see [Mer00], [Mer03], [LM07]). The results obtained have provento be particularly successful in their application to the study of projective homogeneousvarieties over fields. In the case where X and Y are quadrics, several applications of thisapproach were already known quite some time ago. Indeed, it was already apparent fromthe foundational works of A. Pfister in the 1960’s that the study of rational morphismsbetween quadrics has many important applications to the algebraic theory of quadraticforms; for example, to the study of the structure of the Witt ring, or to the constructionof fields which exhibit certain arithmetic properties pertaining to quadratic forms. On theother hand, the rich structure theory of the Witt ring unearthed by Pfister permitted thestudy of rational morphisms between quadrics from a entirely algebraic point of view. Inthe subsequent decades, the algebraic methods were developed and applied intensively tothis area of research. Two of the highlights of this approach are the following results, dueto D. Hoffmann and O. Izhboldin respectively: Theorem 1.1 ([Hof95], Theorem 1) . Let X and Y be anisotropic projective quadrics overa field k of characteristic different from 2. If there exists n ≥ such that dim( Y ) ≤ n − < dim( X ) , then there are no rational morphisms X Y . Theorem 1.2 ([Izh00], Theorem 0.2) . Let X and Y be anisotropic projective quadricsover a field k of characteristic different from 2 with dim( Y ) = 2 n − ≤ dim( X ) for some n ≥ . If there exists a rational morphism X Y , then there also exists a rationalmorphism Y X . Of course, phenomena of this sort are more readily explained nowadays in light ofthe recent advances in the geometric theory (for example, both the above results can be
Mathematics Subject Classification.
Key words and phrases.
Quasilinear forms, Quasilinear hypersurfaces, Rational morphisms, Birationalgeometry. deduced from Rost’s degree formula). Perhaps the most general result available concerningrational morphisms between quadrics is due to N. Karpenko and A. Merkurjev. Beforewe state it, let us recall a couple of standard definitions. If X is an anisotropic projectivequadric over a field k of characteristic different from 2, then the first Witt index of X ,denoted i ( X ), is the largest positive integer i such that there exists a degree 1 cycle ofdimension i − X k ( X ) , where k ( X ) is the function field of X . The Izhboldindimension of X , denoted dim Izh ( X ), is then defined to be the integer dim( X ) − i ( X ) + 1.The theorem of Karpenko and Merkurjev now says: Theorem 1.3 ([KM03], Theorem 4.1) . Let X and Y be anisotropic projective quadricsover a field k of characteristic different from 2. If there exists a rational morphism X Y , then (1) dim Izh ( X ) ≤ dim Izh ( Y ) . (2) dim Izh ( X ) = dim Izh ( Y ) if and only if there exists a rational morphism Y X . If one knows something additional about the first Witt indices of the quadrics involved,then one can start to produce more explicit examples. For instance, Theorems 1.1 and 1.2can both be recovered from Theorem 1.3 modulo the observation that anisotropic quadricsof dimension 2 n − n ≥
0) have first Witt index equal to 1. Although the latterfact was originally proved by D. Hoffmann as a corollary of Theorem 1.1, there are severalalternative proofs which are completely independent of Theorem 1.1 (for example, seethe paper [Kar03] of N. Karpenko, where all possible values of the first Witt index aredetermined).All the above statements include the assumption that the characteristic of the basefield is different from 2 (and historically, this is the form in which they were first stated).But it turns out that all these results can be extended to allow for fields of characteristic2, even when the quadrics involved are not smooth over the base field (see the articles[HL04], [HL06] and [Tot08]). This includes the extreme case of quasilinear quadrics, whichhave no smooth points at all (quasilinear quadrics are those quadrics which are definedby the diagonal part of a symmetric bilinear form over a field of characteristic 2). In fact,several problems which remain open for smooth quadrics have recently been settled forquasilinear quadrics. In particular, we want to mention the results of B. Totaro on thebirational geometry of quasilinear quadrics. In the article [Tot08], Totaro gives a positiveanswer to the “Quadratic Zariski Problem” for quasilinear quadrics: If X and Y areanisotropic quasilinear quadrics of the same dimension over a field such that there existrational morphisms from X to Y and from Y to X , then X and Y are birational. Theproof of this result uses the quasilinear analogue of Theorem 1.3 (due to Totaro) togetherwith the following “Ruledness Theorem”, proved by Totaro in the same article: If X is ananisotropic quasilinear quadric over a field k , then X is birational to X ′ × P i ( X ) − k for anysubquadric X ′ ⊂ X of codimension i ( X ) − i ( X ) is defined in thesame way as it is for smooth quadrics). For non-quasilinear quadrics, the correspondingproblems are still wide open, even in the smooth case.The goal of the present article is to prove analogues of all the above results for the class ofso-called quasilinear p -hypersurfaces, which generalises the class of quasilinear quadrics.More precisely, a quasilinear p-hypersurface is the projective hypersurface defined by adiagonal form of degree p over a field of characteristic p >
0. Forms of the latter type arecalled quasilinear p-forms . For evident reasons, the geometry of quasilinear hypersurfacesis only interesting over non-perfect fields. Due in part to the absence of any smoothpoints on these varieties, one does not have access to the general geometric methods which
ATIONAL MORPHISMS BETWEEN QUASILINEAR HYPERSURFACES 3 have proved to be very successful in the study of projective homogeneous varieties overfields. On the other hand, there are several unusual aspects of the theory of quasilinear p -forms which suggest that a more algebraic approach is feasible. The main result of thepaper is Theorem 5.12. This is an analogue of the main result of [KM03]. The latterresult, due to N. Karpenko and A. Merkurjev, immediately implies their Theorem 1.3as a corollary. Our result should also imply the analogue of Theorem 1.3 for arbitraryquasilinear p -hypersurfaces, but there is a technical obstruction which we have currentlyonly managed to overcome for quasilinear quadrics and cubics. This issue is discussedin §
4. Nevertheless, Theorem 5.12 is already sufficient for several interesting applicationswhich hold for all primes p . For example, in § p , but dueto the technical issue discussed in §
4, we are only able to present partial results on theanalogue of the “Quadratic Zariski Problem”. Although all these results were previouslyknown in the case p = 2, the original proofs do not immediately generalise for largerprimes p , and we are forced to take a different approach. In particular, we provide newproofs of all the above results for the class of quasilinear quadrics.The motivation for this paper is the article [Hof04] by D. Hoffmann, where an extensivestudy of quasilinear p -forms was carried out from an algebraic point of view. It was shownthere that several aspects of the classical theory of quadratic forms over fields carry over infull generality to this new situation. We recall several notions and results from Hoffmann’spaper in § § F will be a non-perfect field of characteristic p > F willdenote a fixed algebraic closure of F . By a scheme we mean a scheme of finite type overa field. By a variety , we mean an integral scheme. A scheme will be called complete if itis proper over the base field. If X is a scheme defined over a field k , the residue field ata point x ∈ X will be denoted by k ( x ). If X is a variety, then we will write k ( X ) for thefunction field of X . If L/k is any field extension, X L will denote the scheme X × k Spec L .Finally, we implicitly assume that all morphisms and rational morphisms of schemes aredefined relative to the given base field.2. Quasilinear p -forms In this section we introduce quasilinear p -forms and discuss some of their basic propertiesand invariants. Everything in this section can be found in the article [Hof04]. Since theproofs of the statements we need are all very short, we include them for the reader’sconvenience. Definition 2.1.
Let V be a finite dimensional F -vector space. A quasilinear p-form (wewill often simply say form for simplicity) on V is a map ϕ : V → F satisfying(1) ϕ ( λv ) = λ p ϕ ( v ) for all λ ∈ F and all v ∈ V .(2) ϕ ( v + w ) = ϕ ( v ) + ϕ ( w ) for all v, w ∈ V .We will say that ϕ is a quasilinear p-form over F if ϕ is a quasilinear p -form on somefinite dimensional F -vector space. In this case, we denote the underlying space by V ϕ .The dimension of ϕ is the dimension of V ϕ , denoted dim( ϕ ). A morphism ϕ → ψ of formsover F will be an F -vector space morphism V ϕ → V ψ which carries ϕ to ψ . If ϕ and ψ areisomorphic, we will write ϕ ≃ ψ . We will say that ϕ is proportional to ψ if there exists STEPHEN SCULLY λ ∈ F ∗ such that ϕ ≃ λψ , where λψ is the form on V ψ defined by v λψ ( v ). If ϕ is aquasilinear p -form over F , then a subform ψ of ϕ is the restriction of ϕ to a subspace of V ϕ .We write ψ ⊂ ϕ . The direct sum ϕ ⊕ ψ and tensor product ϕ ⊗ ψ of forms ϕ, ψ over F aredefined in the obvious way. If L/F is a field extension and ϕ a quasilinear p -form over F ,we write ϕ L for the form over L obtained by the extension of scalars. If ϕ is a quasilinear p -form over F , a vector v ∈ V ϕ is called isotropic if ϕ ( v ) = 0. We say that the form ϕ is isotropic if V ϕ contains a nonzero isotropic vector; ϕ is called anisotropic otherwise. Bycondition (2) in the definition, the subset of isotropic vectors in V ϕ is actually a subspace,and ϕ is isotropic if and only if it has nonzero dimension. This additivity condition alsoimplies that ϕ is “diagonalized” in every basis of V ϕ ; that is, of the form a x p + ... + a n x pn .We will use the notation h a , ..., a n i to denote the quasilinear p -form a x p + ... + a n x pn onthe F -vector space F n in its standard basis. If ϕ is a quasilinear p -form over F , and L/F is a field extension, we define the value set D L ( ϕ ) := { ϕ L ( v ) | v ∈ V ϕ ⊗ F L } . Since ϕ is additive, D L ( ϕ ) is actually an L p -vector subspace of L . Clearly it is finitedimensional of dimension ≤ dim( ϕ ), and we have D L ( ϕ ) = D L ( ϕ L ).The classification of quasilinear p -forms over F is given by the following statement: Proposition 2.2 ([Hof04], Proposition 2.6) . Let ϕ be a quasilinear p -form over F ofdimension n , and let a , ..., a m be a basis for D F ( ϕ ) over F p . Then m ≤ n and there isan isomorphism of forms over F : ϕ ≃ h a , ..., a m i ⊕ h , ..., | {z } n − m i . Proof.
Let W ⊆ V ϕ be the subspace of isotropic vectors. For each 1 ≤ i ≤ m , let v i ∈ V ϕ be such that ϕ ( v i ) = a i , and let U be the subspace of V ϕ spanned by the v i . Since the a i arelinearly independent over F p , the v i are linearly independent over F . Therefore, in orderto prove the statement, it suffices to show that V ϕ = U ⊕ W . Clearly U ∩ W = { } . Onthe other hand, given any v ∈ V ϕ , we can find λ i ∈ F such that ϕ ( v ) = P mi =1 λ pi a i . Then ϕ ( v ) = ϕ ( P mi =1 λ i v i ), so that v − P mi =1 λ i v i is an isotropic vector. Therefore v ∈ U + W ,and the statement is proved. (cid:3) The isomorphism class of a quasilinear p -form ϕ over F is therefore determined by twoinvariants: the dimension of the subspace of isotropic vectors, and the F p -vector space D F ( ϕ ). In particular, the theory is essentially vacuous over perfect fields (i.e. when F = F p ), which is why we assume F to be non-perfect. In the case of anisotropic forms,we get: Corollary 2.3.
Let ϕ and ψ be anisotropic quasilinear p -forms over F . Then ψ is iso-morphic to a subform of ϕ if and only if D F ( ψ ) ⊂ D F ( ϕ ) . If ϕ is an arbitrary quasilinear p -form over F , it follows from Proposition 2.2. that thereis a unique (up to isomorphism) anisotropic subform ϕ an ⊂ ϕ such that ϕ = ϕ an ⊕h , ..., i .The form ϕ an is called the anisotropic part of ϕ . The integer i ( ϕ ) := dim( ϕ ) − dim( ϕ an )is called the defect index of ϕ . By the proof of Proposition 2.2, i ( ϕ ) is nothing elsebut the dimension of the subspace of isotropic vectors in V ϕ . We write ϕ ∼ ψ whenever ϕ an ≃ ψ an for p -forms ϕ , ψ over F . This is obviously an equivalence relation on theset of isomorphism classes of quasilinear p -forms over F . The following consequence ofProposition 2.2 will be used several times in the sequel: ATIONAL MORPHISMS BETWEEN QUASILINEAR HYPERSURFACES 5
Lemma 2.4.
Let ϕ = h a , ..., a n i be a quasilinear p -form over F , and let L/F be anyfield extension. Then there is a subset { a j , ..., a j m } ⊂ { a , ..., a n } such that ( ϕ L ) an ≃h a j , ..., a j m i .Proof. Since D L ( ϕ ) = L p · a + ... + L p · a n , we can find a subset { a j , ..., a j m } ⊂ { a , ..., a n } which constitutes a basis of D L ( ϕ ) over L p . The statement then follows from Proposition2.2. (cid:3) We now introduce a special class of forms. In the theory of quadratic forms over fieldsof characteristic = 2, an important role is played by the class of so-called Pfister forms (namely, tensor products of binary forms which represent 1). The set of isomorphismclasses of n -fold Pfister forms over such a field k is in bijection with the set of pure symbolsin the torsion Milnor K -group K Mn ( k ) / K Mn ( k ). The projective quadric defined by an n -fold Pfister form hh a , ..., a n ii is a splitting variety in the sense that for any extension L/k ,the quadric has an L -rational point if and only if the symbol { a , ..., a n } is divisible by2 in K Mn ( L ). Let us now recall the equal characteristic analogue of the norm residueisomorphism theorem (formerly the Bloch-Kato conjecture), due to S. Bloch, K. Kato andO. Gabber: Theorem 2.5 ([Kat82], [BK86]) . For any field F of characteristic p > , there are iso-morphisms of abelian groups K Mn ( F ) /pK Mn ( F ) ∼ −→ ker( ℘ : Ω nF → Ω nF /d Ω n − F ) { a , ..., a n } 7→ da a ∧ ... ∧ da n a n . Here, (Ω • F , d ) is the de Rham complex of absolute differential forms over F , and ℘ isthe inverse Cartier operator (see [Kat82] for details). It follows from this result that forany a , ..., a n ∈ F ∗ , the pure symbol { a , ..., a n } is divisible by p in K Mn ( F ) if and only if[ F p ( a , ..., a n ) : F p ] < p n . Consider the quasilinear p -form hh a , ..., a n ii := X ≤ j ,...,j n ≤ p − ( n Y i =1 a j i i ) x pj ,...,j n of dimension p n on the F -vector space F p n in its standard basis. Clearly hh a , ..., a n ii ≃hh a ii ⊗ ... ⊗ hh a n ii . By the definition, we have D L ( hh a , ..., a n ii ) = L p ( a , ..., a n ) for everyfield extension L/F . It therefore follows from Proposition 2.2 that the form hh a , ..., a n ii L is isotropic if and only if [ L p ( a , ..., a n ) : L p ] < p n , which in turn holds if and only if { a , ..., a n } = 0 in K Mn ( L ) /pK Mn ( L ). The (projective) hypersurfaces defined by the forms hh a , ..., a n ii , which are called quasi-Pfister forms , may therefore be regarded (in the abovesense) as splitting varieties in the equal characteristic setting. Moreover, it turns out thatthe quasi-Pfister forms exhibit all the same properties as the Pfister quadratic forms in themixed characteristic: they are precisely those forms which are “multiplicative”; up to pro-portionality, the anisotropic quasi-Pfister forms are characterised by the degree to whichthey split over the generic point of the associated hypersurface; they have the “roundness”property, etc. We refer to the article [Hof04] for further details.It will be convenient to record here the following observation concerning the splittingof quasi-Pfister forms over extensions of the base field: Lemma 2.6.
Let π = hh a , ..., a n ii be an anisotropic quasi-Pfister form over F . Let L/F be a field extension such that π L is isotropic. Then there is a proper subset { a j , ..., a j m } ⊂{ a , ..., a n } such that ( π L ) an ≃ hh a j , ..., a j m ii . STEPHEN SCULLY
Proof.
Let m be such that [ L p ( a , ..., a n ) : L p ] = p m . Since π L is isotropic, we have m < n . Now, we may choose a subset { a j , ..., a j m } of { a , ..., a n } such that L p ( a , ..., a n ) = L p ( a j , ..., a j m ). Then the quasi-Pfister form hh a j , ..., a j m ii is anisotropic over L , and wehave D L (( π L ) an ) = D L ( π ) = L p ( a , ..., a n ) = L p ( a j , ..., a j m ) = D L ( hh a j , ..., a j m ii ) . The statement therefore follows from Corollary 2.3. (cid:3)
To the arbitrary form ϕ , we associate a certain anisotropic quasi-Pfister form. Let usfirst recall the following invariant: Definition 2.7 ([HL04], [Hof04]) . Let ϕ be a quasilinear p -form over a field F , and let L/F be a field extension. The norm field of ϕ (over L ) is the field N L ( ϕ ) := L p ( ab | a, b ∈ D L ( ϕ ) ∩ L ∗ ) . Note that if two forms ϕ and ψ are proportional over F , then N L ( ϕ ) = N L ( ψ ) for allextensions L/F . The norm field invariant was first introduced in the context of quasilinearquadratic forms by D. Hoffmann and A. Laghribi in [HL04]. It turns out that this isa birational invariant of quasilinear p -hypersurfaces (see Proposition 4.10), and we willexploit this for the proofs of our main results. A more direct description of the norm fieldis given by the following lemma: Lemma 2.8.
Let ϕ denote the quasilinear p -form h a , ..., a n i over F . Then N F ( ϕ ) = F p ( a a , ..., a n a ) Proof.
Since the norm field does not change when we multiply ϕ by a scalar, we mayassume that a = 1. In this case, it is clear that F p ( a , ..., a n ) ⊆ N F ( ϕ ). But the reverseinclusion also holds, since D F ( ϕ ) = F p + F p · a + ... + F p · a n ⊆ F p ( a , ..., a n ). (cid:3) It follows that N L ( ϕ ) is finite dimensional over its subfield L p for any form ϕ over F and any extension L/F . Moreover, if a , ..., a m ∈ F are such that N F ( ϕ ) = F p ( a , ..., a m ),then we have N L ( ϕ ) = N L ( ϕ L ) = L p ( a , ..., a m ) for all extensions L/F . Note that thedimension of N L ( ϕ ) as an L p -vector space is always a power of p . One may thereforedefine: Definition 2.9.
The integer lndeg L ( ϕ ) := log p ([ N L ( ϕ ) : L p ]) is called the ( logarithmic ) norm degree of ϕ (over L ). Remark . In [Hof04], the norm degree of a form ϕ over F is defined to be the integerndeg F ( ϕ ) := p lndeg F ( ϕ ) . For our purposes, it will be more convenient to take the base p -logarithm; for example, see Theorem 4.2 for a result of S. Schr¨oer which shows that theinteger lndeg F ( ϕ ) determines the size of the singular (non-regular) locus of the projectivehypersurface { ϕ = 0 } .Finally, we define: Definition 2.11 ([HL04], [Hof04]) . Let ϕ be a p -form over F , and let L/F be a fieldextension. If lndeg L ( ϕ ) = m , and a , ..., a m ∈ L ∗ are such that N L ( ϕ ) = L p ( a , ..., a m ),then the anisotropic quasi-Pfister form b ν L ( ϕ ) := hh a , ..., a m ii is called the norm form of ϕ (over L ).By Corollary 2.3, this does not depend on the choice of generators a i up to isomorphism.Note that we have b ν L ( ϕ ) = b ν L (( ϕ L ) an ), and by the proof of Lemma 2.6, both forms are ATIONAL MORPHISMS BETWEEN QUASILINEAR HYPERSURFACES 7 equal to the anisotropic part of b ν F ( ϕ ) over L . By definition, the dimension of b ν L ( ϕ ) isnothing else but p lndeg L ( ϕ ) . If ϕ L is anisotropic, then by Corollary 2.3 and Lemma 2.8, ϕ L is proportional to a subform of b ν L ( ϕ ).3. Quasilinear p -forms over extensions of the base field In this section we record some general observations about the behaviour of quasilinear p -forms and their invariants over extensions of the base field. Again, most of the state-ments here can be found in the article [Hof04].The invariant lndeg F ( ϕ ) defined above gives a useful necessary (but not sufficient)condition for an anisotropic quasilinear p -form to become isotropic over an extension ofthe base field. Proposition 3.1 ([Hof04], Proposition 5.2) . Let ϕ be an anisotropic quasilinear p -formover F , and let L/F be a field extension over which ϕ becomes isotropic. Then lndeg L ( ϕ ) < lndeg F ( ϕ ) .Proof. Recall that for any extension
E/F , the integer p lndeg E ( ϕ ) is equal to the dimensionof the anisotropic part of the norm form b ν F ( ϕ ) over E . Proving the statement thereforeamounts to checking that b ν F ( ϕ ) becomes isotropic over L . But this is evident, because ϕ is proportional to a subform of b ν F ( ϕ ). (cid:3) Corollary 3.2.
Let ϕ be an anisotropic quasilinear p -form over F , and let a ∈ F \ F p . If ϕ F ( p √ a ) is isotropic, then a ∈ N F ( ϕ ) .Proof. Let m = lndeg F ( ϕ ), and let a , ..., a m ∈ F ∗ be such that N F ( ϕ ) = F p ( a , ..., a m ).By Proposition 3.1, the field N F ( p √ a ) ( ϕ ) = F ( p √ a ) p ( a , ..., a m ) = F p ( a , ..., a m , a ) hasdimension ≤ p m − over F p ( a ) = F ( p √ a ) p . It must therefore have dimension p m over F p ,and so a ∈ F p ( a , ..., a m ) = N F ( ϕ ). (cid:3) This allows us to give another characterisation of the norm field for anisotropic forms:
Corollary 3.3.
Let ϕ be an anisotropic quasilinear p -form over F . Let S be the set of all a ∈ F such that ϕ F ( p √ a ) is isotropic. Then N L ( ϕ ) = L p ( S ) for all field extensions L/F .Proof.
By the remarks following Lemma 2.8, it suffices to prove this for L = F . Since N F ( ϕ ) is invariant under multiplying ϕ by a scalar, we may assume that ϕ = h , a , ..., a n i for some a i ∈ F ∗ . Then N F ( ϕ ) = F p ( a , ..., a n ) by Lemma 2.8. Since ϕ F ( p √ a i ) is clearlyisotropic for all 1 ≤ i ≤ n , we have N F ( ϕ ) ⊂ F p ( S ). The reverse inclusion follows fromCorollary 3.2. (cid:3) Let
L/F be a finitely generated field extension. Recall that
L/F is called separablygenerated if L can be realised as a purely transcendental extension of F followed by aseparable algebraic extension. If L/F is separably generated, then there are several ratherdirect ways to see that any anisotropic quasilinear p -form over F remains anisotropic over L . More generally, we have: Proposition 3.4.
Let
L/F be any field extension. Then there exists an anisotropic quasi-linear p -form ϕ over F such that ϕ L is isotropic if and only if L/F is not separablygenerated.Proof.
The extension
L/F is not separably generated if and only if L p /F p is not separablygenerated. By Proposition 4.1 in Chapter VIII of [Lan02] (we refer to the proof ratherthan the statement itself), the latter is true if and only if L p and F are not linearly disjointover F p , which is a precise translation of the condition in our statement. (cid:3) STEPHEN SCULLY
In particular, we have:
Corollary 3.5 ([Hof04], Proposition 5.3) . Let ϕ be an anisotropic quasilinear p -form over F , and let L/F be a separably generated field extension. Then ϕ L is anisotropic. Moreover, lndeg L ( ϕ ) = lndeg F ( ϕ ) .Proof. The first assertion is implicit in Proposition 3.4. The second statement follows byapplying the first statement to the norm form b ν F ( ϕ ). (cid:3) Remark . For finite separable extensions, the first part of this statement may be viewedas a replacement for Springer’s Theorem from the theory of quadratic forms, which saysthat an anisotropic quadratic form remains anisotropic over any odd degree extension ofthe base field.Thus in order to study the isotropy behaviour of quasilinear p -forms over extensions ofthe base field, we are essentially reduced to considering finite purely inseparable extensions.We conclude this section by collecting a couple of simple facts with this in mind. Lemma 3.7.
Let ϕ be a quasilinear p -form over F , and let L/F be a finite extension ofdegree n . If ϕ L is isotropic, then ϕ contains a subform of dimension ≤ n which becomesisotropic over L .Proof. Let µ , ..., µ n be a basis for L over F , and suppose that w ∈ V ϕ ⊗ F L is a nonzeroisotropic vector for ϕ L . Then we can write w = v ⊗ µ + ... + v n ⊗ µ n for some v i ∈ V ϕ . The v i span a nonzero subspace W ⊂ V ϕ of dimension ≤ n , and therestriction ϕ | W becomes isotropic over L . (cid:3) Lemma 3.8.
Let ϕ be an anisotropic quasilinear p -form over F , and let L/F be a degree p extension. Then (1) dim( ϕ L ) an ≥ p dim( ϕ ) . (2) lndeg L ( ϕ ) ≥ lndeg F ( ϕ ) − , with equality if ϕ L is isotropic.Proof. For any extension
E/F , the dimension of ( ϕ E ) an is equal to the dimension of the E p -vector space D E ( ϕ ) by Proposition 2.2. In order to prove (1), we therefore have toshow that dim L p ( D L ( ϕ )) ≥ p dim F p ( D F ( ϕ )). But this is obvious, because p · dim L p ( D L ( ϕ )) = dim F p ( D L ( ϕ )) ≥ dim F p ( D F ( ϕ )) . The inequality in statement (2) now follows by applying (1) to the norm form b ν F ( ϕ ). Inthe case where ϕ L is isotropic, equality holds by Proposition 3.1. (cid:3) Quasilinear p -hypersurfaces In this section we present some basic observations concerning quasilinear p -hypersurfacesand rational morphisms between them. We start with the following lemma: Lemma 4.1.
Let F be a field of characteristic p > . Let f = a x p + a x p + ... + a n x pn ∈ F [ x , ..., x n ] be a polynomial, and assume that a = 0 . Then f is reducible in F [ x , ..., x n ] if and only if a i a ∈ F p for all i ∈ [1 , n ] .Proof. We may assume that a = 1. If a i ∈ F p for all i ∈ [1 , n ], then f = ( P ni =0 p √ a i x i ) p .Conversely, if f is reducible in F [ x , ..., x n ], then it is certainly reducible in F ( x , ..., x n )[ x ].It follows that a x p + ... + a n x pn is a p th -power in F ( x , ..., x n ), and this easily implies that a i ∈ F p for all i ∈ [1 , n ]. (cid:3) ATIONAL MORPHISMS BETWEEN QUASILINEAR HYPERSURFACES 9
Now, let ϕ be a quasilinear p -form over F of dimension d + 2. We consider the projec-tive hypersurface X ϕ := Proj( S • ( V ∗ ϕ ) / ( ϕ )) ⊂ P ( V ϕ ) of dimension d over F . A scheme ofthis type will be called a quasilinear p-hypersurface . By Lemmas 2.8 and 4.1, the scheme X ϕ is integral if and only if lndeg F ( ϕ ) >
0. In particular, X ϕ is integral whenever ϕ isanisotropic. If lndeg F ( ϕ ) >
0, then we let F ( ϕ ) denote the field of rational functions on X ϕ . Note that F ( ϕ ) may be realised as a purely transcendental extension of F followedby a purely inseparable extension of degree p . If L/F is any extension of fields, then X ϕ L is canonically isomorphic to ( X ϕ ) L , and by construction, X ϕ has an L -rational point ifand only if ϕ L is isotropic. In particular, the anisotropic form ϕ becomes isotropic overthe field F ( ϕ ). We will say that X ϕ is isotropic (resp. anisotropic) if ϕ is isotropic (resp.anisotropic), and we define i ( X ϕ ) := i ( ϕ ). By Corollary 3.5, X ϕ is isotropic if and only ifit has a zero cycle of degree 1. Moreover, if X ϕ is isotropic, then Proposition 2.2 shows that X ϕ is a cone over X ϕ an with vertex given by the linear subspace of dimension i ( X ϕ ) − V ϕ . It follows that for any i ≥
0, wehave i ( X ϕ ) > i if and only if X ϕ has a dimension i cycle of degree prime to p (where by degree we mean the degree as a cycle on the ambient projective space).A quasilinear p -hypersurface X over F is a twisted form of the p th infinitesimal neigh-bourhood of a hyperplane in some projective space P nF . In particular, the smooth locus of X is empty. Still, since the base field F is assumed to be non-perfect, it is interesting toask when X is a regular scheme. The following result, due to S. Schr¨oer, shows that thisis rarely the case: Theorem 4.2 ([Sch10], Theorem 3.3) . Let ϕ be a quasilinear p -form over F of dimension ≥ . Then X ϕ is a regular scheme if and only if lndeg F ( ϕ ) = dim( ϕ ) − . If X ϕ is notregular, then the non-regular locus has codimension lndeg F ( ϕ ) in X ϕ . The remainder of this section consists of some general observations concerning ra-tional morphisms between quasilinear p -hypersurfaces. Since an isotropic quasilinear p -hypersurface is a cone over its anisotropic part (and hence stably birational to itsanisotropic part), we may restrict our attention to the anisotropic case. Note that if ϕ and ψ are quasilinear p -forms over F of dimension ≥ F ( ψ ) >
0, then theexistence of a rational map X ψ X ϕ is equivalent to the isotropy of the form ϕ F ( ψ ) .Indeed, given a rational map X ψ X ϕ , the closure of its graph in X ψ × X ϕ pulls back toa rational point on the generic fibre ( X ϕ ) F ( ψ ) of the canonical projection X ψ × X ϕ → X ψ .Conversely, any rational point of ( X ϕ ) F ( ψ ) can be viewed as the generic point of a closedsubvariety of X ψ × X ϕ birational to X ψ over F ; using the other projection X ψ × X ϕ → X ϕ ,we get a rational morphism X ψ X ϕ . We will switch between the algebraic and geo-metric terminology where we feel it is appropriate.It will be useful to observe the following simple fact: Lemma 4.3.
Let ϕ and ψ be anisotropic quasilinear p -forms over F . Then (1) dim( ϕ F ( ψ ) ) an ≥ p dim( ϕ ) . (2) lndeg F ( ψ ) ( ϕ ) ≥ lndeg F ( ϕ ) − , with equality if ϕ F ( ψ ) is isotropic.Proof. Since F ( ψ ) can be realised as a purely transcendental extension of F followed bya degree p extension, this follows from Corollary 3.5 and Lemma 3.8. (cid:3) Now, a basic question here is the following:
Question 4.4.
Let X , Y and Z be anisotropic quasilinear p -hypersurfaces over F . Sup-pose that there exist rational morphisms X Y and Y Z . Does there exist arational morphism X Z ?One approach to this problem is suggested by the following classical result: Proposition 4.5.
Let
X, Y and Z be varieties over a field k with Z complete, and supposethat there are rational morphisms X Y and Y Z . If the image of the generic pointof X under the map X Y is a regular point of Y , then there exists a rational morphism X Z .Proof. This is a standard application of the valuative criterion of properness. More ex-plicitly, let y ∈ Y be the image of the generic point of X under the map X Y .Since y is regular, there is a valuation ring R of the function field k ( Y ) with residue field k ( y ). By the valuative criterion of properness, the map Spec k ( Y ) → Z extends to amorphism Spec R → Z . Passing to the residue field we, get a morphism Spec k ( y ) → Z .Finally, composing this with the natural map Spec k ( X ) → Spec k ( y ) gives a morphismSpec k ( X ) → Z , as we wanted. (cid:3) Corollary 4.6.
Let Y be a complete variety over F , and let ϕ and ψ be anisotropicquasilinear p -forms of dimension ≥ over F such that ψ is proportional to a subform of ϕ . If there exists a rational morphism X ϕ Y , then there exists a rational morphism X ψ Y .Proof. It suffices to treat the case where ψ is a codimension 1 subform of ϕ . In this case, X ϕ is regular at the generic point of X ψ , because the latter is an effective Cartier divisorin X ϕ . The statement therefore follows from Proposition 4.5. (cid:3) Now observe that the statement of Question 4.4 depends not on the variety X , butonly on its generic point. Moreover, the function field F ( X ) can be realised as a purelytranscendental extension of F followed by a purely inseparable extension of degree p . Re-placing the base field F with a suitable purely transcendental extension of it, we thereforereduce to the case where X is just a point (of degree p ). Using Lemma 3.7 and Corollary4.6, we can further reduce to the case where dim( Y ) ≤ p −
2. Taking Proposition 4.5 intoconsideration, we see that Question 4.4 can be settled affirmatively with a positive answerto the following question:
Question 4.7.
Let Y be an anisotropic quasilinear p -hypersurface of dimension ≤ p − F , and let y ∈ Y be a closed point of degree p . Is it true that there exists a regularclosed point z ∈ Y such that F ( z ) ∼ = F ( y ) over F ?Of course, this is trivially true in the case where p = 2. Therefore Question 4.4 has apositive answer for quasilinear quadrics, as was well-known previously. For larger primes p , it was essentially asked in [Hof04] whether Question 4.7 has a positive answer with themuch stronger condition that the point z be the intersection of the hypersurface Y witha line in the ambient projective space (see Question B in § p = 3. To give an explicit example, let a, b ∈ F ∗ be suchthat [ F ( a, b ) : F ] = 3 = 9. Then the anisotropic cubic form ϕ = h , a + ba , b i becomesisotropic over the field F ( √ a ), but one easily checks that ϕ has no 2-dimensional subformswhich become isotropic over the same extension. Nevertheless, the weaker assertion wemake here holds in the case where p = 3: Proposition 4.8.
Question 4.7 (and hence Question 4.4) has a positive answer for p = 2 and p = 3 . ATIONAL MORPHISMS BETWEEN QUASILINEAR HYPERSURFACES 11
Proof. If Y is just a point, then there is nothing to prove. We may therefore assume that p = 3 and dim( Y ) = 1. Let ϕ be 3-dimensional anisotropic form over F which defines thehypersurface Y . By Lemma 2.8, we either have lndeg F ( ϕ ) = 1 or lndeg F ( ϕ ) = 2. In thelatter case, Y is a regular at all of its points by Theorem 4.2, and again there is nothing toprove. We are therefore left with the case where lndeg F ( ϕ ) = 1. Since ϕ becomes isotropicover the residue field F ( y ), it follows from Proposition 3.1 that we have lndeg F ( y ) ( ϕ ) = 0.In other words, the subspace of isotropic vectors for ϕ F ( y ) is 2-dimensional. For dimensionreasons, it follows that every 2-dimensional subform of ϕ over F becomes isotropic over F ( y ). So if z is a closed point of Y defined by any 2-dimensional subform of ϕ , we thereforehave F ( z ) ∼ = F ( y ) over F . Since all such points are regular, the statement is proved. (cid:3) Actually, the proof of Proposition 4.8 shows that for any prime p , Question 4.7 has apositive answer whenever dim( Y ) ≤
1. In particular, for the prime 5 we only need to treatthe case where dim( Y ) = 2 or dim( Y ) = 3. Suppose that dim( Y ) = 2, and let ϕ be a4-dimensional anisotropic form defining Y . By Lemma 2.8, we have 1 ≤ lndeg F ( ϕ ) ≤ F ( ϕ ) = 1 or lndeg F ( ϕ ) = 3. Therefore the only interesting case is where lndeg F ( ϕ ) =2. The following example suggests that things are already rather more complicated in thissituation: Example 4.9.
Let p = 5, and let a, b ∈ F ∗ be such that [ F ( a, b ) : F ] = 5 = 25. Choosea polynomial g ∈ F [ s, t ] of degree ≤ s, t so that the form ϕ = h , a, b, g ( a, b ) i is anisotropic over F . By the definition of ϕ , we have lndeg F ( ϕ ) = 2. Moreover, conditionon the coefficients a, b implies that there are derivations D a , D b : F → F satisfying(1) D a ( a ) = 1, D a ( b ) = 0, and(2) D b ( a ) = 0, D b ( b ) = 1.For indeterminates T , T , T , T , we can extend these to derivations D a , D b : F [ T , ..., T ] → F [ T , ..., T ] by sending the variables to zero. Now, let us identify our form ϕ with thepolynomial T + aT + bT + g ( a, b ) T ∈ F [ T , ..., T ]. Then the derivatives D a ( ϕ ) = T + ∂g∂s ( a, b ) T , and D b ( ϕ ) = T + ∂g∂t ( a, b ) T necessarily vanish at the non-regular points of the hypersurface X ϕ . A direct calculationthen shows that the non-regular locus of X ϕ consists of just one point, with residue fieldisomorphic to L = F ( r ∂g∂s ( a, b ) , r ∂g∂t ( a, b ) , r g ( a, b ) − a ∂g∂s ( a, b ) − b ∂g∂t ( a, b )) . As far as Question 4.7 is concerned, we are only interested in points of minimal degree (inthis case, degree 5), so the first task here is to determine conditions on the polynomial g under which [ L : F ] = 5. One can hope that the restrictions imposed on g are sufficientlystrong to force the existence of more than one L -rational point on the hypersurface X ϕ ;by the proof of Proposition 4.8, this would be enough to provide a positive answer toQuestion 4.7 for the form ϕ . We conclude this section by pointing out the following important consequence of Corol-lary 4.6. It was first proved by D. Hoffmann and A. Laghribi for quasilinear quadraticforms in [HL04], and was later extended to arbitrary quasilinear p -forms by D. Hoffmannin [Hof04]: Proposition 4.10 ([Hof04], Lemma 7.12) . Let ϕ and ψ be anisotropic quasilinear p -forms over F of dimension ≥ . If there exists a rational morphism X ψ X ϕ , then N F ( ψ ) ⊂ N F ( ϕ ) .Proof. Since any two forms which are proportional have the same norm field, we mayassume that ψ = h , a , ..., a n i for some a i ∈ F ∗ . For each i ∈ [1 , n ], let τ i denote thebinary subform h , a i i of ψ . By Corollary 4.6, there are rational maps X τ i X ϕ for all i .In other words, ϕ becomes isotropic over all the fields F ( τ i ) ∼ = F ( p √ a i ). By Corollary 3.2,we therefore have a i ∈ N F ( ϕ ) for all i , and hence N F ( ψ ) = F p ( a , ..., a n ) ⊂ N F ( ϕ ). (cid:3) This result shows that the norm field and norm degree are birational invariants ofquasilinear p -hypersurfaces. We will make use of this in the next section.5. The Izhboldin dimension and the main theorem
In this section, we prove the main result of the paper, Theorem 5.12.Let ϕ be an anisotropic quasilinear p -form of dimension ≥ F . In analogy withthe theory of quadratic forms, we define the integers i ( X ϕ ) = i ( ϕ ) := i ( ϕ F ( ϕ ) )and dim Izh ( X ϕ ) := dim( X ϕ ) − i ( X ϕ ) + 1 . The latter integer will be called the
Izhboldin dimension of X ϕ . Example 5.1.
It follows from the first part of Lemma 4.3 that i ( ϕ ) ≤ dim( ϕ ) − p dim( ϕ )for any anisotropic form of dimension ≥
2. Generally speaking, this bound is sharp.Indeed, if π is an anisotropic quasi-Pfister form of dimension p n , then it follows fromLemma 2.6 that i ( π ) = p n − p n − . Example 5.2.
An algebraic variety X is called incompressible if every rational morphismfrom X to itself is dominant. In the theory of quadratic forms, an important result of A.Vishik says that any anisotropic quadric X over a field of characteristic = 2 with i ( X ) = 1is incompressible. This result has a key role to play in the proof of Theorem 1.3. In oursetting, the corresponding statement is trivial. In fact, if X is an anisotropic quasilinear p -hypersurface over F with i ( X ) = 1, then X only has one rational point over its functionfield F ( X ). In other words, there is only one rational map from X to itself. This map is,of course, the identity.In fact, an anisotropic quasilinear p -hypersurface X is incompressible if and only if i ( X ) = 1, as the following lemma shows: Lemma 5.3.
Let ϕ be an anisotropic quasilinear p -form over F of dimension ≥ , andlet ψ ⊂ ϕ be a subform of codimension ≤ i ( ϕ ) − . Then the form ψ F ( ϕ ) is isotropic.Proof. The point is that the subspace of isotropic vectors for ϕ F ( ϕ ) must intersect theunderlying space of ψ F ( ϕ ) non-trivially for dimension reasons. (cid:3) ATIONAL MORPHISMS BETWEEN QUASILINEAR HYPERSURFACES 13
We are going to show (as a consequence of the main theorem) that it is impossible tofind a subform of codimension larger than i ( ϕ ) − F ( ϕ ).First we will need a couple of lemmas: Lemma 5.4.
Let ϕ be an anisotropic quasilinear p -form of dimension ≥ over F , andlet ψ ⊂ ϕ be a subform of codimension 1. Then ϕ F ( ϕ ) ∼ ψ F ( ϕ ) .Proof. We can write ϕ = h a i ⊕ ψ for some a ∈ F ∗ . But ψ represents a over the field F ( ϕ ),so the statement follows from Proposition 2.2. (cid:3) Lemma 5.5.
Let ϕ be an anisotropic quasilinear p -form of dimension ≥ over F . Thenthere exists a purely transcendental field extension K/F and a subform ψ ⊂ ϕ K of codi-mension i ( ϕ ) − such that i ( ψ ) = 1 .Proof. We may assume that ϕ = h , a , ..., a n i for some a i ∈ F ∗ . Let m < n be such thatdim( ϕ F ( ϕ ) ) an = m +1. Reordering the a i if necessary, we can assume that h , a , ..., a m i F ( ϕ ) is the anisotropic part of ϕ F ( ϕ ) by Lemma 2.4. Now, let T , ..., T n − be indeterminatevariables. Then the function field F ( ϕ ) is F -isomorphic to the field F ( T , ..., T n − )( p q a T p + ... + a n − T pn − + a n ) . Let K = F ( T m +1 , ..., T n − ), and consider the subform ψ = h , a , ..., a m , a m +1 T pm +1 + ... + a n − T pn − + a n i of ϕ K . Then we have ψ K ( ψ ) ∼ h , a , ..., a m i K ( ψ ) by Lemma 5.4. But the field K ( ψ ) is F -isomorphic to F ( ϕ ) by construction, so the form h , a , ..., a m i K ( ψ ) is anisotropic. It follows that i ( ψ ) = 1. (cid:3) Remark . We will show later (see Proposition 6.1) that i ( ψ ) = 1 for any subform ψ ⊂ ϕ of codimension i ( ϕ ) − F . The example constructed above(modulo passing to a purely transcendental extension of F ) will be sufficient for our moreimmediate concerns.In the theory of quadratic forms, Theorem 1.3 is actually deduced as a consequence ofthe following stronger statement. It was first proved over fields of characteristic differentfrom 2 by N. Karpenko and A. Merkurjev. It was extended to smooth quadrics in char-acteristic 2 in the book [EKM08] by R. Elman, N. Karpenko and A. Merkurjev, and toarbitrary quadrics (smooth or otherwise) by B. Totaro. Theorem 5.7 ([KM03] Theorem 3.1, [Tot08] Theorem 5.1) . Let X be an anisotropicquadric over a field k , and let Y be a complete variety over k which has no closed pointsof odd degree. Suppose that Y has a closed point of odd degree over k ( X ) . Then (1) dim Izh ( X ) ≤ dim( Y ) . (2) If dim Izh ( X ) = dim( Y ) , then there exists a rational morphism Y X .Remark . Let X be an anisotropic quadric satisfying i ( X ) = 1. In the terminologyof [Hau11] §
10, Theorem 5.7 says that X is strongly 2-incompressible . For an arbitraryvariety X , the degree to which X fails to satisfy the weaker notion of incompressibility (see Example 5.2) can be measured by the minimum dimension of the image of a rationalmorphism from X to itself. In the case where the variety X is regular, this integer iscommonly referred to as the canonical dimension of X . The first part of Theorem 5.7implies that the canonical dimension of a smooth anisotropic quadric is equal to the Izhboldin dimension dim
Izh ( X ). This includes the previously mentioned result which saysthat an anisotropic quadric X is incompressible if and only if i ( X ) = 1.We will now prove an analogue of Theorem 5.7 for quasilinear p -hypersurfaces. The ap-proach given here is rather different from the one for quadrics given in [KM03] and [Tot08].In fact, the latter approach does not work for quasilinear p -hypersurfaces whenever p > p -hypersurfaces, but is in general false inthe anisotropic case (which is precisely the case needed for the proof). For example, if p >
2, and X is a quasilinear p -hypersurface of dimension 1 (i.e. a curve), then one canshow that CH ( X ) contains nonzero p -torsion if and only if X is regular. The proof whichwe present here makes no use of intersection theory, but rather exploits properties of thenorm field and norm degree invariants introduced earlier. The important observation isthe following lemma: Lemma 5.9.
Let ϕ and ψ be anisotropic quasilinear p -forms of dimension ≥ over F ,and let L be a field such that F ⊂ L ⊂ F ( ψ ) . If the form ϕ F ( ψ ) is isotropic, then ϕ L isisotropic if and only if lndeg L ( ϕ ) < lndeg F ( ϕ ) .Proof. One direction was already proved in Proposition 3.1. The interesting part is the con-verse. So suppose that lndeg L ( ϕ ) < lndeg F ( ϕ ), and suppose for the sake of contradictionthat ϕ L is anisotropic. Then since ϕ F ( ψ ) is isotropic, we have lndeg F ( ψ ) ( ϕ ) < lndeg L ( ϕ ) byProposition 3.1. But this implies that lndeg F ( ψ ) ( ϕ ) ≤ lndeg F ( ϕ ) −
2, which is impossibleby the second part of Lemma 4.3. (cid:3)
Proposition 5.10.
Let X and Y be anisotropic quasilinear p -hypersurfaces over F . Sup-pose that there exists a rational morphism X Y , and let Z denote (the closure of ) itsimage in Y . Then there exists a rational morphism Z X .Proof. Let ψ and ϕ be anisotropic forms over F defining X and Y respectively. We needto show that ψ becomes isotropic over the field F ( Z ). But we have a natural embedding F ( Z ) ⊂ F ( ψ ) over F , and since the form ψ F ( ψ ) is isotropic, Lemma 5.9 shows that it issufficient to check that lndeg F ( Z ) ( ψ ) < lndeg F ( ψ ). Now, since there exists a rational map X Y , we have an inclusion N E ( ψ ) ⊂ N E ( ϕ ) for all extensions E/F by Proposition4.10. Suppose that lndeg F ( Z ) ( ψ ) = lndeg F ( ψ ). Since Z is a subvariety of Y , the form ϕ F ( Z ) is isotropic, and hence(5.1) lndeg F ( Z ) ( ϕ ) < lndeg F ( ϕ )by Proposition 3.1. By part (2) of Lemma 4.3, we also have(5.2) lndeg F ( ψ ) ( ψ ) = lndeg F ( ψ ) − F ( Z ) ( ψ ) − . Now, since N F ( Z ) ( ψ ) ⊂ N F ( Z ) ( ϕ ), it follows from (5.2) that lndeg F ( ψ ) ( ϕ ) ≤ lndeg F ( Z ) ( ϕ ) −
1. But then combining this with (5.1), we get lndeg F ( ψ ) ( ϕ ) ≤ lndeg F ( ϕ ) −
2, which isimpossible by the second part of Lemma 4.3. Hence lndeg F ( Z ) ( ψ ) < lndeg F ( ψ ), and theproof is complete. (cid:3) Corollary 5.11.
Let f : X Y be a rational morphism of anisotropic quasilinear p -hypersurfaces over F , and let Z denote (the closure of ) its image in Y . If i ( X ) = 1 , then X and Z are birational via f .Proof. There exists a rational morphism Z X by Proposition 5.10, and since i ( X ) = 1,the composition X Z X is the identity (see Example 5.2). (cid:3) ATIONAL MORPHISMS BETWEEN QUASILINEAR HYPERSURFACES 15
A point y on a variety Y over a field k will be called separable if the residue fieldextension k ( y ) /k is separably generated. Here is our version of Theorem 5.7: Theorem 5.12.
Let X be an anisotropic quasilinear p -hypersurface over F , and let Y be a complete variety over F which has no separable points. Suppose that there exists arational morphism X Y . Then (1) dim Izh ( X ) ≤ dim( Y ) . (2) If dim Izh ( X ) = dim( Y ) , then there exists a rational morphism Y X .Proof. It follows immediately from Corollary 3.5 that passing to a purely transcenden-tal extension of the base field F changes nothing in the statement. By Lemma 5.5,we may therefore assume that X has a plane section X ′ of codimension i ( X ) − i ( X ′ ) = 1. Moreover, there exists a rational morphism X ′ Y by Corollary 4.6. Sincedim Izh ( X ′ ) = dim Izh ( X ), we may replace X by X ′ and assume that i ( X ) = 1 (or equiv-alently, dim Izh ( X ) = dim( X )). Let Z denote the image of the map X Y . Since Y has no separable points, the field extension F ( Z ) /F is not separably generated. It followsfrom Proposition 3.4 that there exists a rational morphism Z Y ′ for some anisotropicquasilinear p -hypersurface Y ′ over F . Now, since the given map X Z is dominant, wecan consider the composition f : X Z Y ′ . By Corollary 5.11, X is birational to itsimage in Y ′ via f . It follows that X and Z are birational, and part (1) of the statementfollows immediately. Moreover, if we have the equality dim( X ) = dim( Y ), then X and Y are actually birational, whence part (2). (cid:3) Remark . Note that the condition that the variety Y has no separable points is equiv-alent to the condition that Y has no separable closed points. Indeed, this follows fromthe well-known fact that any generically smooth variety has a dense subset of separableclosed points. However, the statement of Theorem 5.12 is still weaker than the analogueof Theorem 5.7. More precisely, our result is weaker than the claim that a quasilinear p -hypersurface with i ( X ) is strongly p-incompressible (see Remark 5.8 and [Hau11] § p -form ϕ becomeisotropic over the field F ( ϕ ). The analogue of the first part of the following statementfor quadratic forms over fields of characteristic different from 2 is due to A. Vishik (see[Vis99], Corollary 3): Corollary 5.14.
Let ϕ be an anisotropic quasilinear p -form over F of dimension ≥ ,and let ψ be a subform of ϕ . Then ψ F ( ϕ ) is isotropic if and only if codim ϕ ( ψ ) ≤ i ( ϕ ) − .In the case where codim ϕ ( ψ ) = i ( ϕ ) − , there is only one rational morphism X ϕ X ψ ,and moreover, it is dominant.Proof. The fact that every subform ψ of codimension ≤ i ( ϕ ) − F ( ϕ ) was proved in Lemma 5.3. The converse follows from the first part of Theorem5.12. If codim ϕ ( ψ ) = i ( ϕ ) −
1, there is only one rational morphism X ϕ X ψ , becauseotherwise the subspace of isotropic vectors for ψ F ( ϕ ) would have dimension ≥ i ( ϕ ) subform of ϕ which becomes isotropic over F ( ϕ ). The mapis dominant by part (1) of Theorem 5.12. (cid:3) We would like to prove an analogue of Theorem 1.3 for quasilinear p -hypersurfaces. Tothis end, the only obstruction is Question 4.4. We will illustrate this with a proof in thecase when p = 2 or p = 3 (where we have a positive answer to Question 4.4 by Proposition4.8). The case where p = 2 was previously proved by B. Totaro in [Tot08]. Theorem 5.15.
Assume that p = 2 or p = 3 , and let X and Y be anisotropic quasilinear p -hypersurfaces over F . Suppose that there exists a rational morphism X Y . Then (1) dim Izh ( X ) ≤ dim Izh ( Y ) . (2) dim Izh ( X ) = dim Izh ( Y ) if and only if there is a rational morphism Y X .Proof. Let Y ′ ⊂ Y be a plane section of codimension i ( Y ) −
1. By Corollary 5.14, we havea dominant rational morphism Y Y ′ . Proposition 4.8 now implies that there existsa rational morphism X Y ′ . By the first part of Theorem 5.12, we get dim Izh ( X ) ≤ dim( Y ′ ) = dim Izh ( Y ), which proves (1). If there also exists a rational morphism Y X ,then the same argument shows that dim Izh ( Y ) ≤ dim Izh ( X ), and hence dim Izh ( X ) =dim Izh ( Y ). On the other hand, if we are given the equality dim Izh ( X ) = dim Izh ( Y ), thenthere is a rational morphism Y ′ X by the second part of Theorem 5.12. Composingwith the dominant rational morphism Y Y ′ , we get a rational morphism Y X ,and this proves (2). (cid:3) Remark . Note that we do not need a positive answer to Question 4.4 in its entiretyto prove the analogue of Theorem 1.3. Indeed, we only need the case where Z is a planesection of codimension i ( Y ) − Y . For this special case, our question is easily seen tobe equivalent to: Question . Let Y be an anisotropic quasilinear p -hypersurface over F . Is is true that i ( Y ) is minimal among all defect indices attained by Y over extensions of the base fieldwhere Y becomes isotropic?It is a well-established fact that i ( X ) satisfies the analogous “generic property” when X is a smooth anisotropic quadric. In our case, we have a positive answer to Question 5.17when p = 2 or p = 3 by Proposition 4.8.6. Further applications to rational morphisms between quasilinearhypersurfaces
In this section we use Theorem 5.12 to prove some more specific results concerning ra-tional morphisms between quasilinear p -hypersurfaces. In particular, we prove analoguesof Hoffmann’s Theorem 1.1 (Theorem 6.10) and Izhboldin’s Theorem 1.2 (Theorem 6.12).Let ϕ be an anisotropic quasilinear p -form over F of dimension ≥
2, let j ∈ [1 , i ( ϕ )],and let ψ ⊂ ϕ be a subform of codimension i ( ϕ ) − j . Then it is easy to see that i ( ψ ) ≥ j .Indeed, it suffices to show that every codimension j − ψ becomes isotropicover the field F ( ψ ) (see Proposition 2.2). But any such subform is isotropic over F ( ϕ ) byLemma 5.3, and hence isotropic over F ( ψ ) by Corollary 4.6. Theorem 5.12 now allows usto prove that equality holds: Proposition 6.1.
Let ϕ be an anisotropic quasilinear p -form over F of dimension ≥ ,let j ∈ [1 , i ( ϕ )] , and let ψ ⊂ ϕ be a subform of codimension i ( ϕ ) − j . Then i ( ψ ) = j .Proof. Suppose that i ( ψ ) > j . Then there is a subform σ ⊂ ψ of codimension j whichbecomes isotropic over F ( ψ ). Let τ be a subform of ψ of codimension j − σ as a codimension 1 subform. Then there exists a rational morphism X τ X σ byCorollary 4.6. Since τ has codimension i ( ϕ ) − ϕ , there is a dominant rationalmorphism X ϕ X τ by Corollary 5.14. But then taking the composition of these mapsgives a rational morphism X ϕ X σ , and this is impossible by Corollary 5.14 (cid:3) ATIONAL MORPHISMS BETWEEN QUASILINEAR HYPERSURFACES 17
Remark . The analogue of Proposition 6.1 for quadratic forms in characteristic differentfrom 2 was proved by A. Vishik in [Vis99] prior to Karpenko and Merkurjev’s Theorem1.3.We will now prove analogues of Theorems 1.1 and 1.2 for quasilinear p -hypersurfaces.First we need to introduce the class of quasi-Pfister neighbours . Definition 6.3.
Let ϕ be a quasilinear p -form over F of dimension ≥
2, and let n be theunique positive integer satisfying p n − < dim( ϕ ) ≤ p n . We say that ϕ is a quasi-Pfisterneighbour if ϕ is proportional to a subform of a quasi-Pfister form of dimension p n . Inthis case, the variety X ϕ will also be called a quasi-Pfister neighbour . Remark . Quasi-Pfister neighbours are analogous to
Pfister neighbours in the theoryof quadratic forms. For example, if ϕ is a neighbour of a quasi-Pfister form π over F ,then for every field extension L/F , ϕ L is isotropic if and only if π L is isotropic. Indeed, ifdim( π ) = p n and π L is isotropic, then Lemma 2.6 implies that the subspace of isotropicvectors for π L has dimension at least p n − p n − , and therefore must intersect the underlyingspace of ϕ L non-trivially.Recall from Example 5.1 that if π is an anisotropic quasi-Pfister form of dimension p n ,then i ( π ) = p n − p n − . Applying Proposition 6.1 to the case of an anisotropic quasi-Pfisterneighbour, we therefore get: Corollary 6.5.
Let ϕ be an anisotropic quasi-Pfister neighbour over F , and let n be theunique positive integer such that p n − < dim( ϕ ) ≤ p n . Then i ( ϕ ) = dim( ϕ ) − p n − . Now, the following observation is key here:
Proposition 6.6.
Let ϕ be an anisotropic quasilinear p -form of dimension ≥ over F . Then there exists a field extension e F /F such that ϕ e F is an anisotropic quasi-Pfisterneighbour.Proof. Let π = b ν F ( ϕ ) be the norm form of ϕ . Recall that ϕ is proportional to a subformof π . Let n be the unique positive integer such that p n − < dim( ϕ ) ≤ p n . If dim( π ) = p n ,then ϕ is already a quasi-Pfister neighbour (of π ). We can therefore assume that dim( π ) ≥ p n +1 . In particular, we have dim Izh ( X π ) ≥ p n − X ϕ ) ≤ p n − X π X ϕ . In otherwords, ϕ remains anisotropic over the field F ( π ). Now, the anisotropic part of π F ( π ) isnothing else but the norm form π ′ = b ν F ( π ) ( ϕ ) of ϕ over F ( π ). We have dim( π ′ ) < dim( π ),and since ϕ F ( π ) is anisotropic, ϕ F ( π ) is proportional to a subform of π ′ . Repeating thisprocedure as many times as is necessary, we eventually produce an extension e F /F overwhich ϕ becomes an anisotropic quasi-Pfister neighbour (of the norm form b ν e F ( ϕ )). (cid:3) Remark . We should point out that the analogous statement for nondegenerate qua-dratic forms is certainly not true in general (see [HI00] for a detailed discussion of thisproblem for quadratic forms over fields of characteristic different from 2).
Corollary 6.8.
Let ϕ be an anisotropic quasilinear p -form of dimension ≥ over F ,and let n be the unique positive integer such that p n − < dim( ϕ ) ≤ p n . Then i ( ϕ ) ≤ dim( ϕ ) − p n − .Proof. By Proposition 6.6, there is an extension e F /F such that ϕ e F is an anisotropic quasi-Pfister neighbour. By Corollary 6.5, we therefore have i ( ϕ e F ) = dim( ϕ ) − p n − . Since wehave a natural embedding F ( ϕ ) ⊂ e F ( ϕ e F ), we get i ( ϕ ) = i ( ϕ F ( ϕ ) ) ≤ i ( ϕ e F ( ϕ e F ) ) = i ( ϕ e F ) = dim( ϕ ) − p n − , which is what we wanted. (cid:3) Example 6.9.
Let ϕ be an anisotropic form of dimension p n + 1 for some n ≥
0. Then i ( ϕ ) = 1.Now we can prove an analogue of Theorem 1.1 for quasilinear p -hypersurfaces: Theorem 6.10.
Let X and Y be anisotropic quasilinear p -hypersurfaces over F . If thereexists n ≥ such that dim( Y ) ≤ p n − < dim( X ) , then there are no rational morphisms X Y .Proof. By part (1) of Theorem 5.12, it is sufficient to show that dim
Izh ( X ) > p n −
2, andthis follows from Corollary 6.8. (cid:3)
Remark . The analogue of Corollary 6.8 for quadratic forms over fields of characteristicdifferent from 2 was originally proved by D. Hoffmann as a corollary of Theorem 1.1 (see[Hof95]). Here the roles are reversed. The difference is that we were able to use Theorem5.12 to prove Corollary 6.6, but, as we have remarked above, the analogue of Corollary6.6 for nondegenerate quadratic forms is false in general. Corollary 6.8 and Theorem 6.10were proved in the case p = 2 by D. Hoffmann and A. Laghribi in [HL06] using differentmethods.We also get the following analogue of Izhboldin’s Theorem 1.2. The case where p = 2was proved using different methods by D. Hoffmann and A. Laghribi in [HL06]. Theorem 6.12.
Let X and Y be anisotropic quasilinear p -hypersurfaces over F with dim( Y ) = p n − ≤ dim( X ) for some n ≥ . Suppose that there exists a rational morphism X Y . Then there exists a rational morphism Y X . If in addition we have dim( X ) = p n − , then X and Y are birational.Proof. Let X ′ ⊂ X be a plane section of dimension p n −
1. By Corollary 4.6, there existsa rational morphism X ′ Y . By Example 6.9, we have that i ( X ′ ) = 1. It then followsfrom Corollary 5.11 that X ′ is birational to Y , whence the result. (cid:3) Note that in the case where dim( X ) = dim( Y ) = p n − n ≥
0, we get thestronger assertion (in comparison with Theorem 1.2) that X and Y are birational. Fornon-quasilinear quadrics, this is still an open problem (see also Conjecture 7.1). Remark . In a recent article [Hau11], O. Haution has shown that any degree p hy-persurface of dimension p − p is strongly p -incompressible. In particular, an anisotropic quasilinear p -hypersurface of dimension p − p -incompressible. For such varieties, thisstatement is stronger than the results established above in the present article (see Remark5.13). The proof uses an extension of K. Zainoulline’s degree formula for the Euler charac-teristic (see [Zai10]) to fields of arbitrary characteristic, and also to non-smooth varietiesof sufficiently small dimension. The Euler characteristic (of the structure sheaf) does not,however, distinguish between degree p hypersurfaces of dimension ≥ p −
1, so this invari-ant can only be used to explain the first level of the “separation” exhibited by Theorems6.10 and 6.12. We remark any smooth degree p hypersurface of dimension p n − n ≥
0) over a field of characteristic = p which has no closed points of degree prime to p is strongly p -incompressible by the degree formulas of A. Merkurjev (generalising those ofRost; see [Mer03]). It is not known at present if the degree formulas used to prove thishold in arbitrary characteristic. ATIONAL MORPHISMS BETWEEN QUASILINEAR HYPERSURFACES 19 Birational geometry of quasilinear hypersurfaces
In this section we consider the extension of the results obtained by B. Totaro on thebirational geometry of quadrics in [Tot08] to quasilinear hypersurfaces of higher degree.An old problem of O. Zariski asks whether two stably birational varieties of the samedimension over a field are actually birational. It is well-known that this is false in general,but in the case where both varieties are smooth anisotropic quadrics, it is still an impor-tant open problem. Using the fact that a smooth isotropic quadric is a rational variety,one easily shows that two smooth anisotropic quadrics X and Y over a field are stablybirational if and only if there exist rational morphisms X Y and Y X . We cantherefore ask the following question for arbitrary quadrics: Conjecture 7.1 (Quadratic Zariski Problem) . Let X and Y be anisotropic quadrics ofthe same dimension over a field k . Suppose that there exist rational morphisms X Y and Y X . Is it true that X and Y are birational? In a series of papers ([Tot07], [Tot08], [Tot09]), B. Totaro has suggested a new approachto this problem by means of a related conjecture concerning rulings on quadrics. We willsay that a variety X over a field k is ruled if X is birational to Y × P k for some variety Y over k . Totaro has observed the following consequence of Vishik’s result stating that aquadric with first Witt index equal to 1 is incompressible: Theorem 7.2 ([Tot08], Corollary 3.2) . Let X be an anisotropic quadric over a field k . If i ( X ) = 1 , then X is not ruled. Moreover he conjectures:
Conjecture 7.3 ([Tot07], Conjecture 3.1) . Let X be an anisotropic quadric over a field k . Then X is ruled if and only if i ( X ) > . This is formulated more precisely as follows:
Conjecture 7.4 ([Tot09], Conjecture 1.1) . Let X be an anisotropic quadric over a field k . Then X is birational to X ′ × P i ( X ) − k for some subquadric X ′ ⊂ X of codimension i ( X ) − . Many results are known on all these problems for quadrics of small dimension. In[Tot08], Totaro proves Conjecture 7.4 for the entire class of quasilinear quadrics, and thenuses it to prove the quasilinear case of Conjecture 7.1. The same approach should workfor quasilinear hypersurfaces of higher degree. The analogue of Theorem 7.2 is trivial inthis setting:
Proposition 7.5.
Let X be an anisotropic quasilinear p -hypersurface over F . If i ( X ) =1 , then X is not ruled.Proof. Suppose that X is birational to Y × P F for some variety Y over F . Then we canconstruct a rational morphism X X as the composition X Y × P F pr Y −−→ Y ֒ → Y × P F X, where the second map is the canonical projection and the third map is the embedding of Y in Y × P F at a rational point of P F . Note that the composition is defined because thethird map embeds Y as an effective Cartier divisor in Y × P F (see Proposition 4.5). Butthe resulting map is not surjective by construction, and this contradicts the fact that theonly rational morphism from X to itself is the identity (see Example 5.2). (cid:3) We can now prove an analogue of Conjecture 7.4 for quasilinear hypersurfaces of higherdegree. Given the results of §
5, the proof for the case p = 2 given by B. Totaro in [Tot08]carries over verbatim to all primes p . We reproduce the argument here for the reader’sconvenience: Theorem 7.6.
Let X be an anisotropic quasilinear p -hypersurface over F , and let X ′ ⊂ X be a plane section of codimension i ( X ) − . Then X is birational to X ′ × P i ( X ) − F .Proof. For simplicity of notation, let us put r = i ( X ). Let ϕ be an anisotropic form over F which defines the variety X , and let ψ ⊂ ϕ be a subform of codimension r − X ′ . By Corollary 5.14, there exists a dominant rational morphism π : X X ′ via which we may view F ( ψ ) as a subfield of F ( ϕ ). In particular, the defectindex of ϕ over F ( ψ ) is no more than r . On the other hand, it is at least this large by asimple application of Corollary 4.6. Hence ϕ has the same defect index over F ( ψ ) as itdoes over F ( ϕ ). Let v , ..., v r − be a basis of the subspace of isotropic vectors for the form ϕ F ( ψ ) . The v i may be regarded as rational morphisms from X ′ to the affine hypersurface { ϕ = 0 } . Define a rational morphism f : X ′ × P r − F X over F by the assignment( x ′ , [ λ : ... : λ r − ]) [ λ v ( x ′ ) + ... + λ r − v r − ( x ′ )] . Now, the identity map X → X corresponds to some isotropic line in the space V ϕ ⊗ F F ( ϕ ).Since ϕ has the same index over F ( ψ ) as it does over F ( ϕ ), there are rational functions f i ∈ F ( ϕ ) such that [ f · ( v ◦ π ) + ... + f r − · ( v r − ◦ π )] is the identity map from X toitself. Define a rational morphism g : X X ′ × P r − F by the assignment x ( π ( x ) , [ f ( x ) , ..., f r − ( x )]) . By construction, the composition f ◦ g is the identity on X . Therefore g is a birationalisomorphism, and the statement is proved. (cid:3) Finally, we make some remarks concerning an analogue of the Quadratic Zariski Problemfor quasilinear hypersurfaces of higher degree. Unfortunately, the obstruction here is againQuestion 4.4. As with Theorem 5.15, we illustrate this with a proof for quasilinear quadricsand cubics (for which we know that Question 4.4 has a positive answer).
Theorem 7.7.
Assume that p = 2 or p = 3 , and let X and Y be anisotropic quasilinear p -hypersurfaces of the same dimension over F . Suppose that there exist rational morphisms X Y and Y X . Then X and Y are birational.Proof. Let X ′ ⊂ X and Y ′ ⊂ Y be plane sections of codimensions i ( X ) − i ( Y ) − X is birational to X ′ × P i ( X ) − F and Y is birational to Y ′ × P i ( Y ) − F . Moreover, we have i ( X ) = i ( Y ) by Theorem 5.15 (here we are using thestatement of Question 4.4). In order to prove the statement, it therefore suffices to showthat X ′ and Y ′ are birational. Now, we have rational morphisms X ′ Y and Y Y ′ by Corollary 4.6 and Lemma 5.3 respectively. By Proposition 4.8, we therefore have arational morphism X ′ Y ′ (again, we are using the statement of Question 4.4). Sincei ( X ) = 1 and dim( X ′ ) = dim( Y ′ ), X ′ and Y ′ are birational by Corollary 5.11. (cid:3) Remark . As with Theorem 5.15, we do not need a positive answer to Question 4.4 inits entirety, but only a positive answer to Question 5.17.Still, using some of the results we have obtained, we can give partial results towards apositive solution to the Zariski problem for all primes p . By Corollary 5.11, the conjectureis true whenever i ( X ) = 1. In particular, it is true whenever dim( X ) = dim( Y ) = p n − ATIONAL MORPHISMS BETWEEN QUASILINEAR HYPERSURFACES 21 for some n ≥ p n −
1. First we need the case of quasi-Pfister neighbours:
Proposition 7.9.
Let X and Y be quasi-Pfister neighbours of the same dimension over F . Suppose that there are rational morphisms X Y and Y X . Then X and Y arebirational.Proof. Let X ′ ⊂ X and Y ′ ⊂ Y be plane sections of codimensions i ( X ) − i ( Y ) − X ′ Y ′ and Y ′ X ′ . Now, let ϕ and ψ be quasi-Pfister formsover F such that X is a neighbour of X ϕ and Y is a neighbour of X ψ . Then there existrational morphisms X ′ X ψ and Y ′ X ϕ by Corollary 4.6. But by Corollary 6.5, X ′ and Y ′ are also neighbours of X ϕ and X ψ respectively. In particular, for any variety Z over F , there exists a rational morphism Z X ′ (resp. Z Y ′ ) if and only if thereexists a rational morphism Z X ϕ (resp. Z X ψ ; see Remark 6.4). Therefore wehave rational morphisms X ′ Y ′ and Y ′ X ′ , and the proof is complete. (cid:3) Remark . For example, any two neighbours of the same quasi-Pfister hypersurfacewhich have the same dimension are birational. Together with Proposition 6.6, Proposition7.9 shows that the analogue of the quadratic Zariski problem is true up to making anextension of the base field which preserves the anisotropy of the quasilinear hypersurfacesinvolved.We conclude with the following result, which settles the Zariski problem in a largenumber of cases:
Proposition 7.11.
Let X and Y be anisotropic quasilinear p -hypersurfaces of the samedimension d over F . Suppose that there exist rational morphisms X Y and Y X ,and let n be the unique non-negative integer satisfying p n < d + 2 ≤ p n +1 . If d ≤ p n + n ,then X and Y are birational.Proof. Let ϕ and ψ be anisotropic forms defining X and Y respectively. By Proposition4.10, we have lndeg F ( ϕ ) = lndeg F ( ψ ). Let us denote this integer by m . If m = n + 1,then X and Y are quasi-Pfister neighbours, and we are done by Proposition 7.9. We cantherefore assume that m ≥ n + 2. Now, let X ′ ⊂ X and Y ′ ⊂ Y be plane sections ofcodimensions i ( X ) − i ( Y ) − X ′ Y ′ and Y ′ X ′ . By Corollary 4.6 and Lemma 5.3,we have rational morphisms X ′ Y and Y Y ′ (resp. rational morphisms Y ′ X and X X ′ ), and by Proposition 4.5 it will be sufficient to prove that Y (resp. X )is regular at the image of the generic point of X ′ (resp. Y ′ ). We may therefore assumethat X and Y are not regular. Note that we have i ( X ′ ) = i ( Y ′ ) = 1 by Proposition 6.1.It therefore follows from Corollary 5.11 that we may also assume that the given rationalmorphisms X ′ Y and Y ′ X are closed embeddings of subvarieties. Now, thenon-regular locus of X (resp. Y ) has codimension m ≥ n + 2 by Theorem 4.2. On theother hand, the Separation Theorem 6.10 implies thatcodim Y ( X ′ ) = d − dim( X ′ ) ≤ d − ( p n − , and similarly codim X ( Y ′ ) ≤ d − ( p n − d ≤ p n + n , then we havecodim Y ( X ′ ) , codim X ( Y ′ ) ≤ n + 1 < m, and so Y (resp. X ) is regular at the generic point of X ′ (resp. Y ′ ). (cid:3) Acknowledgements.
This work is part of my PhD thesis carried out at the Universityof Nottingham. I would like to thank Detlev Hoffmann for suggesting several of the prob-lems discussed in this text and for his guidance during the period in which this work wascompleted. I would also like to thank Olivier Haution and Alexander Vishik for numeroushelpful discussions. I am particularly grateful to Alexander Vishik for comments whichhelped to improve § References [BK86] S. Bloch and K. Kato. p -adic ´etale cohomology. Inst. Hautes ´Etudes Sci. Publ. Math. , (63):107–152, 1986.[EKM08] R. Elman, N. Karpenko, and A. Merkurjev.
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