Geometric properties of commutative subalgebras of partial differential operators
aa r X i v : . [ m a t h . AG ] J u l Geometric properties of commutative subalgebras of partialdifferential operators
Herbert Kurke Alexander Zheglov
Abstract
We investigate further alebro-geometric properties of commutative rings of partial differ-ential operators continuing our research started in previous articles. In particular, we startto explore the most evident examples and also certain known examples of algebraically inte-grable quantum completely integrable systems from the point of view of a recent generaliza-tion of Sato’s theory which belongs to the second author. We give a complete characterisationof the spectral data for a class of ”trivial” rings and strengthen geometric properties knownearlier for a class of known examples. We also define a kind of a restriction map from themoduli space of coherent sheaves with fixed Hilbert polynomial on a surface to analogousmoduli space on a divisor (both the surface and divisor are part of the spectral data). Wegive several explicit examples of spectral data and corresponding rings of commuting (com-pleted) operators, producing as a by-product interesting examples of surfaces that are notisomorphic to spectral surfaces of any (maximal) commutative ring of PDOs of rank one. Atlast, we prove that any commutative ring of PDOs, whose normalisation is isomorphic to thering of polynomials k [ u, t ] , is a Darboux transformation of a ring of operators with constantcoefficients. In this paper we continue the study of algebro-geometric properties of commutative algebras ofpartial differential operators (PDO for short) in two variables started in [23]. Everywhere in thispaper we assume that k is a field of characteristic zero.Recall that one of very complicated questions appearing in the theory of algebraically in-tegrable systems is: how to find explicit examples of certain commutative rings of PDOs orhow to classify them (see [23, Introduction] for an extensive history). This question can also bereformulated in the following way. In [5] the quantum analogue of the classical definition of anintegrable Hamiltonian system was defined. By a quantum completely integrable system (QCIS)on an algebraic variety X the authors understand a pair (Λ , θ ) , where Λ is an irreducible n -dimensional affine algebraic variety, and θ : O Λ → D ( X ) is an embedding of algebras (here thealgebra D ( X ) of differential operators on X is the quantum analogue of the Poisson algebra O ( T ∗ X ) ).By definition, a QCIS S = (Λ , θ ) is said to be algebraically integrable if it is dominated byanother QCIS S ′ with rk ( S ′ ) = 1 (see loc. cit.), where the rank of QCIS is the dimension ofthe space of formal solutions of the system θ ( g ) ψ = g ( λ ) ψ, g ∈ O Λ near a generic point of X . In [5] these definitions were also generalized to the case of integrablesystems on a formal polydisc. Thus, in this case X is Spec( k [[ x , x , . . . , x n ]]) and the symbols O X , k ( X ) , D ( X ) denote respectively k [[ x , . . . , x n ]] , k (( x , . . . , x n )) , O X [ ∂ , . . . , ∂ n ] , where ∂ i = ∂/∂x i . In this situation for n = 1 even the classification of all algebraically integrable1ommutative subalgebras B = θ (Λ) ⊂ D ( X ) in terms of the spectral data is known since thework of Krichever [20], [19]. In [5] the criterion for algebraic integrability of QCIS’s is given interms of the corresponding Galois groups. In this paper we continue to explore geometric properties of commutative rings of PDOs in D = k [[ x , x ]][ ∂ , ∂ ] started in [23] (the restriction n = 2 seems to be basically not essential,but in general case one needs to do some work to generalize a number of statements from ourprevious papers). Recall that even in this case there is still no classification of algebraicallyintegrable (in the above sense) commutative subalgebras in terms of spectral data, though thereis a classification of subalgebras in a completed ring of differential operators (see [39], cf. [23,Introduction]) in terms of Parshin’s modified geometric data (which include algebraic projectivesurface, an ample Q -Cartier divisor, a point regular on this divisor and on the surface, atorsion free sheaf on the surface and some extra trivialisation data). Moreover, up to now onlya few examples of such algebras are known. Probably the first nontrivial (in certain sense, seediscussion below) examples appeared in [8], [9], [10]. The examples were connected with thequantum (deformed) Calogero-Moser systems (cf. [30]). Later the ideas of these constructionswere developed in a series of papers (see e.g. [13], [14], [11]) in order to construct more examples(for review see e.g. [7] and references therein; cf. also [5], [3], [4]). Let’s also mention that theidea to construct a free BA-module (the module consisting of eigenfunctions of the ring of PDO)was developed later by various authors (see e.g. [29], [25], [7]) to produce explicit examples ofcommutative matrix rings of PDO.In [39], [23] several properties of the above mentioned geometric data were investigated.In particular, all algebraically integrable commutative rings of PDOs correspond to rank onegeometric data with X Cohen-Macaulay, C rational and C = 1 , F torsion free of rank oneand Cohen-Macaulay along C . In this paper we strengthen the last property: namely, we showthat any commutative subalgebra of PDOs (satisfying as in [23] certain mild conditions) leadsto a sheaf F on X which is Cohen-Macaulay (theorem 3.1).Cohen-Macaulay rank one torsion free sheaves appearing as sheaves from geometric dataclassifying commutative subalgebras of (completed) operators with fixed spectral surface can beparametrized by a moduli space which is an open subscheme of the projective scheme parametris-ing semistable sheaves with fixed Hilbert polynomial (see remark 2.15). We introduce in thispaper a kind of restriction map ζ from this moduli space to the moduli space of coherent torsionfree rank one sheaves on the divisor C (see section 2.4, remark 2.15) and formulate a conjecturethat this morphism is surjective (remark 2.15). It is important to study this moduli space in or-der to find new examples of algebraically integrable systems or to classify commutative algebrasof PDOs. We hope to return to this question in future works.This moduli space can be thought of as another analogue of the Jacobian of the curve inthe context of the classical KP theory. Recall that in the work [32] Parshin offered to considera multi-variable analogue of the KP-hierarchy which, being modified, is related to algebraicsurfaces and torsion free sheaves on such surfaces as well as to a wider class of geometric dataconsisting of ribbons and torsion free sheaves on them if the number of variables is equal to two(see [37], [22, Introduction]). In the work [22] we described the geometric structure of the Picardscheme of a ribbon. This scheme has a nice group structure and can be thought of as an analogueof the Jacobian of a curve in the context of the classical KP theory. In particular, generalizedKP flows are defined on such schemes (flows defined by the multi-variable analogue of the KP-hierarchy). The disadvantage of the Picard scheme of a ribbon is its infinite-dimensionality. Themoduli space we have mentioned above is finite dimensional. The generalized KP flows are alsodefined on it. It is not difficult to show that it can be embedded into the Picard scheme of aribbon. 2nvestigating already existing examples of commutative algebras mentioned above we provea theorem (3.3) about algebraically integrable commutative rings of PDOs whose affine spectralsurface is rational. Such rings appeared, for example, in papers [13], [14], [11], [4]. In the examplesfrom these papers the normalisation of the affine spectral surface is known to be A . In [4] theauthors gave a method of producing new non-trivial examples of commutative rings of PDOsusing the Darboux transformation. We show in theorem 3.3 that all rings with this propertyof the affine spectral surface are Darboux transformations of rings of operators with constantcoefficients. As a by-product we also give a geometric characterisation of certain completion of A (see theorem 3.2): a completion of A , whose divisor at infinity is an ample irreducible Q -Cartier divisor with self-intersection index 1, is P . This result could be probably provedby classical methods of algebraic geometry using old results of Morrow ([26]) or relatively newresults of Kojima, Takahashi ([18]) (we would like to thank M.Gizatoulline and T.Bandman forpointing out these works), but we used instead only some ideas from our theory of ribbons and(or alternately) the construction of the generalized Krichever-Parshin map.It is reasonable to ask if there are examples of algebraically integrable commutative rings ofPDOs whose spectral surface is isomorphic to a given one. We give here two counterexamples(3.1, 4.1), both for affine and projective spectral surfaces.Another natural question is: how to characterise those commutative algebras which consistof operators not depending on x or x . We call these algebras ”trivial”, because one can easilyconstruct such algebras taking commutative subalgebras of one-variable operators and addinga derivation with respect to another variable. Surprisingly the geometry of spectral data is notso trivial for these algebras. We give a description (theorem 4.1) of such algebras in terms ofgeometric data.At last, we give examples (4.1, 4.2) of surfaces for which it is possible to describe all sheavesfrom the moduli space mentioned above and calculate all corresponding rings of commuting(completed) operators. All these rings are ”trivial”. The paper is organized as follows:In section 2.1 we recall basic definition of geometric data from [39] and give also an alternativedefinition of these data.In section 2.2 we recall the construction of Schur pairs associated with geometric data.In section 2.4 we introduce the restriction map ζ and prove several technical lemmas.In section 2.5 we recall basic definitions and properties of the ring of completed operators,recall the classification theorem from [39] and prove additional technical lemmas needed in therest of the paper.In section 2.6 we recall and prove some properties of Schur pairs corresponding to geometricdata with sheaves whose Hilbert polynomial is fixed. For understanding the map ζ the mostimportant properties are formulated in proposition 2.3. We also formulate a conjecture aboutthe map ζ .In section 3 we prove theorems about CM-property, completion of plane and Darboux trans-formations mentioned above.In section 4 we give the description of ”trivial” algebras and examples. Acknowledgements.
Part of this research was done at the Humboldt University of Berlinduring a stay supported by the Vladimir Vernadskij stipendium of MSU-DAAD (Referat 325-paw, Kennziffer A/12/89240). Part of this research was done at the Max Planck Institute ofMathematics during a research stay of the second author (1-30 September 2013). He would liketo thank the MPI for the excellent working conditions.The first author would like to thank the Department of Differential Geometry and Applica-tions of Moscow State University for hospitality during his stay in Moscow in May 2012, where3ome aspects of this work were planned.We are also grateful to Igor Burban, Antonio Laface and Denis Osipov for their permanentinterest and many stimulating discussions.The second author was partially supported by the RFBR grant no. 14-01-00178-a, 13-01-00664 and by grant NSh no. 581.2014.1.
In this work we usually use standard notation from algebraic geometry used e.g. in the book[16]. We also use some notation from our previous papers [39], [23].On the two-dimensional local field k (( u ))(( t )) we will consider the following discrete valua-tion of rank two ν : k (( u ))(( t )) ∗ → Z ⊕ Z : ν ( f ) = ( m, l ) iff f = t l u m f , where f ∈ k [[ u ]] ∗ + tk (( u ))[[ t ]].(Here k [[ u ]] ∗ means the set of invertible elements in the ring k [[ u ]] .) We also define the discretevaluation of rank one ν t ( f ) = l. In this subsection we recall definitions from [39], [23]; we slightly change some definitions fromloc.cit. to simplify the exposition and to avoid explaining certain technical details.For any n -dimensional irreducible projective variety X over the field k , and any Cartierdivisors E , . . . , E n ∈ Div( X ) on X one defines the intersection index ( E · . . . · E n ) ∈ Z on X (see, e.g., [15], [24, ch. 1.1].) Let ( E n ) = ( E · . . . · E ) be the self-intersection index of aCartier divisor E ∈ Div( X ) on X , and F be a coherent sheaf on X . There is the asymptoticRiemann-Roch theorem (see survey in [24, ch. 1.1.D]) which says that the Euler characteristic χ ( X, F ⊗ O X O X ( mE )) is a polynomial of degree ≤ n in m , with χ ( X, F ⊗ O X O X ( mE )) = rk( F ) · ( E n ) n ! · m n + O ( m n − ), (1)where rk is the rank of the sheaf.There is the cycle map: Z : Div( X ) → WDiv( X ) from the Cartier divisors to the Weildivisors on X (see [23, Appendix A]). If E , E ∈ Div( X ) such that Z( E ) = Z( E ) , then theself-intersection indices ( E n ) = ( E n ) on X (see [23, § + ( X ) is aninjective map to the semigroup of effective Weil divisors WDiv + ( X ) not contained in the sin-gular locus. We will say that an effective Weil divisor C on X , not contained in the singularlocus, is a Q -Cartier divisor on X if lC ∈ Im (Z | Div + ( X ) ) for some integer l > Definition 2.1.
Let C be a Q -Cartier divisor on X . We define the self-intersection index( C n ) on X as ( C n ) = ( G n ) /l n , (2)where G = lC is a Cartier divisor for some integer l > l > lC is a Cartier divisor, then for any other l ′ > l ′ C is a Cartier divisor we have that l | l ′ . Therefore, using the property( E n ) = m n ( E n ) for any E = mE , E ∈ Div( X ) , m ∈ Z we obtain that formula (2) doesnot depend on the choice of appropriate l . 4 efinition 2.2. We call (
X, C, P, F , π, φ ) a geometric data of rank r if it consists of thefollowing data (where we fix the ring k [[ u, t ]] for all data):1. X is a reduced irreducible projective algebraic surface defined over a field k ;2. C is a reduced irreducible ample Q -Cartier divisor on X ;3. P ∈ C is a closed k -point, which is regular on C and on X ;4. π : b O P −→ k [[ u, t ]]is a local k -algebra homomorphism satisfying the following property. If f is a localequation of the curve C at P , then π ( f ) k [[ u, t ]] = t r k [[ u, t ]] and the induced map π : ˆ O C,P = ˆ O P / ( f ) → k [[ u ]] = k [[ u, t ]] / ( t ) is an isomorphism. (The definition of π does not depend on the choice of appropriate f . Besides, from this definition it followsthat π is an embedding, k [[ u, t ]] is a free b O P -module of rank r with respect to π .Moreover for any element g from the maximal ideal M P of O P such that elements g and f generate M P we obtain that ν ( π ( f )) = (0 , r ) , ν ( π ( g )) = (1 ,
0) .)5. F is a torsion free quasi-coherent sheaf on X .6. φ : F P ֒ → k [[ u, t ]] is an O P -module embedding subject to the following condition forany n ≥ k [[ u, t ]] is an O P -module withrespect to π ). By item 2 there is the minimal natural number d such that C ′ = dC is a very ample divisor on X . Let γ n : H ( X, F ( nC ′ )) ֒ → F ( nC ′ ) P be an embedding(which is an embedding, since F ( nC ′ ) is a torsion free quasi-coherent sheaf on X ). Let ǫ n : F ( nC ′ ) P → F P be the natural O P -module isomorphism given by multiplicationto an element f nd ∈ O P , where f ∈ O P is chosen as in item 4. Let τ n : k [[ u, t ]] → k [[ u, t ]] / ( u, t ) ndr +1 be the natural ring epimorphism. We demand that the map τ n ◦ φ ◦ ǫ n ◦ γ n : H ( X, F ( nC ′ )) −→ k [[ u, t ]] / ( u, t ) ndr +1 is an isomorphism. (These conditions on the map φ do not depend on the choice of theappropriate element f .) Remark 2.1.
The rank of the sheaf is greater or equal to the rank of the data, cf. [23, Rem.3.3].If the sheaf F is coherent of rank one, then π is an isomorphism and φ induces the isomorphismˆ φ : ˆ F P ≃ k [[ u, t ]] , see [39, Rem. 3.7]. Note that any two trivialisations ˆ φ , ˆ φ : ˆ F P ≃ k [[ u, t ]]differ by multiplication on an element a ∈ k [[ u, t ]] ∗ . In some cases the conditions on the map φ in last item of the definition can be rewritten in purely algebro-geometrical terms, see proposition2.3 below. In this section we would like to give an alternative definition of the geometric data. This definitionseems to be more ”geometric”.Let’s introduce the following notation: T = Spec k [[ u, t ]] ⊃ T = Spec k [[ u ]] (defined by t = 0 ), O = Spec( k ) ∈ T , R = k [[ u, t ]] , M = ( u, t ) ⊂ R . Definition 2.3.
A geometric data is a triple (
X, j, F ) , where X is an integral projectivesurface, j : T → X is a dominant k -morphism and F ⊂ j ∗ O T is a quasicoherent subsheaf subject to the followingconditions: 5. j ∗ ( T ) = C ⊂ X is a curve (automatically integral), and P = j ( O ) is a point neither inthe singular locus of C nor of X .2. T × X { P } = { O } , T × X C = rT (the fiber product is a subscheme of T and rT is aneffective Cartier divisor on T ), r is called the rank of ( X, j, F ) .3. There exists an effective, very ample Cartier divisor C ′ ⊂ X with cycle Z ( C ′ ) = dC andfor all n > F ⊂ j ∗ O T ) H ( X, F ( nC ′ )) → H ( X, j ∗ O T ( nC ′ )) = H ( T, O T ( ndrT )) = Rt − ndr → Rt − ndr / M ndr +1 t − ndr is an isomorphism.We left to the reader the proof of equivalence of these two definitions. Remark 2.2.
With the data above we have the following properties:1) C is a Q -Cartier divisor and C = ( C ′ · C ) /d = ( C ′ ) /d .2) H ( X, F ) ≃ k (by (3)) hence we have a canonical embedding O X ⊂ F .3) F is a torsion free sheaf on X , and if F is coherent thenrk( F )( C ) = r . Indeed, for F as above we have χ ( F ( nC ′ )) = ( ndr + 1)( ndr + 2)2 . If F is coherent of rank m then F ∼ O mX ( ∼ means that the highest terms of the Hilbertpolynomials of sheaves coincide). For any coherent sheaf G on X the function χ ( G ( nC ′ )) isa polynomial of degree dim( G ) = l with positive leading coefficient ( ∈ Z /l ! ), so χ ( F ( nC ′ )) ∼ mχ ( O X ( nC ′ )) and n d r / m ( C ′ ) / Proposition 2.1.
If the embedding O X,P → F P is an isomorphism, then r = 1 . Furthemore, O X = F if and only if X = P and C is a straight line in P .Proof. If C is defined by f = 0 in a small neighbourhood of P (by (1) O X,P is regular, andalso O C,P = O X,P /f O X,P ), then f R = t r R (by (2)) and F ( nC ′ ) P = ( F P ) f nd . By (3) we have Rt − ndr = H ( X, F ( nC ′ )) ⊕ M ndr +1 t − ndr , so Rt − ndr = F P t − nd + M ndr +1 t − ndr and if F P = O X,P we get R = k [[ u, t r ]] + M ndr +1 (for theproof we may assume that u, t r are generators of ˆ M X,P = M X,P ˆ O X,P ). This is only possiblefor r = 1 . If O X = F we get a canonical basis for each H ( X, O X ( nC ′ )) of the form v ij ,0 ≤ i ≤ i + j ≤ nd , and v ij v hm = v i + h,j + m in H ( X \ C, O X ) =: A ( v ij corresponds to u i t j under the isomorphism in (3)). Thus A = k [ x, y ] with x = v , y = v (then v ij = x i y j ).Since X = Proj ⊕ n ≥ H ( X, O X ( nC ′ )) = Proj( ⊕ n ≥ A n s n ) , where A n = P i + j ≤ nd kx i y j , we get (by substituting x = x ′ /z , y = y ′ /z , s = z d ) ⊕ n ≥ A n s n = k [( x ′ ) i ( y ′ ) j z k | i + j + k = m ] , i.e. X = P with the d -th Veronese embedding. Since C = 1 , we get C is a straight line. Notation: for a morphism of noetherian schemes f : X → Y and a closed subscheme Z ⊂ X , f ∗ Z ⊂ Y isthe closed subscheme defined by the ideal ker( O Y f ∗ → f ∗ O X → f ∗ O Z ) .2 Associated Schur pairs Given a geometric data (
X, C, P, F , π, φ ) of rank r we define a pair of subspaces W, A ⊂ k [[ u ]](( t )) , where A is a filtered subalgebra of k [[ u ]](( t )) and W a filtered module over it, as follows (cf.[39, Def.3.15]):Let f d be a local generator of the ideal O X ( − C ′ ) P , where C ′ = dC is a very ample Cartierdivisor (cf. definition 2.2, item 6). Then ν ( π ( f d )) = (0 , r d ) in the ring k [[ u, t ]] and therefore π ( f d ) − ∈ k [[ u ]](( t )) . So, we have natural embeddings for any n > H ( X, F ( nC ′ )) ֒ → F ( nC ′ ) P ≃ f − nd ( F P ) ֒ → k [[ u ]](( t )),where the last embedding is the embedding f − nd F P φ ֒ → f − nd k [[ u, t ]] ֒ → k [[ u ]](( t )) (cf. definition2.2, item 6). Hence we have the embedding χ : H ( X \ C, F ) ≃ lim −→ n> H ( X, F ( nC ′ )) ֒ → k [[ u ]](( t )).We define W def = χ ( H ( X \ C, F )) . Analogously the embedding H ( X \ C, O ) ֒ → k [[ u ]](( t )) isdefined (and we’ll denote it also by χ ). We define A def = χ ( H ( X \ C, O )) .As it follows from this construction, A ⊂ k [[ u ′ ]](( t ′ )) ⊂ k [[ u ]](( t )) , (3)where t ′ = π ( f ) , u ′ = π ( g ) (see also definition 2.2, item 4). Thus, on A there is a filtration A n induced by the filtration t ′− n k [[ u ′ ]][[ t ′ ]] on the space k [[ u ′ ]](( t ′ )) : A n = A ∩ t ′− n k [[ u ′ ]][[ t ′ ]] = A ∩ t − nr k [[ u ]][[ t ]] (4)We have X ≃ Proj( ˜ A ) , where ˜ A = ∞ L n =0 A n s n (see also [39, lemma 3.3, lemma3.6, th.3.3]). Thesimilar filtration is defined on the space W ⊂ k [[ u ]](( t )) : W n = W ∩ t − nr k [[ u ]][[ t ]] (5)And the sheaf F ≃
Proj( ˜ W ) , where ˜ W = ∞ L n =0 W n s n . Note that we have W nd ≃ H ( X, F ( nC ′ ))by definition 2.2, item 6 and by construction of the map χ . The same pair of subspaces (
W, A ) can be defined also in terms of the alternative definition.Namely, to each geometric datum (
X, j, F ) we associate a pair ( A, W ) with A ⊂ R [ t − ] = k [[ u ]](( t )) , W ⊂ R [ t − ] , where A = H ( X \ C, O X ) ≃ lim −→ n> H ( X, O X ( nC ′ )) embedded via j ∗ : H ( X, O X ( nC ′ )) → H ( X, j ∗ O T ( nC ′ )) = H ( X, j ∗ O T ( ndrT )) = R · t − ndr and analogously for F ⊂ j ∗ O T . A is a filtered subring with the filtration A n = A ∩ R · t − nr ,and W is a filtered A -module with the filtration W n = A ∩ R · t − nr .This pair ( A, W ) determines the geometric data (
X, j, F ) , where X and F are defined asabove, and the morphisms j : T → X , F ⊂ j ∗ O T come from the embeddings A ⊂ k [[ u ]](( t )) , W ⊂ k [[ u ]](( t )) . Here and later in the article we use the non-standard notation Proj for the quasi-coherent sheaf associatedwith a graded module. If M is a filtered module, then we use the notation ˜ M = ∞ L i =0 M i s i for the analog of theRees module, as well as for filtered rings. .3 Category of geometric data In this section we recall definition of a category Q of geometric data from [39] and give itsalternative definition. Formally we need this section only to recall the content of theorem 2.3.We write it for the sake of completeness. Definition 2.4.
We define a category Q of geometric data as follows:1. The set of objects is defined by Ob ( Q ) = [ r ∈ N Q r , where Q r denotes the set of geometric data of rank r .2. A morphism ( β, ψ ) : [( X , C , P , F , π , φ )] −→ [( X , C , P , F , π , φ )]of two objects consists of a morphism β : X → X of surfaces and a homomorphism ψ : F → β ∗ F of sheaves on X such that:(a) β | C : C → C is a morphism of curves and β − ( X \ C ) = X \ C ;(b) β ( P ) = P . (c) There exists a continuous k -algebra isomorphism h : k [[ u, t ]] → k [[ u, t ]] (in a naturallinear topology, where the base of neighbourhoods of zero is generated by the powersof the maximal ideal) such that h ( u ) = u mod ( u ) + ( t ) , h ( t ) = t mod ( ut ) + ( t ) , and the following commutative diagram holds: H ( X \ C , O ) β ♯ −−−−→ H ( X \ C , O ) y χ y χ k [[ u ]](( t )) ˆ h −−−−→ k [[ u ]](( t )) , where ˆ h denotes the natural extension of the map h to a k -algebra k [[ u ]](( t ))automorphism.(d) There is a k [[ u, t ]] -module isomorphism ξ : k [[ u, t ]] ≃ h ∗ ( k [[ u, t ]]) (which is given justby multiplication of a single invertible element ξ ∈ k [[ u, t ]] ∗ ) such that the followingcommutative diagram holds: H ( X \ C , F ) ψ −−−−→ H ( X \ C , β ∗ F ) = H ( X \ C , F ) y χ y χ k [[ u ]](( t )) ˆ ξ −−−−→ h ∗ ( k [[ u ]](( t ))) = k [[ u ]](( t )) . .3.1 Alternative definition of the category We can give an alternative definition of the category as follows. The set of objects is defined asbefore, i.e. an object from Q r denotes the geometric datum ( X, j, F ) of rank r .We define a morphism of two objects ( X , j , F ) → ( X , j , F ) as a pair ( β, ψ ) with β : X → X a dominant morphism of surfaces, ψ : F → β ∗ F a morphism of quasicoherentsheaves subject to the following conditions:1. ( β − C ) red = C
2. there exists h ∈ Aut k ( T ) with h ∗ ( T ) = T , h ∗ ( u ) = u mod ( u ) + ( t ) , h ∗ ( t ) = t mod ( ut ) + ( t ) , such that the diagram T j −−−−→ X y h y β T j −−−−→ X is commutative;3. there exists ξ ∈ Aut O T ( O T ) (i.e. an element from R ∗ ) such that the diagram( F ) P ψ −−−−→ ( F ) P y ∩ y ∩ R ξ −−−−→ R is commutative.The composition with a second morphism ( β ′ , ψ ′ ) is given by ( β ′ , ψ ′ ) ◦ ( β, ψ ) = ( β ′ β, ( β ′ ) ∗ ( ψ ) ψ ′ ) . Remark 2.3.
It is interesting to note that, in general, a morphism of pairs β : ( X , C ) → ( X , C ) induces an embedding H ( X \ C , O X ) ֒ → H ( X \ C , O X ) if and only if ( β − C ) red = C .The ”if” part is obvious, let’s show the ”only if” part. Without loss of generality let d > C ′ = dC and C ′ = dC are effective very ample Cartier divisors.Then β ∗ C ′ = mC ′ + E , where E = 0 or E is an effective Cartier divisor.If A = H ( X \ C , O X ) , q = H ( X \ C , O X ( − E )) (so, q is an invertible ideal), thenSpec( A ) = X \ C , Spec( ∩ n q − n ) = X \ β − ( C ) . If B = H ( X \ C , O X ) , then we have A ⊂ ∩ n q − n , ∩ n q − n is finite over B . If B ⊂ A , then from the exact triple0 → A/B → ( ∩ n q − n ) /B → ( ∩ n q − n ) /A → ∩ n q − n ) must be a finite A -module, a contradiction if E = 0 . Remark 2.4.
The condition in item 2 on h ∗ ( u ) , h ∗ ( t ) is important to establish a categoricalequivalence with the category of Schur pairs from [39]. The reason is that automorphisms ofthe form h ∗ ( u ) = c u , h ∗ ( t ) = c t applied to a Schur pair will lead (after application of thequasi-inverse functor from [39, Th.3.3]), to a datum with another sheaf F (i.e. to a datum notisomorphic to the original one). This effect was known already in the classical KP theory as ascaling transform (cf. [36, § § .4 The restriction map ζ To construct the map ζ mentioned in Introduction we need to extend the constructions fromprevious section to a wider set of sheaves. Let X, C, C ′ , P and O X,P ⊂ R (for the embedding π or for the morphism j : T → X ) be as before. Let also A ⊂ k [[ u ]](( t )) = R [ t − ] be as before.We start with the following remark. Remark 2.5.
Let’s note that we can construct the analogous spaces W n , ˜ W for any torsionfree sheaf F (not only for sheaves from data) endowed with a O P -module embedding F P ֒ → k [[ u, t ]] . The most important example of such sheaf with an embedding is a coherent torsionfree rank one Cohen-Macaulay sheaf F on X (or, more generally, F is locally free of rank oneat P ), where we additionally assume that the rank of data r = 1 (i.e. the embedding π givesan isomorphism O X,P ≃ R ). In this case the stalk F P is a free O P -module. Let φ ′ : F P ≃ O P be a trivialisation; we can define the embedding φ by composing a trivialization φ ′ with theisomorphism π . Note that, if we choose another trivialisation of such sheaf F , then the newspace W will differ from the old one by multiplication on an element a ∈ k [[ u, t ]] ∗ and thespace A will not change. Note also that the property W nd ≃ H ( X, F ( nC ′ )) might not be truein general.Further we will use the following notation. If W ⊂ R [ t − ] is an A -module, we get a filtration W n = t − nr R ∩ W (compatible with the filtration on A ) and a fortiori graded ˜ A -modules˜ A ( i ) ( ˜ A ( i ) k = ˜ A k + i ) , ˜ W ( i ) ( ˜ W ( i ) k = ˜ W k + i )and quasicoherent sheaves on X : B i = Proj( ˜ A ( i )) , F i = Proj( ˜ W ( i ))with B i ⊂ B i +1 , F i ⊂ F i +1 . Note that1. B id ≃ O X ( iC ′ ) , and if W comes from a geometric datum, then F id ≃ F ( iC ′ ) . In general, F nd ≃ F ( dC ′ ) , because by [12, prop.2.4.7] we have F nd = Proj( ˜ W ( nd )) ≃ Proj( ˜ W ( d ) ( n ))and Proj( ˜ W ( d ) ( n )) ≃ Proj( ˜ W ( d ) )( n ) ≃ F ( nC ′ ) for any n .2. If F is a quasicoherent sheaf with an embedding F ⊂ j ∗ O T (equivalently F P ⊂ R )inducing W = H ( X \ C, F ) ⊂ R [ t − ] , then F ( iC ′ ) ⊆ F id .3. If F is a torsion free quasicoherent sheaf and if F P is a free module of rank one, we canfind an embedding F P ⊂ R (by a choice of a generator F P ≃ O X,P ⊂ R ). The resultingsheaves F i do not depend on the choice of the generator, up to isomorphisms compatiblewith the embeddings F ⊂ F i ⊂ F i +1 .4. From (3), (4) it easily follows that the sheaves B i / B i − ≃ Proj( ∞ M n =0 A i + n /A i + n − ) , F i / F i − ≃ Proj( ∞ M n =0 W i + n /W i + n − )are torsion free coherent sheaves on C ≃ Proj( B / B − ) . We mean here and below the pull-backs of the factor-sheaves on C . Note that that these pull-backs arecanonically isomorphic to the sheaves Proj( L ∞ n =0 A i + n /A i + n − ) , Proj( L ∞ n =0 W i + n /W i + n − ) , where the gradedmodules are considered as B / B − -modules. Indeed, for any f ∈ A d and any graded ˜ A -module M we have( M ⊗ ˜ A ( B / B − )) ( f ) ≃ M ( f ) ⊗ ˜ A ( f ) ( B / B − ) ( f ) , where from the pull-backs of the sheaves B i / B i − , F i / F i − are isomorphic to the sheaves Proj( M ) , Proj( N ) on C , where M = ( L ∞ n =0 A i + n /A i + n − ) ⊗ ˜ A ( B / B − ) , N = ( L ∞ n =0 W i + n /W i + n − ) ⊗ ˜ A ( B / B − ) . But the modules M , N are isomorphic to the B / B − -modules( L ∞ n =0 A i + n /A i + n − ) , ( L ∞ n =0 W i + n /W i + n − ) . efinition 2.5. For any torsion free sheaf F endowed with a O P -module embedding F P ֒ → k [[ u, t ]] we define a map (a kind of a ”restriction” map): ζ : F 7→ F / F − (6)from the set of torsion free sheaves on X to the set of torsion free sheaves on C . Remark 2.6.
For sheaves F satisfying the property W nd ≃ H ( X, F ( nC ′ )) for all n ≫ F ≃ F by [33, Lemma 9] and [16, Ch.2, ex. 5.9]. If F is a torsion free sheaf of rankone locally free at P , then by remark 2.5 for another choice of trivialisation at P there areisomorphisms F ′ k ≃ F k for any k . So, in this case definition of ζ don’t depend on trivialisation.In fact, in this case it depends only on F (see remark 2.8 below).On the other hand, for any torsion free sheaf F and any m > → F ⊗ O X O X ( − mC ′ ) → F → F ⊗ O X ( O X / O X ( − mC ′ )) → . Thus the pull-back of the sheaf F on the scheme ( C, i − ( O X / O X ( − mC ′ ))) (where i : C ֒ → X denotes the embedding) is isomorphic to the pull-back of the sheaf F / F − md . Further we will denote the pull-back of a sheaf F on the scheme ( C, i − ( O X / O X ( − mC ′ ))) as F | mC ′ .Note that the scheme ( C, i − ( O X / O X ( − mC ′ ))) is an irreducible scheme since C is irre-ducible. Hence the nilradical of the ring ˜ A/ ˜ A ( − md ) is prime. From (3), (4) it again followsthat Ass ( ˜
W / ˜ W ( − md )) coincides with this nilradical. Therefore, the restriction F | mC ′ is pureof dimension one (cf. [17, p.3]), because any restriction of a non-zero section a ∈ F | mC ′ ( U )(where U is any open subset of C ) to a smaller open subset is not zero. Thus, for any torsionfree sheaf of rank one locally free at P the sheaf F | mC ′ ⊂ F | mC ′ is pure of dimension one.Notably, we have the following property for any torsion free sheaf F such that its restriction F | mC ′ on the scheme ( C, i − ( O X / O X ( − mC ′ ))) is pure of dimension one: Lemma 2.1.
Let F be a torsion free sheaf on X endowed with a O P -module embedding F P ֒ → k [[ u, t ]] satisfying the following condition: if w ∈ W nd , then w ∈ f − nd ( F P ) . Assumethat its restriction F | mC ′ is pure of dimension one for all m > .Then we have H ( X, F ( nC ′ )) ≃ W nd for all n ≥ . Remark 2.7.
The condition on a O P -module embedding from lemma is satisfied for examplefor all rank one torsion free sheaves locally free at P (see remark 2.5) and for coherent sheavesof rank r from the geometric data (where r coincides with the rank of the data), because O P is a regular factorial ring. Other examples see in theorem 3.1. Proof.
By definition of the space W we have W nd = { w ∈ W | f nd w ∈ k [[ u ]][[ t ]] } = { w ∈ W | ν t ( f nd w ) ≥ } . We also have by definition χ ( H ( X, F ( nC ′ )) ⊂ W nd . Let w ∈ W nd , w = 0 . Let’s show that w ∈ χ ( H ( X, F ( nC ′ )) . We have w ∈ χ ( H ( X, F ( mC ′ ))for some m . Since F is a torsion free sheaf and C ′ is a Cartier divisor, we have embeddings χ ( H ( X, F ( kC ′ )) ⊂ χ ( H ( X, F ( nC ′ ))for all k ≤ n . Suppose that m > n . Assume the converse: w / ∈ χ ( H ( X, F ( nC ′ )) . Let b ∈ H ( X, F ( mC ′ )) be the preimage of w : w = χ ( b ) .11here is a neighbourhood U ( P ) of the point P , where the ample Cartier divisor C ′ is defined by the element f d . Since w ∈ W nd , we have w ∈ f − nd ( F P ) , thus b | U ( P ) ∈ Γ( U ( P ) , F ( nC ′ )) and b | U ( P ) = 0 (since F is torsion free). Now we have the following commu-tative diagram: b ֒ → H ( C, F ( mC ′ ) | ( m − n ) C ′ ) ↓ ↓ → Γ( U ( P ) , F ( nC ′ )) → Γ( U ( P ) , F ( mC ′ )) α → H ( U ( P ) ∩ C, F ( mC ′ ) | ( m − n ) C ′ ) , where the vertical arrows are embeddings. Indeed, the right vertical arrow is an embedding since F ( mC ′ ) | ( m − n ) C ′ is pure of dimension one by assumption.But α ( b ) = 0 , a contradiction. Thus, b ∈ H ( X, F ( nC ′ )) .By [23, Cor. 3.1] all sheaves F of rank one appearing in the geometric data from definition2.2 are Cohen-Macaulay along C . As it easily follows from definition 2.2 (item 6) all such sheavesfulfil the property O X ⊂ F , P / ∈ Supp( F / O X ) . Lemma 2.2.
Let F be a torsion free rank one sheaf on X . Assume that F is Cohen-Macaulayalong C .Then for some trivialisation ˆ φ : ˆ F P ≃ k [[ u, t ]] (see remark 2.5) W nd ≃ H ( X, F ( nC ′ )) for all n ≥ , or, equivalently, F ≃ F .Proof. The proof follows immediately from remarks 2.6, 2.7 and lemma 2.1, since F is locallyfree at P . Remark 2.8. If F is a torsion free sheaf of rank one locally free at P , then ζ ( F ) ≃ i ∗ ( F ) ,where i is the same as in remark 2.6. Indeed, F ≃ F by lemma 2.1, remarks 2.6, 2.7. Bythe arguments from footnote 3 i ∗ ( F ) ≃ Proj( ˜ W ⊗ ˜ A ( ˜ A/ ˜ A ( − A/ ˜ A ( − W ⊗ ˜ A ( ˜ A/ ˜ A ( − W / ˜ W ( − ⊗ ˜ A ( ˜ A/ ˜ A ( − i ∗ ( F / F − ) ≃ Proj( ˜
W / ˜ W ( − ⊗ ˜ A ( ˜ A/ ˜ A ( − Corollary 2.1.
For any k ≥ we have H ( X, F k ( nC ′ )) ≃ W nd + k for all n ≥ . The proof is obvious.
In this paper we will work mainly with commutative k -algebras of PDOs B ⊂ D = k [[ x , x ]][ ∂ , ∂ ] that satisfy the following condition: B contains the operators P, Q with constant principal symbols such thatthe intersection of the characteristic divisors of
P, Q is empty. (7)Recall that the symbol σ ( P ) of an operator P ∈ D is called constant if σ ( P ) ∈ k [ ξ , ξ ] .The characteristic divisor is given by the divisor of zeros of σ ( P ) in P n − k . It is unchanged bya k -linear change of coordinates x , . . . , x n . Recall also that any operator Q from the ring B satisfying condition (7) has constant principal symbol (see e.g. [23, Lemma 2.1]) and all suchrings are finitely generated k -algebras of Krull dimension 2 (see e.g. [23, Th.2.1]).In the work [39] was shown that such algebras are a part of a wider set of commutative k -algebras B ′ ⊂ ˆ D , and all algebras from this set can be classified in terms of geometric data fromsubsection 2.1. To explain what is going on we need to recall several definitions and statementsfrom [39]. 12 efinition 2.6. We define the order function on the ring k [[ x , x ]] by the ruleord M ( a ) = sup { n | a ∈ ( x , x ) n } . Definition 2.7. ([39, Sect.2.1.5]) Defineˆ D = { a = X q ≥ a q ∂ q | a q ∈ k [[ x , x ]] and for any N ∈ N there exists n ∈ N such thatord M ( a m ) > N for any m ≥ n } . (8)Define ˆ D = ˆ D [ ∂ ] , ˆ E + = ˆ D (( ∂ − )) . Definition 2.8. ([39, Def.2.12])We say that an operator P ∈ ˆ D has order ord Γ ( P ) = ( k, l ) if P = P ls =0 p s ∂ s , where p s ∈ ˆ D , p l ∈ k [[ x , x ]][ ∂ ] = D , and ord( p l ) = k (here ord is the usual order in the ring ofdifferential operators D ). In this situation we say that the operator P is monic if the highestcoefficient of p l is 1 .We say that an operator Q = P q ij ∂ i ∂ j ∈ ˆ E + satisfies the condition A ( m ) if ord M ( q ij ) ≥ i + j − m for all ( i, j ) .An operator P ∈ ˆ D , P = P p ij ∂ i ∂ j with ord Γ ( P ) = ( k, l ) satisfies the condition A if itsatisfies A ( k + l ) . Definition 2.9. ([39, Def.2.18]) The ring B ⊂ ˆ D of commuting operators is called quasi ellipticif it contains two monic operators P, Q such that ord Γ ( P ) = (0 , k ) and ord Γ ( Q ) = (1 , l ) forsome k, l ∈ Z .The ring B is called 1 -quasi elliptic if P, Q satisfy the condition A . Definition 2.10. ([39, Def.3.4]) The commutative 1 -quasi elliptic rings B , B ⊂ ˆ D are saidto be equivalent if there is an invertible operator S ∈ ˆ D of the form S = f + S − , where S − ∈ ˆ D ∂ , f ∈ k [[ x , x ]] ∗ , such that B = SB S − . Definition 2.11. ([39, Def.3.1]) The subspace W ⊂ k [ z − ](( z )) is called 1 -space, if thereexists a basis w i in W such that w i satisfy the condition A for all i (we identify here andbelow the ring k [ z − ](( z )) with the ring k [ ∂ ](( ∂ − )) via z ↔ ∂ − , z ↔ ∂ − )Using the identification z ↔ ∂ − , z ↔ ∂ − we can extend the definition of the orderfunction ord Γ from definition 2.8 on the field V = k (( z ))(( z )) . Using the anti-lexicographicalorder on the group Z ⊕ Z we define the lowest term LT ( a ) of any series a from V to be themonomial of a with the lowest order. Definition 2.12. ([38]) The support of a k -subspace W from the space V is the k -subspaceSupp( W ) in the space V generated by LT( a ) for all a ∈ W . Definition 2.13. ([39, Def.3.2]) We say that a pair of subspaces (
A, W ) , where
A, W ⊂ k [ z − ](( z )) and A is a k -algebra with unity such that W · A ⊂ W , is a 1 -Schur pair if A and W are 1 -spaces and Supp( W ) = k [ z − , z − ] .We say that 1 -Schur pair is a 1 -quasi elliptic Schur pair if A is a 1 -quasi elliptic ring.Consider the ring ˆ E + = ˆ D (( ∂ − )) . It has a natural action on the space k [ z − ](( z )) viathe isomorphism ˆ E + / ( x , x ) ˆ E + ≃ k [ z − ](( z )) which endows this space with the structure ofa right ˆ E + -module. 13 efinition 2.14. ([39, Def.3.3]) An operator T ∈ ˆ E + is said to be admissible if it is an invertibleoperator of order zero such that T ∂ T − , T ∂ T − ∈ k [ ∂ ](( ∂ − )) . The set of all admissibleoperators is denoted by Adm .An operator T ∈ ˆ E + is said to be 1 -admissible if it is admissible and satisfies the condition A (the definition of the condition A for operators from ˆ E + is literally the same as foroperators from ˆ D ). The set of all 1 -admissible operators is denoted by Adm .We say that two 1 -Schur pairs ( A, W ) and ( A ′ , W ′ ) are equivalent if A ′ = T − AT and W ′ = W T , where T is an admissible operator. Remark 2.9.
In [39] a more general growth condition A α , α ≥ A . This is the only case when the classificationtheorems from [39] (see also below) work.Let’s recall here one more notion from [39].Consider the set in ˆ E + Π = { P ∈ ˆ E + | ∃ m ∈ Z + s. that P satisfies A ( m ) } . It is an associative subring with unity (see [39, Corol.2.2]).We note that Π ⊃ D . Recall that by [39, lemma 2.10, lemma 2.11] it follows that any1 -quasi elliptic ring B belongs to Π . Remark 2.10.
By [39, Lemma 2.11] any two operators with constant coefficients L , L of theform L = ∂ + ∞ X q =1 v q ∂ − q , L = ∂ + ∞ X q =1 u q ∂ − q and satisfying condition A can be obtained as L = S − ∂ S , L = S − ∂ S , where S =1 + S − ∈ k [[ x , x ]][ ∂ ](( ∂ − )) is an invertible zeroth order 1 -admissible operator.On the other hand, as one can easily check, for the operator T = c exp( c x ∂ ) exp( c x + c x ) ∈ ˆ D , (9)where c , c , c , c ∈ k , we have T − ∂ T = ∂ + c , T − ∂ T = ∂ + c ∂ + c c + c . So, any 1 -admissible operator can be written in the form T = ST . Theorem 2.1. ([39, Th.3.2]) There is a one to one correspondence between the classes ofequivalent -quasi elliptic Schur pairs ( A, W ) with Supp( W ) = h z − i z − j | i, j ≥ i and theclasses of equivalent -quasi elliptic rings of commuting operators B ⊂ ˆ D . The proof of the theorem is constructive; the spaces A and W are obtained as follows: A = S − BS , W = k [ z − , z − ] S , where S is a monic operator of special type satisfying thecondition A . It is defined by a pair of normalized operators from B (see [39, § Definition 2.15.
We say that commuting monic operators
P, Q ∈ ˆ E + with ord Γ ( P ) = (0 , k ) ,ord Γ ( Q ) = (1 , l ) are almost normalized if P = ∂ k + k − X s = −∞ p s ∂ s Q = ∂ ∂ l + l − X s = −∞ q s ∂ s , where p s , q s ∈ ˆ D . 14e say that P, Q are normalized if P = ∂ k + k − X s = −∞ p s ∂ s Q = ∂ ∂ l + l − X s = −∞ q s ∂ s , where p s , q s ∈ ˆ D .Recall that by [39, lemma 2.10] any two commuting operators of order (0 , k ) and (1 , l ) canbe normalized by conjugating with an invertible operator S ∈ ˆ D . The space A from theorem2.1 depends only on the choice of the pair of normalized operators from B , and don’t depend onthe choice of the operators S from [39, Lemma 2.11]. If one chooses another pair of normalizedoperators from B , then the resulting Schur pair from theorem 2.1 will be equivalent to thefirst one. The following lemma clarifies the structure of elements in a ring that has a pair ofnormalized operators and in any equivalent ring. Lemma 2.3. i) If the ring B ⊂ Π ∩ ˆ D of commuting operators contains a pair of normalizedoperators P, Q with ord Γ ( P ) = (0 , k ) , ord Γ ( Q ) = (1 , l ) ( k > ), then all operators in B have constant highest coefficients, i.e. if L = P Ns =0 l s ∂ s , then l N is an operator with constantcoefficients. In particular, l N ∈ D (i.e. it has a finite order).Moreover, any operator P ′ ∈ B with ord Γ ( P ′ ) = (0 , m ) has the form P ′ = m X s =0 p ′ s ∂ s , where p ′ m ∈ k and p ′ m − has constant coefficientsand any operator Q ′ ∈ B with ord Γ ( Q ′ ) = (1 , n ) has the form Q ′ = n X s =0 q ′ s ∂ s , where q ′ n = c ∂ + c , c , c ∈ k . ii) If B ′ = S − BS , S ∈ ˆ D is an equivalent -quasi elliptic ring containing a pair ofnormalized operators P ′ , Q ′ with ord Γ ( P ′ ) = (0 , k ′ ) , ord Γ ( Q ′ ) = (1 , l ′ ) ( k ′ > ), then S hasthe form S = c exp( c x ∂ ) exp( c x + c x ) ∈ ˆ D , where c , c , c , c ∈ k (cf. remark 2.10).Proof. i) We have0 = [ P, P ′ ] = k∂ ( p ′ m ) ∂ k + m − + k∂ ( p ′ m − ) ∂ k + m − + [ p k − , p ′ m ] ∂ k + m − + terms of lower degree . (10)Hence ∂ ( p ′ m ) = 0 , i.e. p ′ m don’t depend on x . Then we have0 = [ Q, P ′ ] = [ ∂ , p ′ m ] ∂ m + l +[ ∂ , p ′ m − ] ∂ l + m − +[ q l − , p ′ m ] ∂ l + m − + terms of lower degree . (11)Hence [ ∂ , p ′ m ] = 0 and therefore p ′ m must be an operator with constant coefficients. So, p ′ m ∈ D (and clearly these arguments work for any operator from B ). Since ord Γ ( P ′ ) = (0 , m ) , p ′ m is a constant, and since ord Γ ( Q ′ ) = (1 , n ) , q ′ n must be a linear polynomial. But thenfrom (11) we have [ ∂ , p ′ m − ] = 0 , i.e. p ′ m − don’t depend on x , and from (10) we have ∂ ( p ′ m − ) = 0 , i.e. p ′ m − must be an operator with constant coefficients.ii) We have P ′ = S − ˜ P S , Q ′ = S − ˜ QS for some operators ˜ P , ˜ Q ∈ B . Since S isinvertible, we obviously have S = c ∈ k ∗ mod ( x , x ) . P , ˜ Q are constant-coefficient operators,we must have ord Γ ( ˜ P ) = (0 , k ′ ) and ord Γ ( ˜ Q ) = (1 , l ′ ) . From remark 2.10 we know that thereexists an operator S of the form exp( cx ) such that S − ˜ q l ′ S = ∂ (here ˜ q l ′ is a linearpolynomial with constant coefficients). Then obviously the operator S ′ = SS − don’t dependon x . So, S = S ′ S .From remark 2.9 we know that ˜ P , ˜ Q ∈ Π and from item i) we know that ˜ p k ′ , ˜ p k ′ − areoperators with constant coefficients (and ˜ p k ′ = ∂ k ′ ). Thus, S ′ has the form S ′ = exp( F ( x , ∂ )) , where F is a polynomial in x , ∂ . This polynomial is linear iff ˜ p k ′ − is linear. But if it is notlinear, then the operator ( S ′ ) − ˜ P S ′ will not satisfy the condition A (as ˜ P satisfies A forsome ( k, l ) ), a contradiction. So, it is linear and we are done. Remark 2.11.
As this lemma shows, if there is a pair of normalized operators in B , then anyequivalent ring B ′ that has a pair of normalized operators is obtained from B by conjugationwith an operator of special form, and this conjugation is equivalent to a linear change of variables ∂ ∂ + c∂ + b, ∂ ∂ + d (12)with c, b, d ∈ k . The Schur pair corresponding to such a ring B ′ will be equivalent to the firstone as well.Conversely, if one starts from any Schur pair ( A, W ) in a given equivalence class, then thering B can be constructed as B = SAS − , where S now comes from the analogue of the Satotheorem (see theorem 2.2 below). If ( A ′ , W ′ ) is an equivalent Schur pair, then A ′ = T − AT , W ′ = W T for some 1 -admissible operator T , which can be written (see remark 2.10) in theform T = T ′ T , where T has the form (9), and T ′ = 1 + T − , where T − ∈ ˆ D [[ ∂ − ]] ∂ − .Then it is easy to see that the corresponding Sato operator for the space W ′ from theorem 2.2is S ′ = T − ST ′ T . So, the corresponding ring B ′ = S ′ A ′ ( S ′ ) − = T − BT , i.e. it is obtainedfrom B by the linear change (12). It will automatically contain a pair of normalized operators.To find a pair of normalized operators in a given ring B we need sometimes to replace B by an equivalent ring (see [39, Lemma 2.10]). Theorem 2.2. ([39, Th.3.1]) Let W be a k -subspace W ⊂ k [ z − ](( z )) with Supp( W ) = W .Let { w i,j , i, j ≥ } be the unique basis in W with the property w i,j = z − i z − j + w − i,j , where w − i,j ∈ k [ z − ][[ z ]] z . Assume that all elements w i,j satisfy the condition A .Then there exists a unique operator S = 1 + S − satisfying A , where S − ∈ ˆ D [[ ∂ − ]] ∂ − ,such that W S = W . The Schur pairs from theorem 2.1 one to one correspond to pairs of subspaces in the space k [[ u ]](( t )) via an isomorphism ψ : k [ z − ](( z )) ∩ Π ≃ k [[ u ]](( t )) z t, z − ut − , (13)where k [ z − ](( z )) ∩ Π denotes the k -subspace generated by series satisfying the condition A (see [39, Cor.3.3]). We will denote these pairs by the same letters ( A, W ) . Obviously, W · A ⊂ W . Definition 2.16. ([39, Def.3.5,3.6]) For the ring A ⊂ k [[ u ]](( t )) define N A = GCD { ν t ( a ) , a ∈ A such that ν ( a ) = (0 , ∗ ) } , where ∗ means any value of the valuation. Define˜ N A = GCD { ν t ( a ) , a ∈ A } . We’ll say that the ring A is strongly admissible if there is an element a ∈ A with ν ( a ) = (1 , ∗ )and ˜ N A = N A . 16 efinition 2.17. ([39, Def.3.8]) For 1 -quasi elliptic commutative ring B ⊂ ˆ D we define num-bers ˜ N B , N B to be equal to the numbers ˜ N A , N A , where A is the ring corresponding to B by theorem 2.1 (after applying the isomorphism (13)). We say that B is strongly admissible if A is strongly admissible.For a strongly admissible ring B we define the rank of B asrk( B ) = N B = ˜ N B Remark 2.12.
If the ring B in the definition 2.17 is a ring of PDO’s, then the numbers ˜ N B , N B and the rank can be defined similarly to definition 2.16: N B = GCD { ord( Q ) , Q ∈ B such that ord Γ ( Q ) = (0 , ∗ ) } . Define ˜ N B = GCD { ord ( Q ) , Q ∈ B } , where ord means the usual order in the ring D .Analogously, B is strongly admissible if there is an element Q ∈ B with ord Γ ( Q ) = (1 , ∗ )and ˜ N B = N B . Its rank rk( B ) = N B = ˜ N B .We would like to emphasize that the rank of the ring B defined as N B = ˜ N B is less or equalto the rank of the sheaf of common eigenfunctions of the operators from B (cf. [23, Rem.2.3],this notion of rank is often used in various papers). This follows from [23, Prop. 3.3, Prop. 3.2,Th.2.1].Below we will write rk ( B ) to denote the rank in the second sense. Theorem 2.3. ([39, Th.3.4]) There is a one to one correspondence between the set of classesof equivalent -quasi elliptic strongly admissible finitely generated rings of operators in ˆ D ∩ Π of rank r and the set of isomorphism classes of geometric data M r of rank r . If we have a ring B ⊂ D of commuting PDOs satisfying the property 7, then by [39, Lemma2.6] and by [39, Prop.2.4] (cf. also the beginning of section 3.1 in loc. cit.) there is a linearchange of variables making this ring 1 -quasi elliptic strongly admissible. Moreover, as it followsfrom the proofs of [39, Lemma 2.6, Prop.2.4], almost all linear changes of variables preserve theproperty of the ring to be 1 -quasi elliptic strongly admissible. In particular, for almost all linearchanges we have the following extra property of the operators P, Q from definition 2.9: σ ( P ) = ξ k + k X q =1 h q ξ q ξ k − q , h k = 0; σ ( Q ) = ξ ξ l + l +1 X q =2 c q ξ q ξ l +1 − q , c l +1 = 0 . (14) Remark 2.13.
From the construction in [39, Sec.3] explaining the correspondence betweengeometric data and 1 -quasi elliptic strongly admissible rings it follows that the ring after suchlinear change of variables corresponds to the data with the same surface and divisor, but withprobably other sheaf, other point P and trivializations π, φ (cf. also remark 2.4 and [23, Th.2.1,Prop.3.3]). Remark 2.14.
If the ring B of PDOs is 1 -quasi-elliptic strongly admissible, then, obviously,there exist two operators P, Q as in definition 2.9 with k = l + 1 = ord ( P ) . In this situation,examining the arguments from the proof of Lemma 2.10, item 1 in [39], we see that there aresome β ∈ k and f ∈ k [[ x , x ]] ∗ such that the operators f − ( P + βQ ) f , f − Qf are normalized(in the sense of [39, Def. 2.19]). Thus, in the equivalent class of B we can find a ring of PDOswith a pair of normalized operators.As the arguments from remark 2.11 show, any Schur pair equivalent to the Schur pair as-sociated with B leads to a ring B ′ obtained from B by the linear change of variables (12).Thus, B ′ is also a ring of PDOs! 17here is another nice property of 1 -quasi elliptic subrings of partial differential operatorsclaiming the ”purity” of such rings: Proposition 2.2. ([39, Prop.3.1]) Let B ⊂ D ⊂ ˆ D be a -quasi elliptic ring of commutingpartial differential operators. Then any ring B ′ ⊂ ˆ D of commuting operators such that B ′ ⊃ B is a ring of partial differential operators, i.e. B ′ ⊂ D . We would like to recall that in the classical KP theory there are well known geometric dataclassifying the commutative rings of ordinary differential operators. These data consist of aprojective curve over a field k plus a line bundle (or a torsion free sheave if the curve issingular) plus some additional data (a distinguished point p of the curve plus a formal localparameter at p , and a formal trivialization at p of the sheaf). Also there is a map whichassociate to each such data a pair of subspaces ( A, W ) (”Schur pair”) in the space V = k (( z )) ,where A ! k is a stabilizer k -subalgebra of W in V : A · W ⊂ W , and W is a point ofthe infinite-dimensional Sato grassmannian (see e.g. [28] for details). This map is usually calledas the Krichever map in the literature. In works [34], [33] (see also [31]) Parshin introducedan analogue of the Krichever map which associates to each geometric data (which include inthat works a Cohen-Macaulay surface, an ample Cartier divisor, a smooth point and a vectorbundle) a pair of subspaces ( A , W ) in the two-dimensional local field associated with the flag(surface, divisor, point) k (( u ))(( t )) (with analogous properties). He showed that this map isinjective on such data. In works [33], [31] some combinatorial construction was also given. Thisconstruction helps to calculate cohomology groups of vector bundles in terms of these subspacesand permits to reconstruct the geometric data from the pair ( A , W ) . The difference of this newKrichever-Parshin map from the Krichever map is that the last map is known to be bijective.To extend the Krichever-Parshin map and to make it bijective we introduced in the work [21]new geometric objects called formal punctured ribbons (or simply ribbons for short) and torsionfree coherent sheaves on them, we extended this map on the set of new geometric data whichinclude these objects and showed the bijection between the set of geometric data and the set ofpairs of subspaces ( A , W ) (also called generalized Schur pairs) satisfying certain combinatorialconditions. We also showed that for any given Parshin’s geometric data one can construct aunique geometric data with a ribbon, and the initial Parshin’s data can be reconstructed fromthe new data with help of the combinatorial construction mentioned above. In the work [23] weextended this construction to the modified Parshin’s data from definition 2.2. First let’s recall some properties of the classical Krichever map for torsion free sheaves of rankone (see [27], [28] for details; we change slightly some notation from these papers here). Inthis case the geometric data is a quintet (
C, P, F , u, φ ) , where C is a projective curve, P isa smooth point on C , F is a torsion free sheaf of rank one, u is a local parameter at thepoint P (in particular, there is an isomorphism π : ˆ O C,P ≃ k [[ u ]] ), and φ : ˆ F P ≃ k [[ u ]] isa trivialisation. Obviously, any two such trivialisations differ by multiplication on an element a ∈ k [[ u ]] ∗ . The Schur pair A, W ⊂ k (( u )) is constructed in analogous way as in section 2.1: A is the image of the group H ( C \ P, O C ) in the space k (( u )) (which is obtained using thetrivialisation π ), and W is the image of the group H ( C \ P, F ) in the space k (( u )) (which isobtained using the trivialisation φ ). If we choose another trivialisation of the sheaf F then thenew space W will differ from the old one by multiplication on the element a ∈ k [[ u ]] ∗ and thespace A will not change.In this case we also have the following properties in terms of spaces A, W : F ( nP ) ≃ Proj( ˜ W ( n )) , (15)18here ˜ W = ⊕ ∞ n =0 W n s n , W n = W ∩ u n · k [[ u ]] ; H ( C, F ) ≃ W ∩ k [[ u ]] , H ( C, F ) ≃ k (( u )) W + k [[ u ]] . (16)Recall that all torsion free rank one sheaves with fixed Euler characteristic can be divided intothe union of orbits of the group Pic ( C ) . Namely (see [36, Sec.6]) there are maximal torsionfree sheaves, i.e. sheaves not isomorphic to a direct image of a torsion free sheaf on a (partial)normalisation of the curve C , and not maximal torsion free sheaves, i.e. sheaves isomorphic todirect images of torsion free sheaves on partial normalisations of C . If the sheaf is maximalthen the action of the group Pic ( C ) is free on it. Thus, every orbit of a torsion free sheaf ofrank one is a torsor over Pic ( C ′ ) , where C ′ is a partial normalisation of C . This torsor hasa topology induced by the topology of Pic ( C ′ ) .Further we will need the following fact: if the Euler characteristic of a sheaf F on a projectivecurve C is k ≥ V in the orbit of this sheaf such thatfor each L ∈ VH ( C, L ) = 0 , h ( C, L ) = k, H ( C, L ( − kP )) = 0 , H ( C, L ( − kP )) = 0 (17)for some point P ∈ C . This fact can be proved by induction on k as follows. For any torsionfree sheaf F with Euler characteristic k ≥ Q and n ≫ H ( C, F ( nQ )) = 0 , h ( C, F ( nQ )) = k + n > a ∈ H ( C, F ( nQ ))there is a dense open subset U ⊂ C such that for any P ∈ U the image of an embedding of a in O C,P (with help of any trivialisation) is invertible. Then from properties (15) and (16) itfollows that h ( C, F ( nQ − P )) = n + k − , H ( C, F ( nQ − P )) = 0 . Thus, by induction there exists an open subset U ′ ⊂ C such that for any P ∈ U ′ H ( C, F ( nQ − ( n + k ) P )) = 0 , H ( C, F ( nQ − ( n + k ) P )) = 0 ,H ( C, F ( n ( Q − P ))) = 0 , h ( C, F ( n ( Q − P ))) = k. The rest of the proof follows from [16, Th.12.8] for the morphism f : P ic ( C ) × C → P ic ( C )and the Poincare sheaf P . ( A , W )Now we would like to recall three properties of the pair ( A , W ) . First we recall (see, for exam-ple, [21]) that a k -subspace W in k (( u )) is called a Fredholm subspace ifdim k W ∩ k [[ u ]] < ∞ and dim k k (( u )) W + k [[ u ]] < ∞ .For a k -subspace W in k (( u ))(( t )) , for n ∈ Z let W ( n ) = W ∩ t n k (( u ))[[ t ]] W ∩ t n +1 k (( u ))[[ t ]]be a k -subspace in k (( u )) = t n k (( u ))[[ t ]] t n +1 k (( u ))[[ t ]] . 19 .6.3 The first property Let the pair ( A , W ) be the image of the ribbon’s data corresponding to some geometric data ofrank one from definition 2.2 with F being coherent of rank one (for details see [23, Sec.3.5]).Recall (see definition 6, remark 2.6) that for such rank 1 torsion free sheaves the map ζ (6) wasdefined. Then (see the proof of theorem 1 in [21]) A ( nd ) ≃ the image of the quintet ( C, P, O C ( nC ′ ) , u, id ) in k (( u )) under the Krichever map , (18)where C ′ = dC is the ample Cartier divisor as above (note that O C ( nC ′ ) ≃ ζ ( O X ( nC ′ )) ), and W ( nd + k ) ≃ the image of the quintet ( C, P, ζ ( F k ( nC ′ )) , u, φ ) under the Krichever map , (19)where 0 ≤ k < d and φ is some trivialization of the sheaf ζ ( F k ( nC ′ )) at the point P on C (note that ζ ( F k ( nC ′ )) ≃ ( F k / F k − )( nC ′ ) ). Thus, from one-dimensional KP theory (see (16))we have H ( C, ( F k / F k − )( nC ′ )) ≃ W ( nd + k ) ∩ k [[ u ]] ,H ( C, ( F k / F k − )( nC ′ )) ≃ k (( u )) / ( W ( nd + k ) + k [[ u ]]) (20) Assume that the pair A , W ∈ k (( u ))(( t )) comes from a geometric data of rank one. Then H ( X, O X ( nC ′ )) ≃ A · t nd ∩ k [[ u ]](( t )) ∩ k (( u ))[[ t ]], (21) H ( X, O X ( nC ′ )) ≃ A · t nd ∩ ( k [[ u ]](( t )) + k (( u ))[[ t ]]) A · t nd ∩ k [[ u ]](( t )) + A · t nd ∩ k (( u ))[[ t ]] , (22) H ( X, O X ( nC ′ )) ≃ k (( u ))(( t )) A · t nd + k [[ u ]](( t )) + k (( u ))[[ t ]] . (23)For the proof see remark 3 and lemma 1 from [22] (remark 3 refers for the proof to papers[31, 33], where C was assumed to be a Cartier divisor; in general case it is not difficult toimprove the proof from these papers; nevertheless, we will need this property in our paper onlyfor such cases when C is known to be Cartier). In particular, if C is a Cartier divisor, it followsthat O X ( nC ) ≃ O X,n , ζ ( O X ( nC )) ≃ O C ( nC ) (24)for any n (cf. remark 2.6). : If A is Cohen-Macaulay ring then A = A ∩ k [[ u ]](( t )) , (25)where A, W are the subspaces in k [[ u ]](( t )) constructed above starting from the geometricdata. This claim was proved in [23, Remark 3.4]. In the introduction to the loc.cit. the analogousproperty for the space W was also announced (though imprecise): W = W ∩ k [[ u ]](( t )) . We are going to clarify it here. 20 roposition 2.3.
Let F be a coherent rank one torsion free sheaf on a projective surface X defined over an uncountable algebraically closed field k . Assume that there is an ampleirreducible Q -Cartier divisor C ⊂ X not contained in the singular locus such that C = 1 .Let C ′ = dC be a very ample Cartier divisor. Suppose that the following conditions hold (seeremark 2.5, definition 6): χ ( X, F ( nC ′ )) = ( nd + 1)( nd + 2)2 , H ( C, ζ ( F k )( − ( k + 1) Q )) = H ( C, ζ ( F k )( − ( k + 1) Q )) = 0 for a smooth point Q ∈ C , n ≥ , where ≤ k < d . Theni) there exists a point P ∈ C regular in C and X such that the conditions from item 6 ofdefinition 2.2 hold for a trivialisation ˆ φ : ˆ F P ≃ k [[ u, t ]] , i.e. the homomorphisms H ( X, F ( nC ′ )) → k [[ u, t ]] / ( u, t ) nd +1 are isomorphisms for all n ≥ ; ii) for this trivialisation W = W ∩ k [[ u ]](( t )); iii) for this trivialisation H ( X, F ) ≃ W ∩ ( k [[ u ]](( t )) + k (( u ))[[ t ]]) W ∩ k [[ u ]](( t )) + W ∩ k (( u ))[[ t ]] = 0 , (26) H ( X, F ) ≃ k (( u ))(( t )) W + k [[ u ]](( t )) + k (( u ))[[ t ]] = 0 ; (27) iv) the sheaf F is Cohen-Macaulay on X .Proof. i) For any sheaf F k and m > → ζ ( F k ) → ζ ( F k ) ⊗ O C O X ( mC ′ ) | C → ζ ( F k ) ⊗ O C ( O X ( mC ′ ) | C / O C ) → , since O X ( mC ′ ) | C is an invertible sheaf. Hence we have H ( C, ζ ( F k ) ⊗ O C O X ( mC ′ ) | C ) = 0 forall m ≥ C = 1 (i.e. deg( O X ( C ′ ) | C ) = d ), by the asymptotic Riemann-Roch theoremwe have χ ( ζ ( F k ) ⊗ O C O X ( mC ′ ) | C ) = h ( C, ζ ( F k ) ⊗ O C O X ( mC ′ ) | C ) = md + k + 1 . (28)For each m ≥ U m in C such that H ( C, ζ ( F k ) ⊗ O C O X ( mC ′ ) | C ⊗ O C O C ( − ( md + k + 1) P )) = H ( C, ζ ( F k ) ⊗ O C O X ( mC ′ ) | C ⊗ O C O C ( − ( md + k + 1) P )) = 0 (29)for any P ∈ U m . Therefore, since the ground field is uncountable, there exists a point P ∈∩ ∞ m =0 U m regular in C and X such that these properties hold simultaneously for all m ≥ ≤ k < d . Since for any n ≥ W nd + k /W nd + k − ≃ H ( X, F k ( nC ′ )) /H ( X, F k − ( nC ′ )) ֒ → H ( C, ζ ( F k ) ⊗ O C O X ( nC ′ ) | C ) , For reader who prefer the alternative definition this item can be reformulated as follows. There exist j : T → X with j ( O ) = P ∈ C \ ( C sing ∪ X sing ) and j − ( C ) = T (i.e. R = k [[ u, t ]] ≃ ˆ O X,P , R/tR ≃ ˆ O C,P )such that for a choice of generator of F P as O X,P -module the embedding F ֒ → j ∗ O T (corresponding to( j ∗ F ) O = R ⊗ O X,P
F ≃ ˆ F P ) condition 3 is satisfied, i.e. the maps H ( X, F ( nC ′ )) → R · t − nd / M nd +1 · t − nd areisomorphisms. χ ( H ( X, F ( nC ′ ))) ⊂ W nd by definition, we have that for all n ≫ h ( X, F ( nC ′ )) = χ ( F ( nC ′ )) = ( nd + 1)( nd + 2)2 ≤ dim k ( W nd ) ≤ d − X k =0 n − X m =0 h ( C, ζ ( F k ) ⊗ O C O X ( mC ′ ) | C ) + h ( C, ζ ( F k ) ⊗ O C O X ( nC ′ ) | C ) . (30)By (28) these inequalities are equalities. Therefore, H ( X, F ( nC ′ )) ≃ W nd for any n ≫ F ≃ F and H ( X, F ( nC ′ )) ≃ W nd , W nd + k /W nd + k − ≃ H ( C, ζ ( F k ) ⊗ O C O X ( nC ′ ) | C ) (31)for all n ≥ P .ii) By (16) and (19) we have H ( C, ζ ( F k ) ⊗ O C O X ( nC ′ ) | C ) ≃ W ( nd + k ) ∩ k [[ u ]] . From this and from (31) follows that W = W ∩ k [[ u ]](( t )) .iii) By assumption on the Euler characteristic of the sheaf F and from (31) it follows h ( X, F ) − h ( X, F ) = 0 . From (29) we know that h ( C, ζ ( F k ) ⊗ O C O X ( nC ′ ) | C ) = 0 for any0 ≤ k < d and n ≥ → F k → F k +1 → ζ ( F k +1 ) → ≤ k . So, from the induced long exact cohomological sequences and from (31) we obtain H ( X, F k ) ≃ H ( X, F k +1 ) for any k ≥ H ( X, F k + nd ) = H ( X, F k ( nC ′ )) = 0 for all n ≫ H ( X, F ) = 0 . From item ii) we get W ∩ ( k [[ u ]](( t )) + k (( u ))[[ t ]]) ⊂ W ∩ k [[ u ]](( t )) + W ∩ k (( u ))[[ t ]] , where from W ∩ ( k [[ u ]](( t )) + k (( u ))[[ t ]]) W ∩ k [[ u ]](( t )) + W ∩ k (( u ))[[ t ]] = 0 . By (16) and (19) we have0 = H ( C, ζ ( F k ) ⊗ O C O X ( nC ′ ) | C ) ≃ k (( u )) W + k [[ u ]]for all k ≥ k (( u ))(( t )) W + k [[ u ]](( t )) + k (( u ))[[ t ]] = 0 . iv) By [23, Cor.3.1] the sheaf F is Cohen-Macaulay along C . The same arguments show thatthe sheaves F k are Cohen-Macaulay along C . Consider the Macaulaysation CM ( F ) of thesheaf F (see [23, Appendix B]). Consider the image W ′ = χ ( H ( X \ C, CM ( F ))) in k [[ u ]](( t )) ,where we use the same embedding φ of the sheaf F to define χ (cf. section 2.1). Let’s notethat CM ( F ) | mC ′ = F | mC ′ is pure of dimension one for any m > F is Cohen-Macaulayalong C . Then by lemma 2.1 we have H ( X, CM ( F )( nC ′ )) ≃ W ′ nd for all n ≥ CM ( F ) k are Cohen-Macaulay for all k . Note that CM ( F k ) ≃ CM ( F ) k . Indeed, by definition ofCohen-Macaulaysation we have CM ( F k ) ⊂ CM ( F ) k . If CM ( F k ) CM ( F ) k , then thiswould mean CM ( F k ) − k ( CM ( F ) k ) − k ≃ CM ( F ) . But CM ( F k ) − k ≃ CM ( F ) , since22 M ( F k ) − k ⊂ ( CM ( F ) k ) − k ≃ CM ( F ) and CM ( F k ) − k is a Cohen-Macaulay sheaf contain-ing F (cf. [23, Rem.B.2]).In particular, we can apply the construction from [23, Sec.3.5] and construct a torsion freesheaf N on the ribbon ( C, A ) (the ribbon constructed by our geometric data). Then we canconstruct a space W ′ ⊂ k (( u ))(( t )) by the sheaf N . Since the construction depends only onsections of the sheaves CM ( F k ) along the curve C , we get W ′ = W . Hence from item ii) weobtain F ≃ CM ( F ) . Remark 2.15.
Torsion free sheaves of rank one on the projective surface X with fixed Hilbertpolynomial χ from proposition 2.3 are stable in the sense of standard definition from [17, Ch.2].Stable sheaves are parametrized by a projective scheme M X ( χ ) (see Chapter 4 in loc.cit.).On the other hand, all sheaves we are interested in satisfy the assumptions of lemma 2.2(and in view of theorems 3.1 and 4.1 even more strong assumption: they are Cohen-Macaulayon X ). By [6, Prop.1.2.16] the Cohen-Macaulayness is an open condition. So, it is reasonableto consider an open subscheme M X of the moduli space M X ( χ ) parametrising such sheaves.Then the map ζ from section 2.4 induces the morphism ζ : M X → M C ( g ) , where M C ( g ) is the moduli space of rank one torsion free sheaves of degree g = p a ( C ) on C (cf. [35]). We conjecture that this morphism is surjective (cf. examples at the end of this paper). Recall that by [23, Th.2.1] any commutative ring B ⊂ D of PDOs satisfying the property (7)leads to a datum ( X, C, L ) , where X, C are the same as in definition 2.2 and L is a torsionfree coherent sheaf on X . Let’s assume (see the discussion before remark 2.4) that B is a1-quasi-elliptic strongly admissible ring satisfying the extra property (14).In this case by [23, Prop. 3.3] the datum ( X, C, L ) is isomorphic to the triple ( X, C, F ) (apart of geometric data) from theorem 2.3. If B is of rank one, then by [23, Th. 2.1] we have C = 1 , and by [23, Prop. 3.2] the sheaf F and the geometric data from theorem 2.3 are ofrank one. Theorem 3.1.
Let ( X, C, P, F , π, φ ) be a geometric datum corresponding to a 1-quasi-ellipticstrongly admissible ring B ⊂ D of commuting operators satisfying the properties (7) , (14) .Then F is a Cohen-Macaulay sheaf on X . Remark 3.1.
If the ring B is maximal then by [39, Th.4.1] the surface X is also Cohen-Macaulay. Proof.
By [23, Prop.3.2] the sheaf
F ≃ L is coherent. By [23, Th.2.1, Prop. 3.3] and remark2.2 the rank of the geometric datum is one.Consider the Macaulaysation CM ( F ) of the sheaf F (see [23, Appendix B]). By [23, Cor.3.1] the sheaf F is Cohen-Macaulay along C ; in particular, F P ≃ CM ( F ) P . Consider theimage W ′ = χ ( H ( X \ C, CM ( F ))) in k [[ u ]](( t )) , where we use the embedding φ of the sheaf F to define χ (cf. section 2.1). Then we claim that this image is a finitely generated linearspace over W = χ ( H ( X \ C, F )) : W ′ = h W, w , . . . , w k i , where w , . . . , w k / ∈ W , w , . . . , w k ∈ k [[ u ]](( t )) .23o prove the claim, first of all let’s note that the sheaf CM ( F ) | mC ′ = F | mC ′ is pure ofdimension one for any m > F is Cohen-Macaulay along C . Let’s show that the condition on the space W ′ from lemma 2.1 is satisfied. This conditionis true for elements w from the space W corresponding to the sheaf F , because W nd ≃ H ( X, F ( nC ′ )) . It is also clear that for any element w from W ′ we have w ∈ f − md F P forsome m . Now take any element w ∈ W ′ nd Then we have for all sufficiently big m > f − md w − c w − . . . − c k w k = a ∈ W ( n + m ) d for some c , . . . , c k ∈ k . Thus, f nd w = f ( n + m ) d a + f ( n + m ) d ( c w + . . . + c k w k ) ∈ F P .Now by lemma 2.1 we have H ( X, CM ( F )( nC ′ )) ≃ W ′ nd for all n ≥ n > W ′ nd /W ′ ( n − d ≃ H ( C, CM ( F )( nC ′ ) | C ′ ) = H ( C, F ( nC ′ ) | C ′ ) ≃ W nd /W ( n − d . Obviously, W ′ nd ⊃ W nd for all n . So, our claim is proved.By [39, Th.3.3, Th.3.1] there is a unique operator S satisfying condition A such that ψ − ( W ) = W S (the map ψ is defined in (13)), where W = k [ z − , z − ] . Moreover, B = Sψ − ( A ) S − , where A = χ ( H ( X \ C, O X )) . Since W ′ · A ⊂ W ′ , we have( ψ − ( W ′ ) S − ) · B ⊂ ( ψ − ( W ′ ) S − ) , ψ − ( W ′ ) S − = h W , ˜ w , . . . , ˜ w k i , where ˜ w i = ψ − ( w i ) S − . Each series ˜ w i can be written in the following way:˜ w i = w ′ i + w ′′ i , w ′ i = X k ≥ ,l> ,k + l = q i c kl z − k z l , w ′′ i = X k ≥ ,l> ,k + l P n , Q n satisfy the same property(14) with k, l replaced by kn, ln . For all n ≫ w i we must have˜ w i P n ∈ ( ψ − ( W ′ ) S − ) , ˜ w i Q n ∈ ( ψ − ( W ′ ) S − ) . Direct calculations show that these elements can be represented as˜ w i P n = w ′ i σ ( P ) n + ”lower order terms” , ˜ w i Q n = w ′ i σ ( Q ) n + ”lower order terms” . Hence, since n ≫ w ′ i σ ( P ) n , w ′ i σ ( Q ) n ∈ W . Therefore, w ′ i σ ( P ) n , w ′ i σ ( Q ) n mustbe homogeneous polynomials of orders q i + nk , q i + n ( l + 1) . Since the characteristic schemes of P and Q have no intersection, this mean that w ′ i must be a homogeneous polynomial of order q i . But then since w ′ i / ∈ W and due to the property (14) the polynomials w ′ i σ ( P ) n , w ′ i σ ( Q ) n will contain a nonzero monomial of type cz − a z b / ∈ W for b > w i must be zero, and W ′ = W , where from CM ( F ) = F . As we have mentioned in Introduction, there are examples of algebraically integrable commuta-tive rings of PDOs whose affine spectral surface satisfy the following property: its normalisationis A . The following theorem clarifies what is the normalisation of the projective spectral surface X . Theorem 3.2.
Let X be a projective surface, C ⊂ X — an integral Weil divisor not containedin the singular locus of X which is also an ample Q -Cartier divisor and C = 1 . Assume that X \ C ≃ A .Then X ≃ P , C ≃ P . roof. Since C is not contained in the singular locus of X , we can choose a point P regularon C and on X . Choose local parameters u, t at P such that t is a local equation of C atthe point P and u ∈ O P restricted to C is a local equation of the point P on C .Now we have the natural isomorphism π : ˆ O P → k [[ u, t ]] . Using this isomorphism we can repeat the construction of the subspace A from section 2.1 anddefine A def = χ ( H ( X \ C, O )) .Repeating the arguments from the proof of [39, lemma 3.6] we obtain that for all n ≥ H ( X, O X ( nC ′ )) ≃ A nd , where A l = A ∩ t − l k [[ u, t ]] . Since C = 1 , we must have for all n ≫ A nd /A ( n − d ) = d n + const. (32)Consider any element a ∈ A nd such that ν ( a ) = ( ∗ , nd ) . We claim that ∗ ≤ nd .Indeed, by [23, sec.3.4] there is a canonically defined ribbon ( C, A ) over the field k . Thus,by the proof of [21, Th.1] we can construct the space A in k (( u ))(( t )) which is a generalizedFredholm subspace (see loc. cit. or section 2.6). As it follows from (18), the space A ( nd ) isnaturally isomorphic to a Fredholm subspace in the field k (( u )) obtained as the image of thesheaf O X ( nC ′ ) | C under the Krichever map. For n ≫ H ( C, O X ( nC ′ ) | C ) ≃ A nd /A nd − and by (16)dim( A ( nd ) ∩ k [[ u ]]) = h ( C, O X ( nC ′ ) | C ) = nd + const. (33) Remark 3.2.
Alternately, we can just repeat the construction of the generalized Krichever mapfrom [33] or from [31] in our situation (replacing a Cartier divisor there with a Q -Cartier one)to avoid referring to the theory of ribbons.Since C = 1 , we have that the Euler characteristic χ ( A ( nd )) = nd + const. Now we can apply arguments from the proof of [38, Th.1] to show that ∗ ≤ nd . Assume theconverse. We have a · A (0) ⊂ A ( nd ) . It is easy to see that χ ( a · A (0)) = χ ( A (0)) + ∗ . Now wehave χ ( A ( nd )) = nd + const < ∗ + const = χ ( A (0)) + ∗ = χ ( a · A (0)) ≤ χ ( A ( nd )) , a contradiction.Now note that, since X \ C ≃ A , we have A ≃ k [ p, q ] . So, the space A is generated bymonomials p k q l . Because of the claim and formulas (32) and (33) we conclude that (without lossof generality) ν ( p ) = (0 ,
1) , ν ( q ) = (1 ,
1) (since any other values would make these formulasimpossible). But then A ≃ k [ t − , ut − ] and X ≃ Proj( ⊕ A n ) = P , C ≃ Proj( ⊕ A n /A n − ) = P (cf. the proof of [39, lemma 3.3]). Using the idea from the proof of theorem 3.2 we can give an example of an affinesurface which can not be a spectral surface of any ring B of PDO’s of rank one satisfying theproperty 7. For example, consider the ring A = k [ X , X , X ] / ( F ) , (34)25here F = X X + X + P rq =1 g q X q , and g i ∈ k [ X ] are any polynomials and k is analgebraically closed field.Then (see [2, Ch.VII, §
3, Ex.5]) A is a factorial ring, and Spec( A ) is a rational affine surface.It is easy to see that A is not isomorphic to a polynomial ring k [ u, v ] for generic g i and thatSpec( A ) is a smooth surface. Assume that there exists a ring B ⊂ D of rank one satisfyingthe property 7 and such that B ≃ A . Without loss of generality we can assume that B is a1-quasi-elliptic strongly admissible ring. Since the rank of the ring is one, the rank of the datais also one by the classification theorem 2.3. Then the sheaf F is coherent of rank one by [23,Prop.3.3]. By theorem 3.1 the sheaf F is Cohen-Macaulay. Since Spec( A ) is smooth, F mustbe locally free on Spec( A ) . But since A is a factorial ring, we have Cl ( A ) ≃ Pic( A ) = 0 ,thus F |
Spec( A ) ≃ O Spec( A ) . But then the space W of the corresponding Schur pair must beequal to the space A . Therefore, A ≃ k [ ut − , t − ] (where u, t are the parameters from (13)),a contradiction. Let B ⊂ D be a commutative ring of rank rk ( B ) = 1 satisfying the properties (7) . Assume that the normalization of Spec( B ) is isomorphic to A .Then there exists a PDO F such that F − BF ⊂ k [ ∂ , ∂ ] .More precisely, F = S∂ n , where S is an operator as in the analogue of the Sato theorem2.2.Proof. We can assume without loss of generality that B is a finitely generated 1-quasi-ellipticstrongly admissible ring satisfying the property (14) (see the beginning of section 2.5 and argu-ments before remark 2.13), since our assertion don’t depend on linear changes of coordinates.Since the rank of the ring B is one, the rank of the corresponding geometric data( X, C, P, F , π, φ ) is also one by the classification theorem 2.3. Then the sheaf F is coherent ofrank one by [23, Prop.3.3] and C is a rational curve with C = 1 by [23, Prop.3.2].The assumption about normalization is equivalent to the assumption that the normalizationof Spec( B ) ≃ X \ C is isomorphic to A . Note that this assumption is equivalent to the as-sumption that the normalization of X is isomorphic to P . Indeed, if p : P ≃ ˜ X → X is thenormalization morphism, then p ∗ ( C ) is a rational irreducible curve. Thus, we have p ∗ ( C ) is anample rational Cartier-Weil divisor on P with p ∗ ( C ) = 1 , i.e. p ∗ ( C ) = P . Therefore, thenormalization of Spec( B ) is isomorphic to the complement to P in P , i.e. A . The conversestatement follows from the same arguments together with theorem 3.2.Let ( ˜ X ≃ P , p ∗ ( C ) ≃ P , p ∗ ( P ) = ˜ P ) be the normalization of ( X, C, P ) . Since P isregular, the local rings O X,P and O P , ˜ P are canonically isomorphic.Now we can repeat the arguments from the beginning of the proof of theorem 3.2 to getan embedding of H ( P \ P , O ) to the same space k [[ u ]](( t )) (here u, t are local parametersat the point P ∈ X ). Let’s denote the image by A ′ . As we have seen above, A ′ must be thenormalization of A . The arguments from the proof of theorem 3.2 show that in fact A ′ ≃ k [ p, q ]with the highest terms of the series p, q equal to t − , ut − correspondingly (so, Supp( A ′ ) = k [ ut − , t − ] ).Set A ′′ = ψ − ( A ′ ) (see (13)). Then we have Supp( A ′′ ) = k [ z − , z − ] , and A ′′ is a 1 -space.By [39, Lemma 2.11, 2),3)] there is an operator S such that S − ∂ S = ψ − ( q ) , S − ∂ S = ψ − ( p ) , and S satisfies the condition A . Thus, S ∈ Adm .Now consider the Schur pair ( ψ − ( A ) , W ) from theorem B .Consider the equivalent pair ( A = Sψ − ( A ) S − , W = W S − ) . Then the ring SA ′′ S − = k [ z − , z − ] is the normalization of the ring Sψ − ( A ) S − (in the field k ( z , z ) ⊂ k (( z ))(( z )) ).Thus, all elements of the space Sψ − ( A ) S − are polynomials in z − , z − .26he space W S − is a finitely generated module over Sψ − ( A ) S − . Without loss of gen-erality we can assume that 1 ∈ W S − by taking another equivalent Schur pair ( A , W T ) ifneeded (for an appropriate operator T with constant coefficients; T just changes the trivial-isation φ in definition 2.2, item 6). By construction of the Schur pair given in section 2.1 wehave W ⊂ k ( z , z ) (as this Schur pair corresponds to a pair coming from the geometric datawith an appropriate trivialization φ , and the rank of the coherent sheaf F is one).So, W is generated by a finite number of elements from k ( z , z ) over A . Since W is a 1 -space, we can choose the generators to be the elements satisfying the condition A . Let’s denoteby Q their common denominator. From lemma 3.1 (see below) it follows that ord Γ ( Q ) = (0 , n ) ,where n = ord ( Q ) (here and below we identify z with ∂ − , z with ∂ − ; in this case ord ( Q ) = deg( Q ) , where deg means the usual degree of the polynomial Q in two variables).Consider the equivalent Schur pair ( A , W Q/∂ deg( Q )2 ) (this is a Schur pair since Q/∂ deg( Q )2 isa zeroth order operator with constant coefficients with ord Γ ( Q/∂ deg( Q )2 ) = (0 ,
0) satisfyingcondition ( A ) !). Note that all elements from the space W Q/∂ deg( Q )2 are just polynomials in ∂ , ∂ , ∂ − with constant coefficients, and the order of these polynomials with respect to ∂ − is less or equal to deg( Q ) .Then from the proof of theorem 2.2 in [39] immediately follows that the operator S is a(non-commutative) polynomial in ∂ − of degree with respect to ∂ − not greater than deg( Q ) .By remark 2.14 the ring S A S − is a ring of PDO’s. Then S ∈ k [[ x , x ]][ ∂ ](( ∂ − )) (thisimmediately follows from lemmas 2.9, 2.11 item 2,3 in [39]). Thus, we can set F = S∂ deg( Q )2 . Lemma 3.1.
Assume that the Laurent expansion of the element
P/Q ∈ k ( ∂ , ∂ ) ⊂ k (( ∂ − ))(( ∂ − )) , where P, Q ∈ k [ ∂ , ∂ ] are relatively prime, belongs to k [ ∂ ](( ∂ − )) . Assume that this expansionsatisfies the condition A .Then ord Γ ( Q ) = (0 , ord ( Q )) .Proof. The proof of this lemma is based on several technical routine elementary calculations,and we will hardly use some technical lemmas from [39].Assume the converse. Then Q can be represented as a polynomial in ∂ of the order withrespect to ∂ less than ord ( Q ) , say Q = q n ∂ n − n − X l =0 q l ∂ l , n < ord ( Q ) , where q l ∈ k [ ∂ ] . Let P = m X l =0 p l ∂ l . Now we will prove our lemma in several steps.
Step 1.
First we claim that deg( q n ) + n = ord ( Q ) .Clearly, we always have deg( q n ) + n ≤ ord ( Q ) . Assume that deg( q n ) + n < ord ( Q ) .Let’s show that this will contradict to the condition A for the element P/Q . Since we areworking with series in the field k (( ∂ − ))(( ∂ − )) of pseudo-differential operators with constantcoefficients, we can literally repeat the proofs of lemma 2.8 and corollary 2.1 in [39] to show thatthe statements from these claims remain true also for operators from k (( ∂ − ))(( ∂ − )) .In particular, Q − don’t satisfy the condition A . Indeed, assume that Q − satisfies thecondition A . Then Q − = q − n ∂ − Q ′ , where Q ′ is an operator of the form from [39, Corol.2.1]27atisfying the condition A (by [39, Lemma 2.8]). Then Q = ( Q − ) − = q n ∂ ( Q ′ ) − must satisfythe condition A by [39, Lemma 2.8, Corol.2.1]. But Q don’t satisfy the condition A by ourassumption (namely, the term with the first coefficient q i of Q such that deg( q i ) + i = ord ( Q )will contradict the condition A ), a contradiction.Let P = P + P be any decomposition of P in a sum of two PDOs with constant coefficientssuch that P satisfies the condition A and the degree of P with respect to ∂ is less than m ( P may be zero). Let Q − = Q + Q be any decomposition of Q − in a sum of two pseudo-differential operators from k (( ∂ − ))(( ∂ − )) such that Q satisfies the condition A and thedegree of Q with respect to ∂ is less than − n (since Q − don’t satisfy the condition A , Q is not zero). Denote by α the first coefficient of Q , and by β the first coefficient of P if P = 0 . Now we have two cases: if P = 0 then the product P Q − will not satisfy the condition A , because the coefficient of P Q − containing p m α will not satisfy it; if P = 0 then theproduct P Q − will not satisfy the condition A , because the coefficient of P Q − containing βα will not satisfy it. Thus, P Q − does not satisfy the condition A , a contradiction. Step 2.
Now the idea of the proof is to come to a contradiction with the assumption that q n = const .Obviously, we can multiply the element P/Q on an appropriate degree of ∂ − to make thedegree of its Laurent expansion to be zero. Thus, we can assume without loss of generality that P, Q are polynomials in ∂ − with nonzero free terms p m , q n correspondingly.Now we can write P/Q = ( m X l =0 p l ∂ l − m )( n X l =0 q l ∂ l − n ) − = ( m X l =0 p l q n ∂ l − m )( ∞ X i =0 ( n − X l =0 q l q n ∂ l − n ) i ) . (35)Note that not all q i are divisible by q n . Indeed, otherwise ( q − n Q ) ∈ k [ ∂ , ∂ ] and therefore q − n P = ( P Q − )( q − n Q ) ∈ k [ ∂ ](( ∂ − )) ∩ k (( ∂ − ))[ ∂ ] = k [ ∂ , ∂ ]i.e. P and Q are divisible by q n = const , a contradiction.Note that we can reduce the proof to the case deg( P ) ≤ n − ∂ − ). Indeed, it is easy to see that p m must be divisible by q n . Since P/Q ∈ k [ ∂ ](( ∂ − )) , all expressions of type ( P/Q − a ) ∂ k will again belong to k [ ∂ ](( ∂ − )) forany polynomial a ∈ k [ ∂ ] . Thus, if we take a = p m /q n , then ( P/Q − a ) ∂ = P ′ /Q , wheredeg( P ′ ) < deg( P ) if m ≥ n . Note that P ′ = 0 , since P, Q are relatively prime.Analogously we can reduce the proof to the case deg( P ) = 0 . Indeed, using Euclideanalgorithm, we can always find polynomials a ∈ k [ ∂ ] and F ∈ k [ ∂ , ∂ − ] such that deg( aQ − F P ) < deg( P ) if deg( P ) = 0 . Again ( aQ − F P ) = 0 , since P, Q are relatively prime anddeg( P ) = 0 . Thus, F ( P/Q ) − a = P ′ /Q with deg( P ′ ) < deg( P ) , P ′ = 0 .At last, in the case deg( P ) = 0 the proof follows immediately from (35): P must be divisibleby infinite power of some prime factor of q n , i.e. P = 0 , a contradiction. In this section we give several examples.
First we would like to explain the geometric picture for a class of ”trivial” examples. These areexamples of rings of commuting operators in ˆ D containing, say, the operator ∂ . In this caseall operators obviously don’t depend on x . Nevertheless, the geometry of the correspondingsurfaces and even the naive moduli space of sheaves from geometric data are non-trivial.28ote that, if we have a commutative 1 -quasi-elliptic strongly admissible ring B ⊂ D satis-fying properties (7), (14) and containing the operator ∂ , then after a linear change ∂ ↔ ∂ the ring B will remain 1 -quasi-elliptic strongly admissible and will contain the operator ∂ .So, in particular, the well known example of the quantum Calogero-Moser system (see [30], [5,sec. 5.3] and [39, Ex.4.3]) belongs to this class of ”trivial” examples. We would like to emphasizethat in [5, sec. 5.3] the affine spectral surface of this system was calculated: it is A × H , where H is some hyperelliptic curve. So, by [23, Th.2.1] the projective spectral surface X from thecorresponding geometric data is normal, and singularities appear only on the curve C (whichis rational). Theorem 4.1.
Let B ⊂ ˆ D be a Cohen-Macaulay finitely generated 1-quasi-elliptic stronglyadmissible ring of commuting operators (note that by [39, Th.4.1] any finitely generated 1-quasi-elliptic strongly admissible ring B lies in a Cohen-Macaulay finitely generated 1-quasi-ellipticstrongly admissible ring).Then B contains ∂ if and only if the divisor C of the corresponding geometric data isCartier, the sheaf F is coherent of rank one, O C ( C ) ≃ O C ( P ) and the map H ( X, O X ) → H ( X, O X ( C )) is injective.Moreover, if the ground field k is uncountable and algebraically closed, the sheaf F isCohen-Macaulay on X .Proof. Recall that the surface X corresponding to B is Cohen-Macaulay by [23, Th.3.2].Assume first that B contains ∂ . Since B is 1-quasi-elliptic strongly admissible, this meansthat rk( B ) = 1 . Also this means that for any n ≫ P n ∈ B withord Γ ( P n ) = (0 , n ) . Thus, we can give an approximation of the dimension of the space A n ⊂ A (where, as usual, A means the space of the Schur pair corresponding to the ring B ): dim k ( A n ) ∼ n / n ≫ C = 1(since A nd ≃ H ( X, O X ( ndC )) for all n ≫ B ) = 1 , the rankof the corresponding geometric data is also one by theorem 2.3. Thus, by [23, Prop.3.2] thecorresponding sheaf F is coherent of rank one.Now let’s prove that C is a Cartier divisor. Our arguments will be very similar to thearguments from the proofs of lemma 3.3 in [39] or theorem 2.1 in [23]. Recall that X ≃ Proj ˜ A and the divisor C is defined by the homogeneous ideal I = ( s ) . It is not contained in thesingular locus, since it contains the regular point P . Since ˜ A is a finitely generated k -algebrawith ˜ A = k , by [2, Ch.III, § d ≥ k -algebra˜ A ( d ) = ∞ L k =0 ˜ A kd is finitely generated by elements from ˜ A ( d )1 as a k -algebra (here ˜ A ( d )1 = ˜ A d ).Let’s show that the divisor dC is an effective Cartier divisor. We consider the subscheme C ′ in X which is defined by the homogeneous ideal I d = ( s d ) of the ring ˜ A . The topologicalspace of the subscheme C ′ coincides with the topological space of the subscheme C (as it canbe seen on an affine covering of X ). The local ring O X,C coincides with the valuation ring ofthe discrete valuation on Quot( A ) induced by the discrete valuation ν t on the field k (( u ))(( t )) : O X,C = ˜ A ( I ) = { as n /bs n | n ≥ , a ∈ A n , b ∈ A n \ A n − } .The ideal I induces the maximal ideal in the ring O X,C , and the ideal I d induces the d -thpower of the maximal ideal. Therefore, if we will prove that the ideal I d defines an effectiveCartier divisor on X , then the cycle map on this divisor is equal to dC (see [23, Appendix A]).By [12, prop. 2.4.7] we have X = Proj ˜ A ≃ Proj ˜ A ( d ) . Under this isomorphism the subscheme C ′ is defined by the homogeneous ideal I d ∩ ˜ A ( d ) in the ring ˜ A ( d ) . This ideal is generated by29he element s d ∈ ˜ A ( d )1 . The open affine subsets D + ( x i ) = Spec ˜ A ( d )( x i ) with x i ∈ ˜ A ( d )1 define acovering of Proj ˜ A ( d ) . In every ring ˜ A ( d )( x i ) the ideal ( I d ∩ ˜ A ( d ) ) ( x i ) is generated by the element s d /x i . Therefore the homogeneous ideal I d ∩ ˜ A ( d ) defines an effective Cartier divisor.Now let’s show that the k -algebra ˜ A ( m ) is finitely generated by elements from ˜ A ( m )1 for all m ≫ k -algebra ˜ B ( m ) is finitely generatedby elements from ˜ B ( m )1 for all m ≫ d be such a number that all generators of B lie in B d and for all n ≥ d thereare elements P n ∈ B with ord Γ ( P n ) = (0 , n ) (the same will be true also for the ring A ).Let Σ denote the set of all numbers a ∈ Z + such that there are operators Q in B d withord Γ ( Q ) = ( ∗ , a ) . Since ∂ ∈ B , we have that for any m > d and any a ∈ Σ there areoperators Q in B with ord Γ ( Q ) = ( m, a ) .Now let m > d be any number such that ˜ B ( m ) is finitely generated by elements from˜ B ( m )1 . It suffices to show that ˜ B ( m +1) is also finitely generated by elements from ˜ B ( m +1)1 .To show this it suffices to show that any element from ˜ B ( m +1) k can be represented as a sumof products of elements from ˜ B ( m +1) k − and from ˜ B ( m +1)1 . The space ˜ B ( m +1)1 has two specialoperators: Q = ∂ m +11 and Q with ord Γ ( Q ) = (0 , m + 1) . As it follows from what we havesaid above, for any l ≥ d and any i, j ∈ Z + such that i + j = l there is an element Q ∈ B with ord Γ ( Q ) = ( i, j ) . Thus, any element Q ∈ ˜ B ( m +1) k can be written as a sum of an elementwith the order less than ord Γ ( Q ) and a product of an element from ˜ B ( m +1) k − and Q or Q .By induction we obtain our claim.Now the arguments above for mC and ( m + 1) C (instead of dC ) show that mC , ( m + 1) C are Cartier divisors. But then C must be also a Cartier divisor.Now let ( A , W ) be the pair from k (( u ))(( t )) corresponding to our geometric data (seesection 2.1). As it follows from section 2.1 (namely, from (18), (20) and (25)) A ( n ) ∩ k [[ u ]] ≃ A n /A n − ≃ H ( C, O C ( nC )) , for all n ≫ A ( n ) is the image of ( C, P, O C ( nC ) , u, id ) under the Krichever map (cf. also(24)). From one-dimensional KP theory (see (15), (17)) we have then that A ( n ) · u − n is the imageof the quintet ( C, P, O C ( nC )( − nP ) , u, id ) under the Krichever map. Since ∂ n ∈ B n \ B n − , wehave that t − n u n ∈ A n /A n − . Hence, H ( C, O C ( nC )( − nP )) ≃ A ( n ) · u − n ∩ k [[ u ]] ≃ k, and by Riemann-Roch h ( C, O C ( nC )( − nP )) = g a ( C ) . But then O C ( nC )( − nP ) ≃ O C for all n ≫ O C ( C ) ≃ O C ( P ) .Now we have two possibilities for the number h ( C, O C ( C )) : it is either 1 or 2. If it is equalto 1, then this means that H ( C, O C ( C )) ≃ A (1) ∩ k [[ u ]] ≃ A /A , (36)because ∂ ∈ B \ B and we have always A /A ⊂ A (1) . Note that we always have theembeddings A ∩ ( k [[ u ]](( t )) + k (( u ))[[ t ]]) · t ֒ → A · t ∩ ( k [[ u ]](( t )) + k (( u ))[[ t ]]) , A ∩ k (( u ))[[ t ]] · t ֒ → A · t ∩ k (( u ))[[ t ]] , A ∩ k [[ u ]](( t )) ≃ A · t ∩ k [[ u ]](( t )) . Thus, we have a natural linear map A ∩ ( k [[ u ]](( t )) + k (( u ))[[ t ]])( A ∩ k (( u ))[[ t ]]) + ( A ∩ k [[ u ]](( t ))) −→ A · t ∩ ( k [[ u ]](( t )) + k (( u ))[[ t ]])( A · t ∩ k (( u ))[[ t ]]) + ( A · t ∩ k [[ u ]](( t ))) . (37)30y (22) this map coincides with the map H ( X, O X ) → H ( X, O X ( C )) . Let’s show that thekernel of this map may contain only elements from( A ∩ ( k [[ u ]](( t )) + k (( u ))[[ t ]])) + [( A ∩ k (( u ))[[ t ]]) + ( A ∩ k [[ u ]](( t )))] , (38)where A = A ∩ t − · k (( u ))[[ t ]] . From this and (36) it follows that the map H ( X, O X ) → H ( X, O X ( C )) is injective. Let a ∈ A ∩ ( k [[ u ]](( t )) + k (( u ))[[ t ]]) be a lift of an element fromthe kernel. Then a · t = a + a , where a ∈ ( A · t ∩ k (( u ))[[ t ]]) and a ∈ ( A · t ∩ k [[ u ]](( t ))) .Since a t − ∈ ( A ∩ k (( u ))[[ t ]]) , we have a t − = a − a t − ∈ A ∩ ( k [[ u ]](( t )) + k (( u ))[[ t ]]) . But a t − ∈ A and also gives an element of the kernel.Now assume that h ( C, O C ( C )) = 2 . This means that the image of the sheaf O C in k (( u ))under the Krichever map contains an element of order − k [ u − ] , i.e. C ≃ P . But then the surface X must be smooth along C , hence X mustbe normal since X is Cohen-Macaulay and C is an ample divisor. Then by [1, Th.2.5.19] and[1, Corol.2.5.20] there is an open neighbourhood of C isomorphic to an open neighbourhoodof a line in P . Since ζ ( F ) is a torsion free sheaf and h ( C, ζ ( F ( nC ))) = dim W n /W n − = n + 1 for all n ≫ ζ ( F ) ≃ O C . Since F is Cohen-Macaulay, it is locally freeon the smooth open neighbourhood of C . Since Cl ( P ) = Cl ( P \ Z ) ≃ Z for any closedsubscheme of codimension greater than one, we must have F ≃ O X on this open set (sinceotherwise its restriction on C = P would be not trivial). But then (e.g. by [1, Prop.1.1.6]) W n ≃ H ( X, F ( nC )) ≃ H ( X, O X ( nC )) ≃ A n since X and F are Cohen-Macaulay. Thus, A ≃ k [ a, b ] and X = P . Hence, H ( X, O X ) = 0 and we are done.At last, from formulas (15) and (16) one can easily deduce that the sheaf F fulfils theassumptions of proposition 2.3. Hence it is Cohen-Macaulay on X .Conversely, assume that C is a Cartier divisor, F is a coherent sheaf of rank one, themap H ( X, O X ) → H ( X, O X ( C )) is injective, O C ( C ) ≃ O C ( P ) . Then by [23, Rem.3.3]the rank of the data is one. As we have seen above, the condition on cohomology means thatthe kernel of the cohomology map (37) is zero. This means (see (38)) that all elements from A ∩ ( k [[ u ]](( t )) + k (( u ))[[ t ]]) can be represented as a sum of an element from A ∩ k [[ u ]](( t ))and an element from A ∩ k (( u ))[[ t ]] . In particular, for any element from A (1) ∩ k [[ u ]] thereexists an element from A ∩ k [[ u ]](( t )) with the same support (multiplied by t − ). This meansthat H ( C, O C ( C )) ≃ H ( C, O C ( P )) ≃ A (1) ∩ k [[ u ]] ≃ A /A (since the rank of the data is one). Note that A (1) contains an element of order one (since A (1)is the image of O C ( P ) under the Krichever map). Thus, there is an element in A with theleast term ut − . But this element will give us the operator ∂ after applying the map ψ − andconjugating by the Sato operator from theorem 2.2. Example 4.1.
This is an example of a surface, divisor and point for which we can calculate allpossible geometric data of rank one, corresponding Schur pairs and corresponding algebras ofcommuting operators. More precisely, we start from a ring A , and describe all possible Schurpairs with the ring A as a stabiliser ring. This description is possible due to using concreteformulas from the classical KP theory in dimension one; these formulas also lead to a precisedescription of commuting operators. Notably, we will see that the map ζ restricted to the set ofall sheaves from these geometric data maps this set surjectively to the dense open subset of thecompactified generalized jacobian of the curve C consisting of sheaves with trivial cohomologies.31e will see also that for this surface there are no other rings of commuting PDOs except onering of operators with constant coefficients.Consider the ring A = k h ∂ , ∂ ( ∂ + 3 ∂ ) , ∂ i ⊂ k [ ∂ , ∂ ] . It is easy to see that A ≃ k [ h ][ z, x ] / ( z − x ( x + 3 h ) ) (where ∂ ( ∂ + 3 ∂ ) z , ∂ x , ∂ h ) and that F = k [ ∂ , ∂ ] , where F denote the normalization of A . It is also clear that A is a 1-quasi-elliptic strongly admissible ring.Recall that, having such a ring A , we can construct a part of geometric data, namely thesurface X , the divisor C and the point P (see section 2.1). This part can be described in amore explicit way: we have the embedding˜ A ≃ k h ∂ , ∂ ( ∂ + 3 ∂ ) , ∂ , T i ⊂ ˜ F ≃ k [ ∂ , ∂ , T ] , which induces the normalisation morphism π : Proj( ˜ F ) → Proj( ˜ A ) , and X = Proj( ˜ A ) can beconsidered as a subscheme in the weighted projective space Proj( k [ x, z, h, T ]) , where weights of( x, z, h, T ) are (2 , , ,
1) . Thus, ∂ = z/ ( x + 3 h ) = x ( x + 3 h ) /z where from π ∗ O P = O X + O X ( − ∂ and π ∗ O P / O X ≃ O X ( − ∂ / O X ( − ∂ ∩ O X ≃ O X ( − / O X ( − ∩ O X ∂ and O E = O X / O X ∩ O X (1) 1 ∂ ≃ O X / O X ( − z + O X ( − x + 3 h ) , where E is the singular locus of X (cf. example 3.3 in [39]). So, E = Proj( k [ h, T ]) = P and π ∗ O P / O X ≃ O E ( −
1) , where from H ( X, O X ) = 0 .Let’s note that, if we have a geometric data ( X, C, P, F , ... ) where X, C, P are defined bythe ring A and the sheaf F is coherent of rank 1, then the corresponding Schur pair ( A, W )induces a 1-dimensional Schur pair ( A ′ , W ′ ) , where A ′ = k (( ∂ )) h ∂ , ∂ ( ∂ + 3 ∂ ) i , and W ′ is a space over K = k (( ∂ )) generated by elements from W (thus, A ′ , W ′ ⊂ K (( ∂ − )) ).The Schur pair ( A ′ , W ′ ) corresponds to a one-dimensional geometric quintet ( C ′ , P ′ , F ′ , . . . )(see [27, Th.4.6] or [28], see also section 2.6.1), where C ′ is the nodal curve over K and F ′ is a torsion free rank one sheaf on C ′ with H ( C ′ , F ′ ) = H ( C ′ , F ′ ) = 0 . It is not difficult tosee that the divisor C on the surface X is naturally isomorphic to the nodal curve too, whoseaffine equation (the equation of C \ P ) is ˜ y = y ( y + 3) .On the other hand, all torsion free rank one sheaves on this nodal curve (cf. [35]) as wellas corresponding Schur pairs of one-dimensional geometric data can be explicitly described asfollows (cf. [36, Sec 3]). The nodal curve C ′ can be thought of as a projective line with twoglued points, whose local coordinates are a and − a (with respect to the local coordinate z on P \ P ′ ). It is not difficult to see that in our case a = i √ ∂ , and for the curve C it is equal to i √ ψ ( x, z ) exp ( xz − ) = S ( x, ∂ − )(exp ( xz − )) (where z is a local parameter at a point on the curve).For the unique non-locally free sheaf n ∗ ( O P ) of degree zero (where n : P → C ′ is thenormalization map) the corresponding space W ′ is equal to K [ ∂ ] . This space comes from32 = k [ ∂ , ∂ ] , and the pair ( A, W ) obviously corresponds to the ring A of differential operatorswith constant coefficients.The only locally free sheaf of degree zero which has non-zero cohomologies is O C ′ . For anylocally free sheaf L parametrized (in the moduli space) by an element λ ∈ K ∗ ≃ Pic( C ′ ) , λ = − λ = − O C ′ ) the corresponding space W ′ is equal to K [ ∂ ] · S, where S = (1 + w∂ − ) and w = − a λ exp ( x a ) − exp ( − x a ) λ exp ( x a ) + exp ( − x a ) . Now we can describe those one-dimensional Schur pairs ( A ′ , W ′ ) (over K ) that are inducedby two dimensional Schur pairs ( A, W ) (over k ). It is easy to see that necessary and sufficientconditions for describing such pairs are the following: all elements from the admissible basis in W ′ must belong to k [ ∂ ](( ∂ − )) and satisfy the condition ( A ) . Since A ⊂ A ′ and all elementsfrom A satisfy the condition ( A ) , it is enough to check this property only for the two firstelements from the admissible basis of W ′ . These elements are w = S | x =0 , w = ∂ + ∂ ( w ) | x =0 ∂ − − ( w | x =0 ) ∂ − . Thus we must have − a λ − λ + 1 = P ( ∂ ) , − a λ ( λ + 1) = Q ( ∂ ) , where P, Q are polynomials in ∂ with coefficients in k of degree not higher than 1 and 2correspondingly. Hence, from the first equality we have λ = a − Pa + P ∈ k ( ∂ ) , and the second equality holds for any such λ for any such P . The same formulas show (dueto theorem 4.1) that for all − = λ ∈ k ∗ the sheaf ζ ( F ) (which is defined by the space ⊕ W i +1 /W i ) is a line bundle on C corresponding to λ . Clearly, ζ ( π ∗ ( O P )) ≃ n ∗ ( O P ) . Thus,the map ζ mentioned in the beginning of this example is indeed surjective.On the other hand, for any such λ we can calculate operators from the corresponding ringof operators. In particular, there will be an operator of the form S − ∂ S = ∂ + 2 ∂ ( w ) = ∂ − a λ ( λ exp ( x a ) + exp ( − x a )) . The last term of this operator can not be a polynomial in ∂ , because the exponential functioncan not belong to an algebraic extension of the field of rational functions. So, by remark 2.14there are no rings of PDOs with the projective spectral surface X except the ring A of operatorswith constant coefficients. Example 4.2.
This is another example of a surface, divisor and point for which we can calculateall possible geometric data of rank one, corresponding Schur pairs and corresponding algebras ofcommuting operators. The map ζ mentioned in the previous example will be again surjective.But we will see that for this surface there are many commutative rings of PDOs.Consider the ring A = k h ∂ , ∂ , ∂ i ⊂ k [ ∂ , ∂ ]It is easy to see that A ≃ k [ h ][ z, x ] / ( z − x ) (where ∂ z , ∂ x , ∂ h ) and that F = k [ ∂ , ∂ ] , where F denotes the normalization of A . It is also clear that A is a 1-quasi-elliptic strongly admissible ring. 33sing similar arguments from the previous example one can show that X can be ob-tained from P by glueing one doubled projective line (cf. [23, Sec. 3.6]). Thus, we have also H ( X, O X ) = 0 . Again as in the previous example X is a cone over C which is a cuspidalcurve. So, we can use in this case the same ideas and notation.Now any Schur pair ( A, W ) induces a 1 -dimensional Schur pair ( A ′ , W ′ ) over K , where A ′ = k (( ∂ )) h ∂ , ∂ i . For the unique non-locally free sheaf n ∗ ( O P ) of degree zero the corresponding space W ′ is equal to K [ ∂ ] . This space comes from W = k [ ∂ , ∂ ] , and the pair ( A, W ) obviouslycorresponds to the ring A of differential operators with constant coefficients.The only locally free sheaf of degree zero which has non-zero cohomologies is O C ′ . For anylocally free sheaf L parametrized by an element λ ∈ K ≃ Pic( C ′ ) , λ = 0 ( λ = 0 correspondsto O C ′ ) the corresponding space W ′ is equal to K [ ∂ ] · S, where S = (1 + w∂ − ) and w = 1 λ − x . Now w = S | x =0 = 1+ (1 /λ ) ∂ − . To find those pairs ( A ′ , W ′ ) that are induced by pairs ( A, W )we again must have 1 /λ = P ( ∂ ) for some linear polynomial P . It is not difficult to see thatfor all such λ the spaces W ′ are induced by W and that the map ζ is surjective.The rings of commuting operators will contain two operators: ∂ and S − ∂ S = ∂ + 2 P ( ∂ ) (1 − x P ( ∂ )) . By remark 2.14 and by proposition 2.2 such a ring is a ring of PDO if and only if P ( ∂ )is a constant. Clearly the sheaves corresponding to such rings are the preimages of the sheaf n ∗ ( O P ) . References [1] Badescu L.,
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On rings of commuting partial differential operators , St-Petersburg Math. J,no. 5, 2013, 86-145; (in Russian); e-print arXiv:math-ag/1106.0765v3H. Kurke, Humboldt University of Berlin, department of mathematics, faculty of mathematicsand natural sciences II, Unter den Linden 6, D-10099, Berlin, Germanye-mail: kurke @ mathematik.hu − berlin.de A. Zheglov, Lomonosov Moscow State University, faculty of mechanics and mathematics, depart-ment of differential geometry and applications, Leninskie gory, GSP, Moscow, 119899, Russiae-mail azheglov @ mech.math.msu.sumech.math.msu.su