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THE VARIETY OF POSITIVE SUPERDIVISORS

arXiv:alg-geom/9303007v1 29 Mar 1993THE VARIETY OF POSITIVE SUPERDIVISORSOF A SUPERCURVE (SUPERVORTICES)J.A. Dom´ınguez P´erez, D. Hern´andez Ruip´erez & C. Sancho de SalasDepartamento de Matem´aticas Puras y Aplicadas, Universidad de SalamancaAbstract.

The supersymmetric product of a supercurve is constructed with theaid of a theorem of algebraic invariants and the notion of positive relative superdi-visor (supervortex) is introduced. A supercurve of positive superdivisors of degree1 (supervortices of vortex number 1) of the original supercurve is constructed as itssupercurve of conjugate fermions, as well as the supervariety of relative positive su-perdivisors of degre p (supervortices of vortex number p.) A universal superdivisor isdefined and it is proved that every positive relative superdivisor can be obtained ina unique way as a pull-back of the universal superdivisor.

The case of SUSY-curvesis discussed.1. IntroductionPositive divisors of degree p on an algebraic curve X can be thought as unorderedsets of p points of X, hence, as elements of the symmetric p-fold product SpX.

Thesymmetric p-fold product is the orbit space of the cartesian p-fold product Xp underthe natural action of the symmetric group, and it is thus endowed with a naturalstructure of algebraic variety. In this way, positive divisors of degree p are thepoints of an algebraic variety Divp(X), and this variety is of great importance in thestudy of the geometry of curves, and it also has a growing interest in MathematicalPhysics.From the geometrical side, one has, for instance, the role played by the variety ofpositive divisors of degree p in some classical constructions of the Jacobian variety ofa complete smooth algebraic curve.

The first construction of the Jacobian variety,due to Jacobi and Abel, is of an analytic nature and defines the Jacobian as acomplex torus through the periods matrix. The first algebraic construction is dueto Weil [31] who showed that the algebraic structure and the group law of theJacobian come from the fact that it is birationally equivalent to the variety ofpositive divisors of degree the genus of the curve.Another procedure stemmedfrom Chow [7], who took advantage of the fact that for p high enough the Abelmap (that maps a divisor of degree p into its linear equivalence class) is a projectivebundle, to endow the Jacobian with a structure of projective algebraic group.

Butregardless the method used for constructing the Jacobian, the structure of thevariety of positive divisors of degree p and the diverse Abel morphisms from these1991 Mathematics Subject Classification. 14A22, 14M30, 14D25, 14C05.The first and the second authors acknowledge support received under C.I.C.Y.T.

project PB-88-0379. The third author was partially supported by D.G.I.C.Y.T.

project PS-88-0037Tt bAMS T X

2J.A. DOM´INGUEZ, D. HERN´ANDEZ & C. SANCHOvarieties to the Jacobian, turns out to be a key point in the theory of Jacobianvarieties (see, for instance [17], [25]) and has proved to be an important tool in thesolution of the Schottcky problem [26].From a physical point of view, the variety of positive divisors of a complexcomplete smooth curve X (a compact Riemann surface) is the variety of vorticesor solutions to the vortex equations ([5], [10].) For every holomorphic line bundleL on X endowed with a hermitian metric, there is a Yang–Mills–Higgs functionalY MHτ(∇, φ) defined on gauge equivalence classes of pairs (∇, φ) where ∇is anunitary connection, byY MHτ(∇, φ) =Z(|F∇|2 + |∇φ|2 + 14 |φ ⊗φ ∗−τ Id|2)dµwhere F∇is the curvature, ∇φ the covariant derivative, and τ is a real parameter(see [5].

)Bradlow’s theorem states that for large τ, gauge equivalence classes of solutions(∇, φ) to the vortex equationY MHτ(∇, φ) = 2πpτ ,where p is the degree of L with respect to the K¨ahler form, correspond to divisors ofdegree p on X. In this correspondence, a solution (∇, φ) corresponds to the divisorgiven by the set of centres of the vortices appearing with multiplicity given by themultiplicity of the magnetic flux.There is no similar theory for supersymmetric extensions of the vortex equations(supervortices,) and in fact only very little work on supervortices or supersymmetricextensions of the Bogonolmy equations has been done (see [20].) This paper willprovide a first step in that direction, by providing the right supervariety of positivesuperdivisors or supervortices on a supercurve.This paper is organized as follows:The supersymmetric product SpX for a supercurve X of dimension (1, 1) isconstructed in Part 2 as the orbit ringed space obtained through the action of thesymmetric group on the cartesian p-fold product of X .

It is far from trivial thatthe resulting graded ringed space is a supervariety of dimension (p, p), a statementwhich is shown to be equivalent to an invariant theorem. It should be stressed thatthis theorem is no longer true for supercurves of higher odd dimension, but ourresult covers the most important cases such as SUSY-curves.In Part 3 the notion of positive relative superdivisor of degree p for a rela-tive supercurve X × S →S is given.

The classical definition cannot be extendedstraightforwardly to supercurves if we wish that superdivisors could be obtained aspull-backs of a suitable universal superdivisor.For ordinary algebraic curves, positive divisors of degree 1 are just points. Thenovelty here is that for an algebraic supercurve X , positive relative superdivisors ofdegree 1 (supervortices of vortex number 1) are are not points of X (see [Ma3],) butrather they are points of another supercurve X c with the same underlying ordinary(bosonic) curve.Actually, if we think of X as a field of fermions on a bosoniccurve, the supercurve X c is the supercurve of conjugate fermions on the underlyingbosonic curve.This is proven in Part 4, that also contains the representability theorem forpositive relative superdivisors of degree p on a supercurveThe theorem means

THE VARIETY OF POSITIVE SUPERDIVISORS . .

.3that positive relative superdivisors of degree p are the points of the supersymmetricp-fold product SpX c of the supercurve X c of conjugate fermions. This property isstated in the spirit of Algebraic Geometry in terms of the functor of the points:the precise statement is that the functor of the positive relative superdivisors ofdegree p of a supercurve X of odd dimension 1, is the functor of the points ofthe supersymmetric p-fold product SpX c. This means that every positive relativesuperdivisor of X × S →S can be obtained in a unique way as the pull-back of acertain universal positive superdivisor through a morphism S →SpX c. We obtainin that way what is the right structure of algebraic superscheme the ‘space’ ofpositive superdivisors of degree p on a supercurve can be endowed with.The case of supersymmetric curves (SUSY-curves) is particularly important,both by historical and geometrical reasons.

We prove that for a supercurve X , theexistence of a conformal stucture on X is equivalent to the existence of an isomor-phism between X and the supercurve X c of conjugate fermions. In other words, asupercurve X is a SUSY-curve, if and only if, X is isomorphic with X c. In this casethe universal positive superdivisor of degree 1 is Manin’s superdiagonal ([4], [23])and we recover from a clearer and more general viewpoint Manin’s interpretation ofthe relationship between points and positive divisors of degree 1 for SUSY-curves,and some connected definitions ([28], [29].

)Summing up, the space of supervortices of vortex number p (positive superdi-visors of degree p) on a supercurve X of odd dimension 1, is an algebraic super-variety of dimension (p, p). This algebraic supervariety is the supervariety SpX cof ‘unordered families’ of p conjugate fermions.

Moreover, only for SUSY-curvessupervortices of vortex number p are ‘unordered families’ of p points of X .This theory can be extended straightforwardly to SUSY-families parametrizedby a ordinary algebraic scheme.The results of this paper only in the case of SUSY-curves were stated (withoutproofs) in [8].2. Supersymmetric products1.

Definitions.A suitable reference for schemes theory is [14]; the general theory of schemes inthe supergeometry (superschemes) can be found in [22] and [27].Let X = (X, A) be a graded ringed space, that is, a pair consisting of a topolog-ical space X endowed with a sheaf A of Z2-graded algebras. Let us denote by Jthe ideal A1 + A21.Definition 1.

A superscheme of dimension (m, n) over a field k, is a graded ringedspace X = (X, A) where A is a sheaf of graded k-algebras such that:(1) (X, O = A/J ) is an m-dimensional scheme of finite type over k.(2) J /J 2 is a locally free O-module of rank n and A is locally isomorphic toVO(J /J 2).Definition 2. A superscheme X = (X, A) over a field k is said to be affine ifthe underlying scheme (X, O = A/J ) is an affine scheme, that is, if there is ahomeomorphismX ∼→Spec(Γ(X, O))and O is the sheaf on X defined by localization on the basic open subsets of thespectrum

4J.A. DOM´INGUEZ, D. HERN´ANDEZ & C. SANCHOIf X = (X, A) is an affine superscheme, and A = Γ(X, A), we shall simply writeX = Spec A for it.Let us consider the productX g = (Xg, A⊗g)where Xg denotes the cartesian product X × g).

. .

× X, and A⊗g = A ⊗g). .

. ⊗A.The symmetric group Sg acts on X g by graded automorphisms of superschemesaccording to the ruleσ: Xg →Xg(x1, .

. ., xg) 7→(xσ(1), .

. ., xσ(g))σ∗: A⊗g →σ∗A⊗gf1 ⊗· · · ⊗fg 7→Yiσ(j)(−1)|fi||fj|fσ(1) ⊗· · · ⊗fσ(g)(1)where | | stands for the Z2-degree.

This action reduces to the ordinary action of Sgon the scheme (Xg, O⊗g). Then, we have the orbit space SgX, a natural projectionp: Xg →SgX, and an invariant sheaf Og = OSg on SgX, whose sections on an opensubset V ⊆SgX areOg(V ) = {f ∈O⊗g(p−1(V )) | σ∗f = f for every σ ∈Sg} .It is well-known that if (X, O) is a projective scheme, the ringed space (SgX, Og)is a scheme, the symmetric p-fold product of (X, O) ([30], Prop.19.

)Let us consider the sheaf Ag = (A⊗g)Sg of graded invariants on SgX defined asabove by lettingAg(V ) = {f ∈A⊗g(p−1(V )) | σ∗f = f for every σ ∈Sg}for every open subset V ⊆SgX.2. The case of supercurves.Definition 3.

A supercurve is a superscheme X of dimension (1, n) over a field k.Let X be a smooth proper supercurve, that is, a supercurve such that (X, O) isproper and smooth.Theorem 1. Let X = (X, A) be a smooth proper supercurve of odd dimensionn > 0.

The graded ringed space SgX is a superscheme if and only if n = 1, that is, ifand only if X is a superscheme of dimension (1, 1). In that case, SgX = (SgX, Ag)is a superscheme of dimension (g, g) that will be called the supersymmetric g-foldproduct of X .Proof.

Let us notice that (X, O) is projective (it has very ample sheaves,) so thatthe ringed space (SgX, Og) is a scheme as we mentioned above (in fact, it is smooth,which is no longer true for higher dimensional X )

THE VARIETY OF POSITIVE SUPERDIVISORS . .

.5As there is a natural projection Ag →Og, we have only to ascertain if Agis locally the exterior algebra of a locally free Og-module. We can thus assumeA = VO(N ), N being a free rank n O-module.Let us write Ni = O ⊗· · · ⊗↓i)N ⊗· · · ⊗O and M = N1 ⊕· · · ⊕Ng.

Now, if¯O = O⊗g and ¯A = A⊗g, we have¯A = VO(N ) ⊗O g). .

. ⊗OVO(N ) ∼→V¯O(M) .The symmetric group Sg acts on M byσ(n1 + · · · + ng) = nσ(1) + · · · + nσ(g)and this action provides an action σ: ¯A →¯A on the exterior algebra ¯A = V¯O(M),given byσ(m1 ∧· · · ∧mp) = σ(m1) ∧· · · ∧σ(mp)This action of Sg on ¯A is actually equal to the one defined in (1), because bothcoincide on V1¯O(M) = M and are morphisms of graded algebras.If we denote by MSg the Og-module consisting of the invariant sections of M,the proof of Theorem 1 will be thus completed with the followingLemma 1.

The natural morphism of sheaves of graded Og-algebras over SgX,φ: VOg(MSg) →(V¯O(M))Sg = Ag ,is an isomorphism if and only if n = 1.Proof. The proof is a computation of invariants in the exterior algebra of a freemodule over a commutativering, which allows us to use standard methods ofCommutative Algebra (all the results that we shall use can be found, for instance,in [1].

)Let us start with the case n = 1.a) We can assume that X = Spec O, where O is a semilocal ring with g maximalideals p1, . .

., pg, and then, that N = O · e, Ni = ¯O · ei (where ei = 1 ⊗· · · ⊗↓i)e ⊗· · · ⊗1) and M = ¯O · e1 ⊕· · · ⊕¯O · eg.Let us notice, first, that φ is an isomorphism if and only if it is an isomorphismwhen localized at every maximal ideal p of Og. On the other hand, p correspondsto a divisor D = x1 + · · · + xg and the fibre of p: Xg →SgX over this pointconsists of the family (x1, .

. ., xg) (some of the points can be equal) together withits permutations.

It follows that we are reduced to consider only the localization ofO at these particular points (x1, . .

., xg).b) We can assume that O = k[t] and N = k[t] · e.Since the completion morphism (Og)p ֒→[(Og)p is a faithfully flat morphism,we are reduced to show that φ is an isomorphism after completing Og at everymaximal ideal.Let t ∈O be an element that takes different values (λ1, . .

., λg) at the points (x1,. .

. , xg) and such that t −λi is a parameter at xi (that is, it generates the maximalideal of the local ring Opi.) Then, k[t] is a subring of O and moreover, given twodifferent maximal ideals pi ̸= pj, the maximal ideals ¯pi = pi ∩k[t] and ¯pj = pj ∩k[t]of k[t] are also different

6J.A. DOM´INGUEZ, D. HERN´ANDEZ & C. SANCHOLet I = p1 ∩· · · ∩pg, ¯I = ¯p1 ∩· · · ∩¯pg be the intersection ideals and O ֒→bO,k[t] ֒→dk[t] the faithfully flat morphisms of completion with respect to the ideals I,¯I, respectively.

Then we havedk[t] ∼→gYi=1dk[t]pi ∼→gYi=1bOpi ∼→bO ,so that, if the theorem is true for k[t] and the module k[t] · e, it is also true fordk[t] = bO and dk[t]· e = bO · e and then it will be true for O and N = O · e by faithfullflatness.c) The case O = k[t] and N = k[t] · e.Now, for every 0 ≤p ≤g we haveVp(M) = Li1<···

.p. .

.i1. .

.ip. .

..Then, an invariant element m = Pi1<···

., p}. In particular, (N1)Sg−1 ∼→MSg.Since N1 ∧· · · ∧Np = ¯O · e1 ∧· · · ∧ep, we have(N1 ∧· · · ∧Np)Sp×Sg−p ∼→¯O−Sp×Sg−pwhere ¯O−Sp×Sg−p stand for the subset of those f ∈¯O such that (σ × µ)(f) =sign(σ) · f for every (σ × µ) ∈Sp × Sg−p.

Taking p = 1, we obtainMSg ∼→N Sg−11∼→¯OSg−1 ,and the original morphismφ: VpOg(MSg) →(Vp¯O(M))Sg = Agis now the morphism¯φp: Vp ¯OSg−1 →¯O−Sp×Sg−pdescribed by¯φp(f1 ∧· · · ∧fp) =XSsign(µ)σµ(1)(f1) . .

.σµ(p)(fp)

THE VARIETY OF POSITIVE SUPERDIVISORS . .

.7where σi is the transposition of 1 and i.Let us proof that ¯φp is an isomorphism: There is a commutative diagram¯OSg−1 ⊗Og · · · ⊗Og ¯OSg−1T−−−−→¯O1×Sg−pHyyH′VpOg ¯OSg−1¯φp−−−−→¯O−Sp×Sg−pwhere T(f1 ⊗· · · ⊗fp) = σ1(f1) . .

. σp(fp), H(f1 ⊗· · · ⊗fp) = f1 ∧· · · ∧fp andH′(f) = Pµ∈Sp sign(µ)(µ × 1)(f).As O = k[t], ¯O = k[t1, .

. ., tg], and if we denote(s1, .

. .

, sg) = symmetric functions of (t1, . .

., tg)(¯s1, . .

. , ¯sg−1) = symmetric functions of (t2, .

. ., tg)(s′1, .

. ., s′g−p) = symmetric functions of (tp+1, .

. ., tg)we haveOg = k[s1, .

. .

, sg]¯OSg−1 = k[t1, ¯s1, . .

. , ¯sg−1] ∼→Og[t1]¯O1×Sg−p = k[t1, .

. ., tp, s′1, .

. ., s′g−p] ∼→Og[t1, .

. ., tp] .If follows that ¯O−Sp×Sg−p can be identified with the pth skew-symmetric tensors ofthe Og-module ¯OSg−1 = Og[t1] and the previous diagram readsOg[t1] ⊗Ogp).

. .

⊗Og Og[t1]∼−−−−→Og[t1, . .

., tp]HyyH′VpOg Og[t1]¯φp−−−−→VpOg Og[t1]where now H′ is the skew-symmetrization operator, finishing the proof of the ifpart.To complete the proof, we have to show that if n > 1, ¯φp is not an isomorphism.Let us write N = Lnj=1 N j with N j of rank 1, and Mj = Lgi=1(O ⊗· · · ⊗↓i)N j ⊗· · · ⊗O) so that M = Lnj=1 Mj.Then(V¯O(M))Sg =Mp1+···+ps=p(Vp1¯O M1 ⊗· · · ⊗Vps¯O Ms)SgVOg(MSg) =Mp1+···+ps=pVp1Og(M1)Sg ⊗· · · ⊗VpsOg(Ms)Sg .By the case n = 1, we haveVpiOg(Mi)Sg ∼→(VpiOg(Mi))Sgand then VOg(MSg) = Lp1+···+ps=p(Vp1Og M1)Sg ⊗· · · ⊗(VpsOg Ms)Sg .But there are invariant elements in the tensor product (Vp1¯O M1⊗· · ·⊗Vps¯O Ms)Sgwhich cannot be written as tensor products of invariant elements. This means thatthe morphism φp is not an isomorphism in this case.

8J.A. DOM´INGUEZ, D. HERN´ANDEZ & C. SANCHOCorollary 1.

If (z, θ) are graded local coordinates on a supercurve X of dimension(1, 1), a system of graded local coordinates for SgX is given by (s1, . .

., sg, ς1, . .

., ςg)(s1, . .

., sg) are the (even) symmetric functions of (z1, . .

., zg) and (ς1, . .

., ςg) arethe odd symmetric functions defined by ςh = Pgi=1 σi(θ1¯sh−1).3. Positive superdivisorsFrom this point, calligraphic types are reserved to graded ringed spaces and thestructure ring sheaf of any ringed space will be denoted by O with the name of theringed space as a subscript.

For instance, X = (X, OX ) or simply X will mean agraded ringed space, whereas (X, OX) or X will represent the underlying ordinaryringed space.1. The universal divisor for an algebraic curve.This section is devoted to summarize the theory of the variety of positive divisorsand the universal divisor for a (ordinary) smooth proper algebraic curve X, andto show that the universal property still holds when the space of parameters is asuperscheme.

Suitable references are [13] or [16].In that case, positive divisors of degree g are unordered families of g points, andthey are then parametrized by the space of such families, that is, by the symmetricproduct SgX. This can be made precise through the notion of relative divisor.If S is another scheme, positive relative divisors of X × S →S of degree g aresubschemes Z →X such that OZ is a locally free OS-module of rank g. There isa nice positive relative divisor Zu of degree g of X × SgX →SgX, whose fibre ona point (x1, .

. ., xg) ∈SgX is the divisor x1 + · · · + xg of X defined by it.

Zu iscalled the universal divisor because the mapHom(S, SgX) →DivgS(X × S)φ 7→(1 × φ)−1(Zu)where DivgS(X × S) denotes the set of positive relative divisors of degree g, is oneto one. This means that each positive divisor can be obtained as a pull-back ofthe universal divisor; this statement is known as representability theorem for thesymmetric product.But it turns out that the above theory is still true when a superscheme is allowedas the space of parameters, once the corresponding notion of positive relative divisorhas been established.Definition 4.

Let X be an ordinary smooth curve and (S, OS) a superscheme. Apositive relative divisor of degree g of X × S →S is a closed sub-superscheme Z ofX × S of codimension (1, 0) defined by a homogeneous ideal J of OX×S such thatOX×S/J is a locally free OS-module of rank (g, 0).The ideal J of a positive relative divisor of degree g is then locally generated byan element of typef = zg −a1zg−1 + · · · + (−1)gag ,(2)where the ai’s are even elements in OS, and OX×S/J is a free OS-module withbasis (1, z, .

. ., zg−1).The representability theorem now reads

THE VARIETY OF POSITIVE SUPERDIVISORS . .

.9Theorem 2. Let X be a smooth proper curve over a field k and Zu the universaldivisor.

The mapHom(S, SgX) →DivgS(X × S)φ 7→(1 × φ)−1(Zu) ,(3)where DivgS(X × S) denotes the set of positive relative divisors of degree g, is oneto one for every superscheme S.Proof of the representability theorem for ordinary schemes applies with onlyminor changes to this case.There are two key points for the proof of this theorem. The first one is theconstruction of the universal divisor, which can be done as follows: If πi: Xg →Xis the ith projection and ∆i is the positive relative divisor of X×Xg →Xg obtainedby pull-back of the diagonal ∆⊂X × X throughout 1 × πi: X × Xg →X × X wecan prove there is a unique positive relative divisor Zu of X × SgX →SgX suchthat(1 × p)−1Zu = ∆1 + · · · + ∆gwhere p: Xg →SgX is the natural projection.

This divisor Zu is the universaldivisor.The second key point is the so-called ‘determinant morphism’ S →SgZ, Z beinga positive relative divisor of degree g because its composition with SgZ →SgXprovides the inverse mapping of (3) (See [Iv].) The determinant morphism for thelocally free OS-module of rank (g, 0) OZ is defined as follows: Each element b inthe invariant sheaf (OZ)g = (O⊗gZ )Sg acts on the OS-module VOS OZ of rank (1, 0)as the multiplication by a well-determined element det(b) in OS.

This gives riseto a morphism of sheaves (OZ)g →OS, and to a morphism of schemes S →SgZ.The determinant morphism provides the inverse mapping of (3) because if b is aneven element in OZ, bi = 1 ⊗· · · ⊗↓i)b ⊗· · · ⊗1 ∈O⊗gZ , and we denote by si(b) thesymmetric functions of b1, . .

., bg, we have thatai = det(si(b))(i = 1, . .

., g) ,where zg −a1zg−1 + · · · + (−1)gag is the characteristic polynomial of b acting onOZ by multiplication (compare with (2).)2. Positive superdivisors on supercurves.The above discussion is based on a trivial but important point: positive divisorsare families of points.

Even in the relative case, positive relative divisors of degree 1are ‘S-points,’ that is, sections of X × S →S, and by this reason, positive divisorsof degree g are parametrized by the symmetric product SgX and the universaldivisor.For a supercurve X = (X, OX ), a similar notion could be done, by defining pos-itive relative superdivisors of X × S →S (S being an arbitrary superscheme,) asclosed sub-superschemes of X × S of codimension (1, 0) flat over the base super-scheme.This definition has two drawbacks. The first one is that ‘S-points’ are not su-perdivisors in that sense because they have codimension (1, 1) and not codimension(1, 0) as superdivisors does ([23].) The second one is that we cannot ensure thatthey are pull backs of a suitable universal superdivisor

10J.A. DOM´INGUEZ, D. HERN´ANDEZ & C. SANCHOWe have thus modified the notion of positive relative superdivisors in order tofulfill the second requirement as follows:Let (X , OX) be a smooth supercurve, and (S, OS) a superscheme.Definition 5.

A positive relative superdivisor of degree g of X ×S →S is a closedsub-superscheme Z of X × S of codimension (1, 0) whose reduction ˆZ = Z ×X Xis a positive relative divisor of degree g of X × S →S (see Definition 4. )Even with our definition, ‘S-points’ are not superdivisors, but as we shall seeafterwards, there is a close relationship between them, at least for SUSY-curves.Positive relative superdivisors can be described locally in a rather precise way inthe case of a smooth supercurve of dimension (1, 1).In this case, the natural morphism OX →OX induces an isomorphism (OX )0 ∼→OX, so that OX is a module over OX, there exists a canonical projection X →Xand OX is in a natural way an exterior algebra OX = VOX L, where L = (OX )1 isa line bundle over the ordinary curve X.Lemma 2.

Let X be a smooth supercurve of dimension (1, 1).A closed sub-superscheme Z of X × S of codimension (1, 0) defined by a homogeneous idealJ of OX×S is a positive relative superdivisor of degree g if and only if the followingconditions hold:(1) OZ = OX×S/J is a locally free OS-module of dimension (g, g). (2) If (z, θ) is a system of graded local coordinates, J can be locally generatedby an element of typef = zg −(a1 + θb1)zg−1 + · · · + (−1)g(ag + θbg)where the ai’s are even and the bj’s are odd elements in OS.Proof.

Let Z be a positive relative superdivisor of degree g defined by a homoge-neous ideal J of OX×S and let us consider a system of relative local coordinates(z, θ). Then, the reduction ˆZ = Z ×X X is a positive relative divisor of degree gof X × S →S defined by the image ˆJ of J by the morphism π: OX×S →OX×S,so that an element f ∈J generates J if and only if ˆJ is generated by ˆf = π(f).Since ˆJ defines a positive relative divisor of degree g of X × S →S, then ˆJ hasa generator of type ˆf = zg −a1zg−1 + · · · + (−1)gag where the ai’s are even ele-ments in OS (see equation (2),) and O ˆZ = OX×S/ ˆJ is a free OS-module with basis(1, z, .

. ., zg).

This means that O ˆZ ∼→OS[z]/( ˆf). It follows that there is a genera-tor of J of the form f = ˆf +θ · d and that d ≡q(z) (mod ˆJ) for certain polynomialq(z) of degree less than g. In consequence, the element ˆf + θq(z) generates J andis of the predicted type.

An easy computation now shows that OZ is a rank (g, g)free OS-module with basis (1, z, . .

., zg−1, θ, θz, . .

., θzg−1).The converse is straightforward.3. The functor of positive superdivisors on a supercurve.Let (X , Z) be a supercurve.

For every superscheme S let us denote by DivgS(X ×S) the set of positive relative superdivisors of degree g of X × S →S. If ϕ: S′ →Sis a morphism of superschemes, and Z is a positive relative divisor of degree g ofX × S →S, (1 × ϕ)−1Z is a positive relative divisor of degree g of X × S′ →S′.In categorial language this essentially means thatS →Divg (X × S)

THE VARIETY OF POSITIVE SUPERDIVISORS . .

.11is a functor.We whish to show that when X has dimension (1, 1), the above functor is rep-resentable in a similar sense to that of Theorem 2. A proof is given in the nextsection.4.

The representability theorem for positive superdivisorson a supercurve of dimension (1, 1)In what follows, we consider only supercurves X = (X, OX ) which are smooth,proper and of dimension (1, 1). This last condition means that the structure sheafOX is canonically isomorphic with OX ⊕L for certain line bundle L on the ordinaryunderlying curve X.1.

The supercurve of positive superdivisors of degree 1.Let S = (Spec B, B) an affine superscheme and Z = (Z, OZ) ֒→X × S →Sa relative superdivisor of degree 1. The structure sheaf OZ is a quotient of thestructure sheaf (OX ⊕L) ⊗k B of X × Spec B.

We also have that OZ ∼→B ⊕Lwhere L is the image of L ⊗k B in OZ, since O ˆZ ∼→B, because Z is a superdivisorof degree 1. Moreover, L = L ⊗OX B, where B is an OX-algebra trough the naturalmorphism f: OX →O ˆZ ∼→B, so that it is a locally free rank 1 B-module.It is now clear that the superdivisor Z is characterized by the morphism f: OX →B together with a morphism ˜f: OX →B ⊕L extending f. That is, Z is defined bya morphism f: OX →B and a derivation ∆: OX →L0 = L ⊗OX B1.

But ∆canbe understood as an element f∆∈HomOX(κX, L0) ∼→HomOX(κX ⊗OX L−1, B1)(where κX is the canonical sheaf of X,) so that the couple (f, ∆) is equivalent to agraded ring morphism g: OX ⊕(κX ⊗OX L−1) →B.The above discusion remains true for arbitrary (non affine) superschemes S. Thismeans that the supercurve X c = Spec(OX ⊕Lc), where Lc = κX ⊗OX L−1, willrepresent the functor of superdivisors of degree 1 of the supercurve Spec(OX ⊕L).The universal divisor, Zu1 ֒→X × X c, will be the divisor corresponding to theidentity morphism Id: S = X c →X c. One can compute this superdivisor as aboveand obtain that it is the closed subsuperscheme whose ideal sheaf is the kernel ofthe graded ring morphism:¯∂: (OX ⊕L) ⊗k (OX ⊕Lc) = VOX⊗kOX[(L ⊗k OX) ⊕(OX ⊗k Lc)] →VOX(L ⊕Lc)given by a ⊗b 7→a · b ⊕b · d(a) on OX ⊗k OX (taking into account that b · d(a) is alocal section of κX ∼→L ⊗OX Lc) and as the natural morphisms on the remainingcomponents. Moreover, if U ⊂X is an affine open subset and z ∈OX(U) is a localparameter, and if L is trivial on U, L|U ∼→θ·OX |U, then Lc is trivial on U generatedby θc = ωθ ·dz, ωθ ∈Γ(U, L−1) being the dual basis of θ.

If U = Spec(OX ⊕L) ⊂Xand Uc = Spec(OX ⊕Lc) ⊂X c, the restriction of the universal superdivisor Zu1 toU × Uc is given by the local equation:z1 −z2 −θ ⊗θc = 0(4)where z1 = z ⊗1 and z2 = 1 ⊗z.The above discussion can be summarized as follows:Let X(X OO⊕L) be a smooth proper supercurve of dimension (1 1)

12J.A. DOM´INGUEZ, D. HERN´ANDEZ & C. SANCHODefinition 6.

The supercurve of positive divisors of degree 1 on X is the super-curve of dimension (1,1) defined as X c = (X, OX ⊕Lc) where Lc = κX ⊗OX L−1.This supercurve is also called the supercurve of conjugate fermions on X .Definition 7. The universal positive superdivisor of degree 1 is the relative super-divisor Zu1 of X × X c →X c defined by the ideal sheaf Ker ¯∂earlier considered.

If(z, θ) are graded local coordinates for X , the corresponding local equation of Zu1 isz1 −z2 −θ ⊗θc = 0 where z1 = z ⊗1, z2 = 1 ⊗z and θc = ωθ · dz.Theorem 3. The morphism of functors:Θ: Hom(S, X c) →Div1S(X × S)ϕ 7→(1 × ϕ)−1(Zu1 ) .is a functorial isomorphism.By this representability theorem, the supercurve X c of conjugate fermions para-metrizes positive superdivisors of degree 1 on the original supercurve X .Thatmeans that positive superdivisors of degree 1 on X are not points of X as it happensin the ordinary case, but rather points of another supercurve X c with the sameunderlying ordinary curve X.2.

Positive superdivisors of degree 1 on a SUSY-curve.This section will explore the relationship between points and positive superdivi-sors of degree 1 for a SUSY-curve (Supersymmetric curve.) This relationship wasfirstly described by Manin (see [23],) but it can be enlightened by means of the su-percurve of positive superdivisors of degree 1 defined above.

Let us start by recallingsome definitions and elementary properties of SUSY-curves. More details can befound in Manin [21], [22], [23], [24], Batchelor and Bryant [3], Falqui and Reina[9], Giddings and Nelson [11], [12], Bartocci, Bruzzo and Hern´andez Ruip´erez [2],Bruzzo and Dom´ınguez P´erez [6], or LeBrun, Rothstein, Yat-Sun Poon and Wells[18], [19].Let S = (S, OS) be a superscheme.Definition 8.

A supersymmetric curve or SUSY-curve over S, is a proper smoothmorphism X = (X, OX ) →S of superschemes of relative dimension (1, 1) endowedwith a locally free submodule D of rank (0, 1) of the relative tangent sheaf TX/S =DerOS(OX ) such that the composition mapD ⊗OX D[ , ]−→DerOS(OX ) →DerOS(OX )/Dis an isomorphism of OX -modules (see, for instance, [19]. )If X = (X, OX, D) is a SUSY-curve, X can be covered by affine open subsetsU ⊆X with local relative coordinates (z, θ) such that D is locally generated byD = ∂∂θ + θ ∂∂z.

These coordinates are called conformal.There is a natural isomorphism D∗∼→BerOS(OX ) and a ‘Berezinian differential’∂: Ω1→D∗≃BerO (O )

THE VARIETY OF POSITIVE SUPERDIVISORS . .

.13which is nothing but the natural projection induced by the immersion D →TX/S.In conformal coordinates ∂is described by ∂(df) =dz ⊗∂∂θ· D(f), wheredz ⊗∂∂θdenotes the local basis of BerOSOX determined by (z, θ) (see [15], [23]. )If (X, OX , D) is a (single) SUSY-curve, that is, a SUSY-curve over a point, wehave that OX = VOX (L), and there are isomorphisms D ⊗OX OX ∼→L−1 andL ⊗OX L ∼→κX .This isomorphism if often called a spin structure on X. Conversely, a spin structureinduces a conformal structure, so that a conformal structure on a proper smoothsupercurve is equivalent to a spin structure on it.Now, there is a geometrical characterization of SUSY-curves in terms of super-divisors:Theorem 4.

Let X be a supercurve of dimension (1, 1). Then X is a SUSY-curveif and only if there is an isomorphism of supercurves X ∼→X c between X and thesupercurve of positive superdivisors of degree 1 (conjugate fermions) on it inducingthe identity on X.

Moreover, there is a one-to-one correspondence between suchisomorphisms and spin structures on X.Proof. If OX = OX ⊕L, then the structure sheaf of X c is OX ⊕(L−1 ⊗OX κX),so that an isomorphism X ∼→X c inducing the identity on X is nothing but a OX-module isomorphism L−1⊗OX κX ∼→L, that is, an isomorphism L⊗OX L ∼→κX.Theorem 3 and the former result, mean that for SUSY-curves, S-points areequivalent to relative positive superdivisors of degree 1 on X × S →S, as Maninclaimed in [23], and the universal relative positive superdivisor of degree 1, gives inthis case nothing but Manin’s superdiagonal:Let X be a SUSY-curve.If ∆denotes the ideal of the diagonal immersion∆: X ֒→X ×X , the kernel of the composition ∆→∆/∆2 ∼→∆∗Ω1X∂−→∆∗Ber(OX )is a homogeneous ideal I of OX×X thus defining a sub-superscheme ∆s called thesuperdiagonal.Lemma 3.

(Manin, [23]) The superdiagonal ∆s = (X, OX×X /I) is a closed sub-superscheme of codimension (1, 0). In conformal coordinates (z, θ), it can be de-scribed by the equationz1 −z2 −θ1θ2 = 0where as usual z1 = 1 ⊗z and z2 = z ⊗1.According to Lemma 2, the superdiagonal is a positive superdivisor.

A simplelocal computation shows that actually we have:Theorem 5. Let X be a SUSY-curve, ψ: X ∼→X c the natural isomorphism betweenX and the supercurve of positive superdivisors of degree 1 (conjugate fermions,) and1 × ψ: X × X ∼→X × X c the induced isomorphism.

Then∆s = (1 × ψ)−1(Z1) ,that is, the isomorphism ψ: X ∼→X c given by the spin structure transforms by in-verse image the universal positive superdivisor of degree 1 into Manin’s superdiag-onal.

14J.A. DOM´INGUEZ, D. HERN´ANDEZ & C. SANCHO3.

The superscheme of positive superdivisors of degree g.Let X be a smooth proper supercurve of dimension (1, 1) as above.Definition 9. The superscheme of positive superdivisors of degree g of X is thesupersymmetric product SgX c of the supercurve X c of positive superdivisors ofdegree 1.The universal superdivisor Zug of X × SgX c is constructed as follows: let usconsider the natural projectionsπi: X × X c ×g· · · × X c →X × X c(x, xc1, .

. .

, xcg) 7→(x, xci) ,the positive superdivisors of degree 1, Zi = πi(Zu1 ) ⊂X × (Qgi=1 X c) and thepositive superdivisor of degree g, Z = Z1 + · · · + Zg.Lemma 4. There exists a unique positive relative superdivisor Zug of degree g ofX × SgX c →SgX c, such that π∗(Zug ) = Z, where π is the natural morphismπ: X × (gYi=1X c) →X × SgX c .Proof.

One has only to prove that Zug = π(Z) is the desired superdivisor. Thiscan be done locally, so that we can assume that X = Spec A is affine and the linebundles L and κX are trivially generated respectively by θ and dz.

Then, the localequation of Zu1 is z ⊗1 −1 ⊗z −θ ⊗θc = 0 (see equation (4),) and Z is thesuperdivisor defined by the equation0 =gYi=1(z −zi −θθci) = zg −(s1 + θ · ς1)zg−1 + · · · + (−1)g(sg + θ · ςg) ,where zi = π∗i (1 ⊗z), θci = π∗i (1 ⊗θc) and si, ςi are the even and odd symmetricfunctions corresponding to z and θc (see Corollary 1. )It follows that this lastequation is also the local equation of Zug in X × SgX c and one can readily checkthat π∗(Zug ) = Z.4.

The representability theorem.This paragraph will justify the above definitions by displaying the representabil-ity theoremTheorem 6. The pair (SgX c, Zug ) represents the functor of relative positive su-perdivisors of degree g of X , that is, the natural map:φ: Hom(S, SgX c) →DivgS(X × S)f 7→(1 × f)∗Zug ,is a functorial isomorphism for every superscheme S.Proof.1) φ is injective:

THE VARIETY OF POSITIVE SUPERDIVISORS . .

.15Let U = Spec A ⊂X be an open subscheme of the underlying ordinary curve X,such that κX and L are trivial generated respectively by dz, θ. Let us consider theaffine open sub-superschemes U = Spec(A⊕θ·A) ֒→X and Uc = Spec(A⊕θc·A) ֒→X c, where θc = dz ⊗ωθ ∈Γ(U, κX ⊗L−1) = Γ(U, Lc).Now, SgUc ֒→SgX c is an affine open sub-superscheme and the symmetric func-tions si(z), ςi(z, θc) (i = 1, .

. ., g) is a graded system of parameters for the gradedring Sgk(A ⊕θc · A).

Let us denote it simply by si, ςi.The family of the affine open sub-superschemes SgUc so obtained (when U rangeson the affine open subschemes of X where κX and L are trivial) is an open cov-ering of SgX c by affine open sub-superschemes such that the universal positivesuperdivisor of U × SgUc →U is the closed sub-superscheme ZuU defined by theequationzg −(s1 + θ · ς1)zg−1 + · · · + (−1)g(sg + θ · ςg) = 0 .Then one has that for these affine open sub-superschemes the mapφU: Hom(S, SgUc) →DivgS(U × S)f 7→(1 × f)∗ZuU ,is injective: In fact, we can assume that S is affine S = Spec B. Now, the morphismsf: S →SgUc are determined by the inverse images of the symmetric functions si,ςi.

But these inverse images are determined by (1 × f)∗ZuU since the coefficients ofthe characteristic polynomial of z ⊗1 acting by multiplication on the B[θ]-moduleO(1×f)∗ZuU are (−1)i(f ∗(si) + θ · f ∗(ςi)). This allows us to conclude.A straightforward consequence of this fact is that the map φ of the statement isinjective for every superscheme S.2) φ is an epimorphism:It is sufficient to prove that given a relative positive superdivisor of degree g,Z ⊂X ×S →S, for every geometric point p ∈S there exist an open neighbourhood,V ⊂S, and a morphism fV: V →SgX c such that (1 × fV)∗(Zug ) = Z ∩(X × V) =ZV, for, in that case, these morphisms define a morphism f: S →SgX c fulfilling(1 × f)∗(Zug ) = Z by virtue of the former paragraph.

Let π: X × S →S be thenatural projection and U = Spec A ⊂X an affine subscheme where κX and L aretrivial and such that (with the notation of the beginning of this section) the affineopen sub-superscheme U ⊂X contains the superdivisor π−1(p) ∩Z ֒→X . Then,W = S −π(Z −Z ∩(U × S)) is open, because π is a proper morphism, and itcontains the point p ∈S.

Let V = Spec B ⊂S be an affine open sub-superschemecontaining p and contained in W. By construction, if we put ZV = Z ∩π−1(V),then ZV is a relative positive divisor of degree g of U × V →V, so that it is affineZV = Spec C. Let dz, θ be generators of κX and L, as usual. Now, according tothe definition of superdivisor, the ring C of ZV is a locally free module over B[θ]of rank g and C = C/θ · C is the ring of an ordinary divisor of degree g of U ⊂X.Let us consider the morphism fV: Spec B = V →SgUc = Spec Sgk(A ⊕θc · A)induced by the ring morphism f ∗V: Sgk(A ⊕θc · A) →B defined, by means of thedeterminant morphism, as follows: Let SgkA →B be the determinant morphismdefined by the quotient ring C of A⊗kB.

This morphism endows B with a structureof SgkA-algebra. But, by Lemma 1, one has Sgk(A ⊕θc · A) = VSgkA M for a certainfree SgkA-module M generated by the odd symmetric functions ςi; then, by theuniversal property of the exterior algebra defining f ∗is equivalent to giving a

16J.A. DOM´INGUEZ, D. HERN´ANDEZ & C. SANCHOhomogeneous morphism of degree zero of SgkA-modules, M →B.

This morphismis actually characterized by the images of the functions ςi (i = 1, . .

., g,) and wedefine these images as the odd coefficients of the characteristic polynomial of z ⊗1acting on the B[θ]-module C by multiplication; this means that if the characteristicpolynomial is zg−(a1+θ·b1)zg−1+· · ·+(−1)g(ag+θ·bg), then we define f ∗V(ςi) = bi.Moreover, one also has that f ∗V(si) = ai and then (1 × f)∗V(ZuU) is the relativepositive superdivisor of degree g of U × V →V defined by the equationzg −(a1 + θ · b1)zg−1 + · · · + (−1)g(ag + θ · bg) .On the other hand, this is the characteristic polynomial of z ⊗1 acting by multipli-cation on the structure ring of ZV, so that this polynomial vanishes on ZV, whichmeans that ZV is contained in (1×f)∗V(ZuU). Since both positive superdivisors havethe same degree, they are equal, thus finishing the proof.5.

The case of SUSY-curves.If X is a SUSY-curve, there exists an isomorphism ψ: X ∼→X c between X andthe supercurve of positive superdivisors of degree 1, as we proved in subsection 4.1.Then we have an isomorphism SgX →SgX c between the supersymmetric productof X and the superscheme SgX c of positive superdivisors of degree g on X, so thatthe representability theorem now reads (see [8]):Theorem 7. Let X be a SUSY-curve.

The supersymmetric product SgX representsthe functor of positive superdivisors on X , that is, there exists a universal relativepositive superdivisor Zug of degree g of X × SgX →SgX such that the natural mapφ: Hom(S, SgX ) →DivgS(X × S)f 7→(1 × f)∗Zug ,is a functorial isomorphism for every superscheme S.Moreover, since 1 × ψ: X × X ∼→X × X c transforms by inverse image the uni-versal positive superdivisor of degree 1 into Manin’s superdiagonal, the universalsuperdivisor of X × SgX →SgX for SUSY-curves is constructed as in Lemma 4with Manin’s superdiagonal playing the role of Zu1 .Summing up, only for SUSY-curves, ‘unordered families of g points’ (the pointsof SgX ) are equivalent to ‘superdivisors of degree g’ (the points of SgX c.)Acknowledgments. We thank J.M.

Mu˜noz Porras for many enlightening com-ments about Jacobian theory and the geometry of the symmetric products, J.M.Rabin for drawing references [28] and [29] to our attention and J. Mateos Guilartewho first introduced us to the vortex equations. We also thank the anonymousreferee for some helpful suggestions to improve the original manuscript.References1.

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