Notes on Jacobian Varieties: a Brief Survey

Notes on Jacobian varieties: a brief survey

Juliana Coelho (UFF)

www.professores.uﬀ.br/jcoelho

March 28, 2017

Abstract This is an expanded version of the notes writen for the School “Ge- ometry at the Frontier I”, in Pucón,Chile (2016). The aim was to give an introduction to the subject of Jacobians and compactified Jacobians of smooth and singular curves. There is no new material here and an effort was made to provide references to the statements made, as virtually no proof is provided.

1 Jacobians of smooth curves: divisors

We start with a smooth projective curve C over the complex numbers C, which means that C can be seen as a smooth algebraic projective variety of dimension 1 over C or alternatively, as a complex compact Riemann surface. Throughout these notes we will shift between the two equivalent interpretations of the word “curve”, using at each time the one that makes our lives easier.

Figure 1: A curve seen as a dimension 1 (algebraic) variety or a Riemann surface

The idea of divisors is to try to imitate the group structure of the projective smooth plane cubic curve C whose aﬃne part is given by the equation y2 = x3 + x2 + 1. (See Figure 2.) Given P,Q ∈ C, there exists a unique point R ∈ C such that P , Q and R are colinear. Note that if P = Q then we must consider the line to be tangent to C at this point. (The existence and unicity of R follows from B´ezout’stheorem [F, Section 5.3].) To turn this into an abelian group structure we need only to introduce an identity element, and it turns out that we can simply choose any point O ∈ C as the identity. Now if S is the point of C such that R, O and S are colinear, then we set P + Q = S.

1 Figure 2: Group structure on C

We will not include this proof here, but it is not hard to show that this indeed defines an abelian group structure, i.e., that O is the identity element, each element has an inverse, and the operation is associative and commutative. (See [Si, Chapter III.2], [Sh, Chapter III.3] or [F, Chapter 5.6].) The above construction relies on the fact that any line cuts the curve C in exactly three points, with multiplicity. As we mentioned, this follows from Bézout’stheorem that implies that a plane curve of degree d will intersect any line in exactly d points, with multiplicity. This shows we cannot mimic the above procedure to define a group structure on a curve of higher degree. In fact, more holds true, and the only smooth curves that admit a group structure that is compatible with the geometry of the curve are the elliptic curves which, in the case of plane curves, are exactly those of degree 3. (For more on projective plane curves see [F, Chapter 5], and for more on elliptic curves see [Si].) Well, there is no group structure, but if we insist on adding up points of a given curve C, we can do this using divisors. A (Weil) divisor on C is a finite sum of integer multiples of points of C, i.e., is a sum of the form X D = nP P where nP ∈ Z, P ∈ C and only finitely many of the nP ’s are non-zero. The degree of the divisor D is X deg(D) = nP and we consider the set

n X Div (C) = {D = nP P | deg(D) = n}.

If we compare this situation to that of the cubic curve above, we see that we need a way to determine when two divisors should be considered “equal”. Indeed, in the case of the cubic curve of Figure 2, if we choose points P, Q, P 0,Q0 of the curve such that the line through P and Q and the line through P 0 and Q0 meet at a point of the curve, then by the definition of the group structure, we should have P +Q = P 0 +Q0. It is then to be expected that D = P +Q and D0 = P 0 +Q0 should also be considered to be equivalent as divisors. The appropriated equivalence is called linear equivalence and there is a purely algebraic way to define it. We will not include this definition in these notes, we will rather use the Abel theorem as the definition. (For more on divisors and linear equivalence, see [Sh, Chapter III].)

2 Again let C be a smooth projective curve and ﬁx a point Q ∈ C. The Abel 0 0 theorem states that, given points P1,...,Pn,P1,...,Pn ∈ C, then the divisors P 0 P 0 D = Pi and D = Pi are linearly equivalent if and only if

0 0 Z P1 Z Pn Z P1 Z Pn ω + ... + ω = ω + ... + ω, Q Q Q Q where ω is any global holomorphic (or regular) differential 1-form in C. (See [Sh, Chapter III] for the definition of differentials.) The problem with this statement is that these integrals are not well defined, since we haven’t fixed a path from 0 Q to the points Pi and Pi , for i = 1, . . . , n.

Figure 3: Paths on a curve C

Now, the way around this problem is not to ﬁx paths going from Q to any other point of C, but rather to “mod out” by the integrals over the cycles. Indeed, if γ1 and γ2 are two paths from Q to a point P as in Figure 3, then γ = γ1 − γ2 is a cycle, i.e., a closed path, and we have Z Z Z ω = ω + ω. γ1 γ2 γ So if we consider the set of the integrals of ω over all cycles on C, then we see that the integrals of ω over γ1 and γ2 are equal modulo this set. In fact, we do not even need to consider the full set of all cycles for this to work. Recall that two cycles are homotopic when, roughly speaking, they can be continuously deformed into each other. In particular, the integrals of any regular diﬀerential 1-form over two homotopic cycles are equal. We then need only to consider the set Γ of all cycles on C, modulo homotopia.

Figure 4: γ1 is homotopic to γ2 but not to γ3

3 The set Γ is then an abelian group – with the operation being composition (or addition) of cycles – generated by the 2g cycles α1, . . . , αg, β1, . . . , βg seen in Figure 5, where g = g(C) is the genus of C, deﬁned as the number of holes of C. (It is interesting to try to picture on your mind how the group operation works, and how to decompose a given cycle in terms of αi’s and βi’s.)

Figure 5: Generators of Γ

Now consider the set Ω1(C) of the global holomorphic differential 1-forms on C. This set has a structure of complex vector space of dimension g, and hence 1 we may choose generators ω1, . . . , ωg ∈ Ω (C). The set Z Z g Λ = ω1,..., ωg ∈ C γ ∈ Γ γ γ is then a lattice in Cg with 2g generators given by Z Z Z Z ω1,..., ωg and ω1,..., ωg αi αi βi βi for 1 ≤ i ≤ g. The quotient g J(C) = C Λ is called the Jacobian variety of C. By this definition, it is a complex torus of dimension equal to the genus g of C. More holds true, and J(C) is actually a principally polarized abelian variety (see [BL, Chapter 4.1] for this definition). Fix a point Q ∈ C. The map

α : C → J(C) R P R P P 7→ Q ω1,..., Q ωg is then well deﬁned and is called the Abel map of C. It can be extended to a map deﬁned on the set of divisors by

α(n) : Divn(C) → J(C) P P nP P 7→ nP α(P ).

Since J(C) is a torus, it is an abelian group and this adition makes sense. The Abel theorem can then be restated by saying that two divisors D and D0 of same degree on C are linearly equivalent if and only if α(n)(D) = α(n)(D0). The Abel theorem implies that, for any n ∈ Z, the Jacobian J(C) is isomorphic as an abelian group to the Picard group deﬁned as the quotient P icn(C) of

4 Divn(C) by the linear equivalence. Note that this isomorphism is not canonical, as it depends on the ﬁxed point Q. However, for n = 0 the isomorphism can be shown to be independent of Q. (See [BL, Chapter 11.1] or [GH, Chapter 2.2] for more on Jacobians and the Abel theorem.) Remark 1. The Abel map α is actually a morphism and Abel’s theorem implies that this morphism is an embedding. This map then is a canonical way of puting the curve – which in general does not have a structure of a group, as we saw – inside a variety with a natural group structure. Moreover, the Jacobi inversion theorem states that the induced morphism given by

P1,...,Pn ∈ C 7→ α(P1) + ... + α(Pn) ∈ J(C) is surjective for n ≥ g (see [GH, Chapter 2.2]). This means that α(C) generates the Jacobian J(C), i.e., every point of the Jacobian can be writen as a ﬁnite sum of points of α(C). Remark 2. A less naive way to deﬁne the Jacobian variety of C is

0 ∗ g H (C, ΩC ) C J(C) = =∼ , H1(C, Z) Λ

0 ∗ 0 1 where H (C, ΩC ) is the dual vector space of H (C, ΩC ) = Ω (C), that is, it is 0 the space of linear functions from H (C, ΩC ) to C; and H1(C, Z) = Γ is seen as 0 ∗ a lattice in H (C, ΩC ) by identifying the closed cycles γ ∈ H1(C, Z) with the linear functions Z 0 ω ∈ H (C, ΩC ) 7→ ω ∈ C. γ This lattice has 2g generators and these generators can be identiﬁed with the closed cycles α1, . . . , αg, β1, . . . , βg of Figure 5.

2 Jacobians of smooth curves: line bundles

Most of the material in this section can be found on [BL], [H], [Sh] and [Sh2].

Let C be a smooth projective curve. The Jacobian J(C) can be viewed as a moduli space of line bundles, more precisely, J(C) parameterizes isomorphism classes of line bundles of degree 0 (it could be any n, but the relation for n = 0 is canonical). This follows from the fact that divisors on a smooth variety are associated to line bundles, and this assocation takes linear equivalence classes of divisors to isomorphism classes of line bundles in a unique way (and vice-versa, i.e., an isomorphism classes of line bundles is associated to an unique linear equivalence classes of divisors). We will not construct this association here, see for instance [Sh2, Section IV.1.4]. What we want in these notes is simply to explore this point of view, and for that we need some deﬁnitions. A vector bundle of rank r over C is an algebraic variety E together with a morphism π : E → C such that:

−1 1. for every P ∈ C, the ﬁber EP = π (P ) is a vector space of dimension r;

5 2. for every P ∈ C there exists an open neighborhood U ⊂ C of P such that π−1(U) =∼ U × Cr and the following diagram is commutative:

π−1(U) =∼ U × Cr π ↓ ↓ p1 U = U

where p1 is the projection on the ﬁrst factor. We’ll usually refer to a vector bundle as E, and omit the morphism π. A line bundle L over C is simply a vector bundle of rank r = 1. For example,

OC = C × C together with the morphism π = p1 is called the trivial line bundle. Also, since C is smooth, there is a line bundle TC whose ﬁber at a point P is the tangent space to C at P .

Figure 6: Line bundle L on a curve C

A morphism between two vector bundles π : E → C and π0 : E0 → C is a morphism of varieties f : E → E0 such that 1. the following diagram is commutative

f E → E0 π ↓ ↓ π0 C = C

0 2. for each P ∈ C, the fiber f|EP : EP → EP is a linear transformation of vector spaces. Now we know what an “isomorphism class of line bundles” is, it is simply the set of all line bundles that are isomorphic to a given one. But what is the degree of a line bundle? Well, since we mentioned that every line bundle is associated to a divisor (or rather, a linear equivalence class of divisors), we simply define the degree of a line bundle L as the degree of its associated divisor. (This works because all divisors on a given class have the same degree.) Alternativelly, one could also define degree of a line bundle using the Euler characteristics χ(C, ·) by seting deg(L) = χ(C,L)−χ(C, OC ). In particular, this shows that the trivial line bundle has degree deg(OC ) = 0.

6 Since the Jacobian variety is an abelian variety, the natural question now is how to interpret its group structure in terms of line bundles. As such, the group operation is given by the tensor product of bundles, wich is actually defined fiberwise. So we should first define tensor product of vector spaces. The tensor product V ⊗V 0 of vector spaces V and V 0 is the set of finite sums P 0 0 0 a(v ⊗ v ), where a ∈ C, v ∈ V and v ∈ V , satisfying: • (av) ⊗ v0 = v ⊗ (av0) = a(v ⊗ v0),

0 0 0 • (v1 + v2) ⊗ v = v1 ⊗ v + v2 ⊗ v and

0 0 0 0 • v ⊗ (v1 + v2) = v ⊗ v1 + v ⊗ v2. The tensor product V ⊗ V 0 is a vector space of dimension dim(V ⊗ V 0) = dim(V ) × dim(V 0) and we have V ⊗ C =∼ V =∼ C ⊗ V . (See [Ei, Section A2.2].) Now, given two line bundles L and L0, there is a line bundle L ⊗ L0 called 0 the tensor product, whose ﬁber over a point P of C is LP ⊗ LP . It should be easy to see that OC ⊗ L = L for every line bundle L, that is, the trivial line bundle OC is the identity element of the tensor product operation. Moreover, we have that deg(L ⊗ L0) = deg(L) + deg(L0) (1)

From now on we will identify a point on J(C) with any line bundle in the isomorphism class associated to it. We will therefore speak of “a line bundle L ∈ J(C)” Remark 3. The polarization of J(C) as a principally polarized abelian variety can also be described in terms of line bundles. Indeed, the polarization is given by the theta divisor Θ on J(C), which is isomorphic to the variety parameterizing (isomorphism classes of) line bundles of degree g(C) − 1 over C having at least one non trivial section, where a section of a line bundle L is a morphism of line bundles OC → L.

3 Jacobians of singular curves

Now we shift our focus to singular curves. One may ask why should we care about that. Well, because smooth curves can specialize to singular ones!

Figure 7: Smooth curve specializing to a nodal curve

7 We’ll only deal with nodal singularities, that is, the ones that can be locally described by an equation of the form xy = 0. Now, if C is singular, then there exists a so called normalization map

ν : Cν → C, where Cν is a smooth curve called the normalization of C and ν is a surjective morphism. (See [Sh, Section 5.3] for a more precise deﬁnition of normalization.) For a nodal curve C, that is, a singular curve with only nodal singularities, the normalization Cν can be obtained by “disconnecting” C at the singular points.

Figure 8: Normalization map for a nodal curve

In this picture we have ν(Q1) = ν(Q2) = P and ν is an isomorphism elsewhere. Now, what is the genus of a singular curve C? Should we expect it to still count the “holes” of C? We deﬁne, as in the smooth case,

0 g(C) = dim H (C, ΩC ).

If C is irreducible and nodal (which is the case we’ll be working with), then g(C) does count the holes of C and we have

g(C) = g(Cν ) + #{nodes of C}. (2)

If C has worst singularities than nodal ones, then the above does not hold true, as these singularities should be counted with some multiplicity greater than one (see [H, Chapter IV, Exercise 1.8]). What about the Jacobian of a singular curve C? Well, moduli theory still works and there exists a variety J(C) parameterizing line bundles of degree 0 on C. Moreover, J(C) has a group structure given by the tensor product and is thus a group variety or algebraic group. Unfortunatelly, that’s where the good news end, since J(C) is not an abelian variety. If we try to mimic the construction of the Jacobian variety of a smooth curve seen on Section 1, we get into trouble with the closed cycles associated to the “holes” where C is singular.

Figure 9: The cycle β is homotopic to the node P

8 Remark 4. Alternatively, one can see that the Jacobian J(C) of a singular curve C is not an abelian variety by considering smooth points {Pt} on C approaching a singular point P , as in Figure 10. The line bundles associated to the smooth points Pt (seen as divisors on C) should specialize to a line bundle associated to the singular point P , but there is no such line bundle since P does not deﬁne a (Cartier) divisor on C. (This is because P is not locally principal, meaning it can’t be locally described by a single parameter. Indeed, since P is a node, it is given locally by the equation xy = 0, and thus it must have two local parameters: x and y.) This shows that J(C) is not compact and hence it is not an abelian variety.

Figure 10: Smooth points specializing to a singular one

Let’s now ﬁx an irreducible nodal curve C. What is the structure of the Jacobian variety of C? We will see that in this case J(C) is an “extension of an abelian variety by a torus”. To this end we must consider the pullback map relative to the normalization of C. Let ν : Cν → C be the normalization map. For a line bundle π : L → C, the pullback of L is a line bundle πν : ν∗(L) → Cν deﬁned by

∗ ν ν ν (L) = C ×C L = {(Q, l) ∈ C × L | ν(Q) = π(l)} and such that πν is just the projection on the ﬁrst factor. Note that we have a commutative diagram ν∗(L) → L πν ↓ ↓ π Cν →ν C This gives us the pullback map

ν∗ : J(C) → J(Cν ), which is actually a morphism of group varieties, that is, a morphism preserving the group structure of the Jacobians, since

ν∗(L ⊗ L0) = ν∗(L) ⊗ ν∗(L0).

(See [Sh2, Section 1.2] for more on pullback of vector bundles.) We now describe the ﬁbers of the pullback map (see [C2, 1.1.4] for a more detailed description). We’ll show that for any line bundle M ∈ J(Cν ) the set

(ν∗)−1(M) = {L ∈ J(C) | ν∗(L) =∼ M} can be identiﬁed with (C∗)δ, where δ is the number of nodes of C. For each 1 2 ν node Pj of C let Qj and Qj be the points of C corresponding to it by the

9 normalization map (see Figure 8), so that

−1 1 2 ν (Pj) = {Qj ,Qj }.

ν 1 2 Since C is obtained from C by identifying Qj and Qj for each j, then the ∗ −1 line bundles L on (ν ) (M) are obtained by identifying the ﬁbers M 1 and Qj M 2 . Since M is a line bundle, these ﬁbers are isomorphic as vector spaces to Qj C. Hence, L is obtained by giving an isomorphism of vector spaces between (=∼ M 1 ) and (=∼ M 2 ). (See Figure 11.) Now, such an isomorphism is C Qj C Qj ∗ simply given by multiplication by a nonzero scalar, that is, by cj ∈ C . Thus ∗ for each node Pj we get a cj ∈ C . Since C has δ nodes, then giving a line bundle L on (ν∗)−1(M) is equivalent to giving c ∈ (C∗)δ.

Figure 11: The gluing of a line bundle

We showed that (ν∗)−1(M) can be identiﬁed with (C∗)δ. Actually, more ∗ holds true, since ν is a morphism of group varieties and OC is the identity ele- ∗ −1 ment in J(C), we can consider the kernel (ν ) (OC ). This kernel is isomorphic as a multiplicative group variety to (C∗)δ and we thus get the exact sequence of multiplicative group varieties

∗ ∗ δ i ν ν 0 → (C ) → J(C) → J(C ) → 0, (3) where i is the composition of the isomorphism mentioned above with the inclusion of the kernel of ν∗ in the Jacobian of C. Recall that any morphism from C∗ (more generally, any rational map from any projective space) to an abelian variety is constant (see for instance [BL, Proposition 4.9.5]). The sequence above tells us that this is not the case for the Jacobian of a nodal curve C. In particular, we ﬁnd one more time that J(C) is not an abelian variety. A semi abelian variety is an algebraic group X that ﬁts in an exact sequence of algebraic groups of the form

0 → T → X → A → 0, where T is a torus and A is an abelian variety. In this case we also say that X is the extension of an abelian variety by a torus. By (3), the Jacobians of

10 irreducible nodal curves are semi abelian varieties. (See [BSU] for more on semi abelian varieties.) Remark 5. We just described the structure of the Jacobian of an irreducible nodal curve. What may not be clear is why we needed the curve to be nodal or irreducible. Well, it turns out that we do not really need the irreducibility condition. Note that if C is nodal but not irreducible, then the normalization Cν is disconnected and its connected components are the normalizations of the irreducible components of C. We can still proceed as before and we get that J(C) is still an extension of an abelian variety (which will be the product of the Jacobians of the normalizations of the irreducible components of C) by a torus of the form (C∗)δ−γ+1, where δ is the number of nodes and γ is the number of irreducible components of C (see [C2, 1.1.4]). What about the condition on the singularities of C? As we mentioned, the formula (2) is only true for nodal curves. From that fact, it should be clear by counting dimensions that the sequence (3) is wrong when C is not nodal. But since we mentioned that (2) can be fixed by counting the singularities with some multiplicity, we might think that it would be just a matter of “adjusting” the dimension of the torus (C∗)δ to count the singularities with their multiplicities. Unfortunately, that is not the case. The reasoning we did on Figure 11 is only possible because on a nodal singularity, the two branches of the curve meet transversally, so there is no extra data coming from it when we do the identification of the fibers of the line bundle M to get a line bundle on C. If these branches do not meet transversally, then the “gluing” of the curve must be taken into account as well. However, all is not lost since, by Chevalley’s theorem, the Jacobian of a singular (not necessarily nodal) curve is still the extension of an abelian variety by an affine algebraic group (see [BSU, Theorem 1.1.1]). The nature of the affine algebraic group, however, varies with the type of singularities on C. There is a very nice discussion on this subject with some examples of different types of singularities on [K, Section 2].

4 Compactiﬁed Jacobians of nodal curves

We just saw that the Jacobian variety of an irreducible nodal curve C is not compact. This is a bad thing because it means that sequences of line bundles on C may fail to have a line bundle as a limit, as we saw on Remark 4. This problem can be fixed by giving a compactification of J(C), that is, a compact (projective) variety J(C) containing J(C) as an open and dense subset. Usually, when compactifying moduli spaces, we look for a so called modular compactification, that is, a compactification that is also a moduli space. Well, there actually exist several modular compactifications of the Jacobian and, broadly speaking, J(C) usually parameterizes a type of object over some curve. For simplicity, we will consider C to be an irreducible curve and we will focus on two compactifications: Caporaso’s and Esteves’. For more on compactified Jacobians, see [A] and [K]. Esteves’ compactification works for any type of singularity and it has the advantage that it parameterizes objects over the original curve C. On the other hand, the objects are a bit more complicated than line bundles. In fact, Esteves’ compactified Jacobian parameterizes torsion-free rank-1 degree-0 sheaves on C. If C is nodal, then these are actually less scary than they sound as, roughly

11 compactification curve objects Esteves’ C not line bundles Caporaso’s not C line bundles speaking, a torsion-free rank-1 sheaf on a nodal curve is the tensor product of a line bundle with the ideal sheaf of some nodes of C. The ideal sheaf of a node P of C is precisely the object to solve the problem we got on Remark 4, that is, it is the limit of the line bundles associated to smooth points {Pt} whose limit is P . In particular it is clear from this description that the Jacobian of C is contained in Esteves’ compactification. (See [Es] for more details.) Caporaso’s compactification, on the other hand, is defined only if C is nodal. The upside is that it does parameterize line bundles, just not on the original curve. It actually parameterizes line bundles on a “blow-up” of C. We will not give a more general definition of blow-up, see [Sh] or [H] for this. What we need is a lot simpler. A blow-up of a nodal curve C is a nodal curve C˜ obtained by disconnecting some of the nodes P of C (as in the normalization map of Figure 1 −1 8) and inserting a P to connect the points Q1 and Q2 of ν (P ). (See Figure 12.)

Figure 12: Blow-up of a nodal curve at two nodes

The added P1’s are called the exceptional components of C˜. The irreducible components of C˜ are the normalization Cν of C and the exceptional components. Moreover, there is a map η : C˜ → C given by contracting all exceptional components, so that η restricted to Cν is juts ν. Caporaso’s compactiﬁed Jacobian of a nodal irreducible curve C parameterizes degree-0 line bundles L on the blow-up C˜ of C at all of its nodes, such that

1 the restriction L|P to each exceptional component has degree 0 or 1. (See [C] or [C2].) It is not clear from this definition how one can see the Jacobian of C as a subset of Caporaso’s compactification. This can be done using pullbacks. Indeed, if M is a line bundle on C, then the pullback L = η∗(M) is a line 1 ˜ 1 bundle on C such that L|P has degree 0 for every exceptional component P . This gives an inclusion of J(C) in Caporaso’s compactified Jacobian.

12 Remark 6. The description of Caporaso’s compactification given above differs slightly from the one on [C] or [C2]. Indeed, on these references, the line bundles L are considered on blow-ups C˜S at a subset S of the set of nodes of C, and the 1 restrictions L|P to the exceptional components are required to have degree 1. Caporaso’s compactified Jacobian as described on [C] or [C2] parameterizes all degree-0 line bundles on all blow-ups C˜S satisfying the above condition. Well, the two descriptions are actually equivalent since, if ηS : C˜ → C˜S is the map given by contracting all exceptional components not associated to a node of S, ˜ ∗ and M is any line bundle on CS, then the restriction of the pullback L = ηS(M) to any exceptional component of C˜ not contracted by ηS has degree 0. Remark 7. In this section we have set the curve C to be irreducible. This is not really necessary. Indeed, Esteves’ compactification works for any curve, but there is a technical condition on the degrees of the restrictions of the torsion-free rank-1 sheaves to the components of C. (See [Es, Section 1.2].) It should be noted that what we described as Esteves’ compactification was done previously by Altman and Kleiman in the case of irreducible curves on [AK]. Esteves’s constribution was to extend the construction to the reducible case. Caporaso’s compactification also works for a nodal curve C with more components, as long as C is stable. However, as in Esteves’s case, there is a technical condition on the degrees of the restrictions of the line bundles to the components. (See [C2, Section 1.3].) A stable curve is a nodal curve where the genus-0 components (i.e., the P1’s) are required to meet the rest of the curve in at least 3 nodes. The name “stable” comes from the construction of a compactification for the moduli of smooth curves as a GIT quotient. The GIT or Geometric Invariant Theory is often used to construct moduli spaces (Caporaso’s construction uses GIT; Esteves’ does not) and is concerned with taking quotients of varieties under group actions. Now, the set of isomorphism classes of smooth curves of genus g is parameterized by a variety Mg, and this variety is not compact since, as we saw on Figure 7, sequences of smooth curves can have a singular curve as a limit. There is a modular compactification Mg of this moduli space, called the Deligne-Mumford compactification, and it parameterizes precisely the stable curves of genus g. Both constructions of Mg and Mg use GIT. Given the infinite types of singularities a curve can have, it is a surprising fact that a compactification of the space Mg of smooth curves can be achieved just by allowing the curves to have the simplest type of singularity there is, i.e., a nodal singularity. What this means is that, if a family of smooth curves specializes to a singular non-stable curve, regardless of how awfull the singularities of this curve are, then we can always modify the family on a proccess called stable reduction so that it specializes to a stable curve instead. Hence when taking limits of smooth curves, we need only to consider stable curves. (See [N] for more on GIT quotients. See [HM] for more on stable curves and moduli of curves.) Lastly, it is worth mentioning that both Caporaso’s and Esteves’ compactifications were constructed for families of curves. In fact, Caporaso’s compactification is done over Mg, meaning that there exists a variety J g which, roughly speaking, is the union of the compactified Jacobians of all stable curves. There is a map from J g to Mg and the fiber of this map over a point corresponding to a stable curve C is the compactified Jacobian of this C.

13 Now, what about the structure of Esteves’ and Caporaso’s compactifications? From Section 3 we know we shouldn’t expect them to be abelian varieties. But can they still be semi abelian ones? Or at least algebraic groups? Do we still have a group structure on these compactifications? Sadly, no. We can still do a tensor product of the objects parameterized by both compactifications, but these tensor products will not be associated with points of the compactifications. Indeed, let C be an irreducible nodal curve, for simplicity. First we examine Caporaso’s compactification. Let C˜ be the blow-up at all the nodes of C. Then the points of the compactification correspond to line bundles on C˜ having degree 0 or 1 on the exceptional components. Now, by the aditivity of the degree seen in Equation (1), if we make the tensor product of two such line bundles L and L0 both having degree 1 on a given exceptional component, then the product L⊗L0 will have degree 2 on this component, and hence it will not give a point of the compactification. Note however that if L has degree 0 on every exceptional component then L ⊗ L0 does give a point of the compactification, regardless of the L0 chosen. Since these L’s correspond to the line bundles on the Jacobian J(C), we get that J(C) acts on Caporaso’s compatification. In Esteves’s compactification the situation is a bit more subtle. As we mentioned, Esteves’ compactification of the Jacobian of C parameterizes, roughly speaking, degree-0 tensor products of line bundles with the ideal sheaf of some nodes of C. Clearly, when we tensor two such objects, we obtain again a line bundle tensored with the ideal sheaf of some nodes of C. What we did not mention before is that for this description to give a point in Esteves’ compactification, we need to take distinct nodes. (The key fact here is that the square of the ideal sheaf of a node – i.e., the tensor product of it with itself – is not torsion-free.) So, as in Caporaso’s case, we once again do not have the tensor product operation defined on the whole compactification, but the Jacobian J(C) acts on the compactification. Therefore neither Esteves’ or Caporaso’s compactification is a group variety, but they both are G-spaces for the group G = J(C), since the Jacobian acts on both of them. Actually, more holds true, and they can bee seen as limits of abelian varieties. More precisely, there is a moduli space Ag parameterizing principally polarized abelian varieties of dimension g. As is the case with the moduli of curves Mg, the space Ag is not compact. But unlike the moduli of curves, which has as a widely accepted compactification the Deligne-Mumford space Mg of stable curves, for the moduli Ag of abelian varieties there is no such canonical or agreed upon choice. There are actually a great number of compactifications for it, see [G] for a detailed survey on Ag and its compactifications. It is a consequence of [A] that both Caporaso’s and Esteves’ compactifications define a point in one of these compactifications of Ag. Remark 8. It should not come as a surprise that Caporaso’s and Esteves’ compactifications of the Jacobian are limits of abelian varieties. Indeed, we mentioned on Remark 7 that they were constructed for families of curves. This means that when we consider a family of smooth curves specializing to a singular curve (stable for Caporaso’s compactification; not necessarily so, for Esteves’), then the compactified Jacobians of these smooth curves specialize to the compactified Jacobian of the singular curve. But the compactified Jacobian of a smooth curve is the Jacobian variety, as it is already compact! Hence we get

14 that the compactiﬁed Jacobian of the singular curve is a limit of the Jacobians of the smooth curves of the family.

References

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