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Notes on Jacobian Varieties: a Brief Survey

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Notes on Jacobian Varieties: a Brief Survey Notes on Jacobian varieties: a brief survey Juliana Coelho (UFF) [email protected]ff.br 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 Puc´on,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 2 C, there exists a unique point R 2 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 2 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 B´ezout'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 2 Z, P 2 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) = fD = nP P j deg(D) = ng: 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 2 C. The Abel 0 0 theorem states that, given points P1;:::;Pn;P1;:::;Pn 2 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 ! + ::: + ! = ! + ::: + !; Q Q Q Q where ! 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 ! = ! + !: γ1 γ2 γ So if we consider the set of the integrals of ! over all cycles on C, then we see that the integrals of ! 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 !1;:::;!g 2 Ω (C). The set Z Z g Λ = !1;:::; !g 2 C γ 2 Γ γ γ is then a lattice in Cg with 2g generators given by Z Z Z Z !1;:::; !g and !1;:::; !g 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 2 C. The map α : C ! J(C) R P R P P 7! Q !1;:::; Q !g 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 2 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 2 C 7! α(P1) + ::: + α(Pn) 2 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).
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