Let $C$ be a smooth curve of genus $g$, and let us consider its $d$-th symmetric product $\textrm{Sym}^d(C)$ and its Jacobian $J(C)$. Fixing a point $p\_0 \in C,$ there are two maps $$u\_d\colon C\_d \to J(C), \quad u\_d(D):=\mathcal{O}\_C(D-dp\_0),$$ $$i\_{d-1} \colon \textrm{Sym}^{d-1}(C) \to \textrm{Sym}^d(C), \quad i\_{d-1}(D):=D+p\_0.$$ Then the following two divisor classes in the Néron-Severi group of $\textrm{Sym}^d(C)$ are independent on $p\_0$: $$\theta:=u\_d^\*(\Theta), \quad x:=i\_{d-1} (\textrm{Sym}^{d-1}(C)),$$  

where $\Theta$ is the theta divisor in the Néron-Severi group of $J(C)$. It is well-known that, *if $C$ is with general moduli,* (i.e., it defines a very general point in the moduli space $\mathcal{M}\_g$), then the following holds:



> 
> **(1)** the Néron-Severi group of $\textrm{Sym}^d(C)$ is generated by $\theta$ and $x$;
> 
> 
> **(2)** the class $\delta$ of the diagonal of $\textrm{Sym}^d(C)$ is given by $$\delta = 2((d+g-1)x - \theta).$$
> 
> 
> 


Now, it is not difficult to produce counterexamples (for instances, curves with non-trivial correspondences) showing that **(1)** does not hold if $C$ is not with general moduli. So let me ask the following



> 
> **Question.** Is **(2)** true for *every* smooth curve of genus $g$?
> 
> 
> 


A. Kouvidakis, in Lemma 7 of his paper *[Divisors on symmetric product of curves](http://www.jstor.org/stable/2154312?seq=1#page_scan_tab_contents)* (Trans. Amer. Math. Soc. **337**), claims that **(2)** is due to MacDonald, and refers to Arbarello-Cornalba-Griffiths-Harris'book *Geometry of Algebraic Curves I*, Proposition 5.1 p. 358 for the proof.


Actually, the (much more general) result quoted in the book is apparently stated without assumptions on $C$, but the proof is based on a Lemma whose statement begins with "*Let $C$ be a curve with general moduli*", and whose content is essentially **(1)**. So I am a bit confused.


My impression is that the answer to the question should be *yes* and that some kind of specialization argument should do the job. Any reference to the relevant literature will be particularly appreciated. I am primarily interested in the case $d=g=2$, but I would like to know the answer to the general question, too.



The class of the diagonal in the symmetric product of a smooth curve
This identity might be explicitly given in one of the following articles of Arthur Mattuck (Mattuck does prove many identities), but I could not find it.


MR0142553 (26 #122)  

Mattuck, Arthur  

Symmetric products and Jacobians.  
 
Amer. J. Math. 83 1961 189–206.   

14.20 (14.51) 


MR0136608 (25 #76)   
 
Mattuck, Arthur   

On symmetric products of curves.   

Proc. Amer. Math. Soc. 13 1962 82–87.   
 
14.10 (14.20) 


Just to repeat your notation, for every $d\in \mathbb{Z}$, define $u\_d:C\_d\to \text{Pic}^d\_{C/k}$ to be the Abel map. Define $i\_{d-1,d}:C\_{d-1}\to C\_d$ to be the morphism adding $\underline{p}\_0$ to an effective divisor. Define $j\_{d-1,d}:\text{Pic}^d\_{C/k}$ to be the morphism that twists an invertible sheaf by $\mathcal{O}\_C(\underline{p\_0})$. The morphism $j\_{d-1,d}$ is an isomorphism. 


The image $\Theta = u\_{g-1}(C\_{g-1})\subset \text{Pic}^{g-1}\_{C/k}$ is an effective, reduced Cartier divisor. For every $d$, define $\Theta\_d\subset \text{Pic}^d\_{C/k}$ to be the unique Cartier divisor such that $j\_{d-1,d}^\*\Theta\_d$ equals $\Theta\_{d-1}$. Define $\Delta\_d \subset C\_d$ to be the (reduced) Cartier divisor parameterizing effective divisors on $C$ that are nonreduced (i.e., at least one pair of points comes together). Finally, denote $H\_d = i\_{d-1}(C\_{d-1})$ as an effective Cartier divisor on $C\_d$. 


The basic identity is that $i\_{d+1,d}^\*(\underline{\Delta\_{d+1}})$ equals $\underline{\Delta\_d} + 2\underline{H\_d}$ as effective Cartier divisors on $C\_d$. Set-theoretically this is clear. The fact that the coefficient of $\underline{H\_d}$ equals $2$ follows from a local analysis: it is the fact that the quotient morphism $q:C^2\to C\_2$ is simply branched along the diagonal (i.e., reduce to the case that $d$ equals $1$). By definition, $i\_{d+1,d}^\*u\_{d+1}^\* \Theta\_{d+1}$ equals $u\_d^\*\Theta\_d$. 


Finally, the normal sheaf of the embedding $i\_{d,d+1}$ equals $T\_{C,p\_0}\otimes\_k \mathcal{O}\_{C\_d}(\underline{H}\_d)$. To see this, consider the one-parameter family of deformations of $i\_{d,d+1}$ obtained by varying $p\_0\in C$. In particular, a first-order deformation in $T\_{C,p\_0}\setminus\{0\}$ defines a section of the normal sheaf of $i\_{d,d+1}$. Away from $H\_d$, it is straightforward to see that this section is nonzero. Since $H\_d$ is irreducible, it follows that the normal sheaf equals $T\_{C,p\_0}\otimes\_k \mathcal{O}\_{C\_d}(m\underline{H}\_d)$ for some integer $m$. Working locally at a generic point of $H\_d$, $m$ equals $1$ since $C^2 \to C\_2$ is simply branched along the diagonal (the same reason for the coefficient $2$ in the previous paragraph).


That is all the necessary preparation. Because of the computations of $i\_{d+1,d}^\*$ on various divisors, on divisor classes we have the relation $$ i\_{d+1,d}^\* \mathcal{O}\_{C\_{d+1}}(-\underline{\Delta}\_{d+1}-2\underline{\Theta}\_{d+1} +2((d+1)+g-1)\underline{H}\_{d+1})) \cong $$ $$ T\_{C,p\_0}^{\otimes 2(d+g)}\otimes\_k \mathcal{O}\_{C\_d}(-\underline{\Delta}\_d -2 \underline{H}\_d - 2\underline{\Theta}\_d + 2(d+g)\underline{H}\_d). $$ Thus the claim for $d+1$ implies the claim for $d$.


Finally, when $d > 2g-2$, by Abel's Theorem / Jacobi Inversion and Riemann-Roch, the morphism $u\_d$ is a smooth, projective morphism whose geometric generic fibers are projective spaces of relative dimension $d-g$. Moreover, $H\_d$ gives a relative hyperplane class. Thus, for every invertible sheaf $\mathcal{L}$ on $C\_d$ that has degree $0$ on a fiber, $(u\_d)\_\*\mathcal{L}$ is an invertible sheaf on $\text{Pic}^d\_{C/k}$, and the adjunction map $$u\_d^\*(u\_d)\_\*\mathcal{L} \to \mathcal{L},$$ is an isomorphism. Now let $f:C\to \mathbb{P}^1$ be a tamely ramified, finite, flat morphism of degree $d> 2g-2$ (such morphisms exist by Riemann-Roch and Bertini type arguments). Let $\widetilde{f}:\mathbb{P}^1\to C\_d$ be the morphism associated to the graph of $f$ in $C\times \mathbb{P}^1$ considered as a Cartier divisor of relative degree $d$ over $\mathbb{P}^1$. Then $\widetilde{f}^\*H\_d$ is the reduced divisor $f(p\_0)$. By Riemann-Hurwitz, $\widetilde{f}^\*\Delta\_d$ is the branch divisor and has degree $2(d+g-1)$. Thus, the invertible sheaf from the previous paragraph has degree $0$ on fibers of $u\_d$.


Thus, for every $C$, for every $p\_0$, for every $d\geq 0$, there exists a unique invertible sheaf $\mathcal{L}\_d$ on $\text{Pic}^d\_{C/k}$ such that $j\_{d,d+1}^\*\mathcal{L}\_{d+1}$ is isomorphic to $\mathcal{L}\_d$, such that
$$u\_d^\*\mathcal{L}\_d \cong \mathcal{O}\_{C\_d}(-\underline{\Delta}\_d - 2\underline{\Theta}\_d + 2(d+g-1)\underline{H}\_d),
$$ for all $d$, and such that $$\mathcal{L}\_d = (u\_d)\_\* \mathcal{O}\_{C\_d}(-\underline{\Delta}\_d - 2\underline{\Theta}\_d + 2(d+g-1)\underline{H}\_d),$$
for all $d>2g-2$. The claim is that $\mathcal{L}\_d$ is an invertible sheaf on $\text{Pic}^d\_{C/k}$ that is algebraically equivalent to zero.


By considering the case that $d=1$, it is already clear that $\mathcal{L}\_d$ has degree $0$ on the Abel image of $C$. Moreover, since the construction works relatively in families, via properness and separatedness of the (components) of the relative Picard scheme, it suffices to prove for $C$ very general that every invertible sheaf on $\text{Pic}^1\_{C/k}$ that has degree $0$ on the Abel image of $C$ is algebraically equivalent to zero. This follows, for instance, from the Franchetta conjecture (proved over the complex numbers by John Harer, and extended to positive characteristic by Stefan Scröer).


**Edit.** You do not need to use Franchetta's conjecture. If every correspondence on $C$ has valence, then you can use $d=2$ to prove that $u\_2^\*\mathcal{L}\_2$ is algebraically equivalent to zero. By the theorem of the cube, already $u\_2^\*$ is injective on Picard groups. So it suffices to prove that there exists a curve $C$ such that every correspondence has valence. That is much easier than Franchetta's conjecture.


**Second Edit.** If you specialize to the case $d=g-1$, which again is sufficient to prove the claim when $g\geq 3$ so that $d\geq 2$, you can eliminate the dependence on $p\_0$. Once you do that, the claim reduces to the following claim: for the universal divisor $\mathcal{D}\subset C\times C\_{g-1}$, the following invertible sheaf on $C\_d$ is algebraically equivalent to zero, $$\mathcal{B}\_C := \text{det}(R^1\text{pr}\_{1,\*}\mathcal{O}(-\underline{\mathcal{D}}))^{\otimes 2}(-\underline{\Delta}).$$ Since this invertible sheaf is defined over the field of definition of $C$, in fact, it must be that $\mathcal{B}\_C$ is isomorphic to the structure sheaf. There may be a direct proof that $\mathcal{B}\_C$ is trivial using Grothendieck-Riemann-Roch (which would prove your result without specialization).


**Final Edit.** If you apply Grothendieck-Riemann-Roch to compute the class of $\text{det}(R^1\text{pr}\_{1,\*}\mathcal{O}(-\underline{\mathcal{D}}))$, it gives that the Cartier divisor class of the square of this line bundle equals the pushforward via the finite flat morphism, $$\text{pr}\_{1,\mathcal{D}}:\mathcal{D} \to C\_{d-1},$$ of the relative canonical divisor class. Via the pullback sequence of sheaves of differentials associated to $\text{pr}\_{1,\mathcal{D}}$, there is an effective representative of the relative divisor class supported on the ramification locus of $\text{pr}\_{1,\mathcal{D}}$. That maps to the diagonal $\Delta\_{g-1}$ in $C\_{g-1}$. By the same local analysis as above, the multiplicity equals $1$. This proves that $\mathcal{B}\_C$ is isomorphic to the structure sheaf without using specialization. That proves that $u\_{g-1}^\*\mathcal{L}\_{g-1}$ is algebraically equivalent to zero. When $g\geq 3$, that implies your result without specialization.


**Final, Final Edit.** Francesco Polizzi points out that the formula does indeed follow from Formulas (1) and (5) of Mattuck's article "On symmetric products of curves" listed above.


