# Divisors on the symmetric product of an elliptic curve

Assume that $$C$$ is an elliptic curve and $$C_p$$ is the $$p$$-fold symmetric product. Let $$\beta:C_p\to C$$ be defined by the addition on the elliptic curve. Let $$u\in C$$ be the zero in the additive group of $$C$$. Then $$\beta^*u=:F$$ is a divisor on $$C_p$$. I want to calculate $$H^0(C_p,nF)$$. The paper Symmetric products of elliptic curves and surfaces of general type with $$p_g = q = 1$$ by Cantanese and Ciliberto(Link) indicates that $$C_p$$ is a projective bundle over $$C$$ and its Theorem 1.17 gives an explicit answer, but I don't understand how the leading coefficient is determined. Another confusion is that it indicates that all divisors on $$C_p$$ are algebraically equivalent to $$mD+nF$$, where $$D=u+C_{p-1}$$, but we know that the Picard group of a projective bundle is the direct sum of the picard group of the base and the integer $$\mathbb{Z}$$. We have $$Pic(C)=C\oplus\mathbb{Z}$$. Therefore we should have $$Pic(C_p)=Pic(C)\oplus \mathbb{Z}$$. I think the symbol $$\mathbb{Z}$$ in $$Pic(C)$$ stands for $$\mathbb{Z}F$$, and the symbol $$\mathbb{Z}$$ in $$Pic(C_p)$$ stands for $$\mathbb{Z}D$$. Why the Picard group of $$C_p$$ can be generated by only two elements? If there was any misunderstanding, please indicate it to me.

• "algebraically equivalent" $\neq$ "linearly equivalent". The authors claim that the Neron-Severi group is generated by two elements, not the Picard group. Oct 12, 2020 at 13:30

For the second part, as what Francesco Polizzi indicated, the group generated by $$F$$ and $$D$$ is not the Picard group of $$C_p$$. For the calculation of the cohomology, we have $$nF=n\beta^*u=\beta^*(nu).$$ Therefore $$H^0(C_p,nF)=H^0(C_p,\beta^*\mathcal{O}_C(nu))=H^0(C,\beta_*\beta^*\mathcal{O}_C(nu)).$$ By projection formula, $$\beta_*\beta^*\mathcal{O}_C(nu)=\beta_*\mathcal{O}_{C_p}\otimes\mathcal{O}_C(nu).$$ We are left to show that $$\beta_*\mathcal{O}_{C_p}=\mathcal{O}_{C}.$$ As indicated by the paper I mentioned in the question, $$C_p$$ is in the form $$\mathbb{P}(E)$$ for some vector bundle $$E$$ on $$C$$. For any affine open subset $$U\cong\mathrm{Spec}(A)\subset C$$, $$\beta_*\mathcal{O}_{C_p}(U)\cong\mathcal{O}_{\mathbb{P}(E)}\bigl(\operatorname{Proj}\operatorname{Sym}E(U)\bigr)\cong E^0(U)=\mathcal{O}_C(U).$$