Abelian fibrations on S[n]

Let $\text{S}\text{→}\text{φ}\text{P}{}^1$ be an elliptic fibration on a K3 surface S. Then the composition $\text{S}{}^{\mathrm{n]}}\text{→}\text{π}\text{S}{}^{\mathrm{(n)}}\text{→}\text{sym}{}^{\mathrm{n}}\text{φ}\text{P}{}^{\mathrm{n}}$ gives an Abelian fibration on S^n]. Let E be the exceptional divisor of π, then symⁿφ°π(E) is of dimension n?1. We prove the inverse in this Note. To cite this article: B. Fu, C. R. Acad. Sci. Paris, Ser. I 337 (2003).