Résume | We consider a complete non-singular curve $C$ and its Jacobian $J$. In the study of algebraic cycles on $J$ we have a number of interesting structures at our disposal. E.g., in addition to the usual intersection product $\cdot$, the Chow ring $\textrm{CH}(J)$ carries a second ring structure: the Pontryagin product $*$. Also we have the Fourier transform $\mathcal{F} : \textrm{CH}(J) \to \textrm{CH}(J)$ that exchanges the two products. Now we can make cycle classes on $J$ by starting with the class of $C \subset J$, using all operations $n^*$, $\mathcal{F}$, $\cdot $ and $*$, and taking linear combinations. The resulting subring $\mathcal{T}(C) \subset \textrm{CH}(J)$ is called the \textit{tautological ring of $C$}. In our talk we shall try to explain some general results of Beauville and Polishchuk; in particular it is known that the tautological ring is finitely generated, with an explicit set of generators. The main theme of my talk will be the connections between geometric properties of the curve $C$ and the structure of the tautological ring. E.g., extending an older result of Colombo and van Geemen, recent results of Herbaut and van der Geer and Kouvidakis show that the existence of linear systems of a given rank and degree translate into relations between the generators of $\mathcal{T}(C)$ modulo algebraic equivalence. We shall also discuss if, and how, one can lift such results to the full Chow ring of $J$. |