### A universal property of the Cayley-Chow space of algebraic cycles

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In a previous paper we described the collection of homological equivalence relations on a curve of genus $\ge 2$ as the set of integral solutions of certain algebraic equations. In the present paper we improve one argument of the previous paper, and we study the equations more closely for a curve of genus 2.

Morphisms on a curve may be seen as homology classes in the self product. We describe these classes as belonging to an intersection: the locus of integral points of an algebraic set in the complex homology group, and the locus of effective divisor classes. We write down explicit equations for the algebraic set, and in the case of genus three we compute a few explicit solutions over the rationals.

In the present paper, it is established in any characteristic the validity of a classical theorem of Enriques', stating the linearity of any algebraic system of divisors on a projective variety, which has index 1 and whose generic element is irreducible, as soon as its dimension is at least 2.

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