F. Enriques' claimed in 1904 that, given a smooth projective surface $S$ with irregularity $q>0$, what we call nowadays the Picard scheme of $S$ is an abelian variety of dimension $q$.

Enriques' algebraic proof was considered controversal and led to many disputes among the geometers of the Italian school (in fact, they called Enriques' claimed result the fundamental theorem in the theory of irregular algebraic surfaces, or also the theorem of completeness of the characteristic series).

Actually, Enriques' proof contained lacunae, as well as the subsequent proof by Severi. The first correct proof, using transcendental methods, was given by Poincaré in 1910. For a correct algebraic proof in characteristic $0$ it was necessary the wait for the work of Grothendieck, 50 years later.

Today it is known that the result is false in positive characteristic, as shown by Igusa in 1955. In fact, he constructed a smooth projective surface $S$ with $\mathrm{Pic}^0(S)$ non reduced, and hence not an abelian variety.

F. Enriques' claimed in 1904 that, given a smooth projective surface $S$ with irregularity $q>0$, what we call nowadays the Picard scheme of $S$ is an abelian variety of dimension $q$.

Enriques' algebraic proof was considered controversal and led to many disputes among the geometers of the Italian school (in fact, they called Enriques' claimed result the fundamental theorem in the theory of irregular algebraic surfaces, or also the theorem of completeness of the characteristic series).

Actually, Enriques' proof contained lacunae, as well as the subsequent proof by Severi. The first correct proof, using transcendental methods, was given by Poincaré in 1910. For a correct algebraic proof in characteristic $0$ it was necessary to wait for the work of Grothendieck, 50 years later.

Today it is known that the result is false in positive characteristic, as shown by Igusa in 1955. In fact, he constructed a smooth projective surface $S$ with $\mathrm{Pic}^0(S)$ non reduced, and hence not an abelian variety.

F. Enriques' claimed in 1904 that, given a smooth projective surface $S$ with irregfularity $q>0$, what we call nowadays the Picard scheme of $S$ is an abelian variety of dimension $q$.

Enriques' algebraic proof was considered controversal and led to many disputes among the geometers of the Italian school (in fact, they called Enriques' claimed result the fundamental theorem in the theory of irregular algebraic surfaces, or also the theorem of completeness of the characteristic series).

Actually, Enriques' proof contained lacunae, as well as the subsequent proof by Severi. The first correct proof, using transcendental methods, was given by Poincaré in 1910. For a correct algebraic proof in characteristic $0$ it was necessary the wait for the work of Grothendieck, 50 years later.

Today it is known that the result is false in positive characteristic, as shown by Igusa in 1955. In fact, he constructed a smooth projective surface $S$ with $\mathrm{Pic}^0(S)$ non reduced, and hence not an abelian variety.

F. Enriques' claimed in 1904 that, given a smooth projective surface $S$ with irregularity $q>0$, what we call nowadays the Picard scheme of $S$ is an abelian variety of dimension $q$.

Enriques' algebraic proof was considered controversal and led to many disputes among the geometers of the Italian school (in fact, they called Enriques' claimed result the fundamental theorem in the theory of irregular algebraic surfaces, or also the theorem of completeness of the characteristic series).

Actually, Enriques' proof contained lacunae, as well as the subsequent proof by Severi. The first correct proof, using transcendental methods, was given by Poincaré in 1910. For a correct algebraic proof in characteristic $0$ it was necessary the wait for the work of Grothendieck, 50 years later.

Today it is known that the result is false in positive characteristic, as shown by Igusa in 1955. In fact, he constructed a smooth projective surface $S$ with $\mathrm{Pic}^0(S)$ non reduced, and hence not an abelian variety.