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There were quite a few different questions, so forgive me if my answer is somewhat fragmented.

The Néron-Severi group $NS(X)$ (divisors modulo algebraic equivalence) is finitely generated over any field for any non-singular projective variety $X$, this is Severi's theorem of the base (at least for the case of characteristic zero). In what follows however I am assume that the variety is defined over $\mathbb{C}$ for simplicity. Some answers to your other questions:

• In the case of curves, $NS(X)\cong\mathbb{Z}$, with the isomorphism given by the degree map.

• More generally, there is a natural morphism $NS(X) \to H^2(X,\mathbb{Z})$ which is injective on the free part of the group. This can then be used to get an upper bound for the Picard number.

• Torsion in $NS(X)$ naturally lives inside $H_1(X,\mathbb{Z})$, so if this is torsion free then so is $NS(X)$. This is the case free for abelian varieties (over $\mathbb{C}$ say), since $H_1(X,\mathbb{Z})$ can identified with a lattice $\Lambda \subset \mathbb{C}^g$ such that $X\cong\mathbb{C}^g/\Lambda$.

• Finally, torsion divisors can never be ample because they are numerically trivial, and ampleness is preserved under numerical equivalence.

There were quite a few different questions, so forgive me if my answer is somewhat fragmented.

The Néron-Severi group $NS(X)$ (divisors modulo algebraic equivalence) is finitely generated over any field for any non-singular projective variety $X$, this is Severi's theorem of the base (at least for the case of characteristic zero). In what follows however I am assume that the variety is defined over $\mathbb{C}$ for simplicity. Some answers to your other questions:

• In the case of curves, $NS(X)\cong\mathbb{Z}$, with the isomorphism given by the degree map.

• More generally, there is a natural injective morphism $NS(X) \to H^2(X,\mathbb{Z})$. This can then be used to get an upper bound for the Picard number.

• Torsion in $NS(X)$ also naturally lives inside $H_1(X,\mathbb{Z})$, so if this is torsion free then so is $NS(X)$. This is the case free for abelian varieties (over $\mathbb{C}$ say), since $H_1(X,\mathbb{Z})$ can identified with a lattice $\Lambda \subset \mathbb{C}^g$ such that $X\cong\mathbb{C}^g/\Lambda$.

• Finally, torsion divisors can never be ample because they are numerically trivial, and ampleness is preserved under numerical equivalence.

There were quite a few different questions, so forgive me if my answer is somewhat fragmented.

The Néron-Severi group $NS(X)$ (divisors modulo algebraic equivalence) is finitely generated over any field for any non-singular projective variety $X$, this is Severi's theorem of the base (at least for the case of characteristic zero). In what follows however I am assume that the variety is defined over $\mathbb{C}$ for simplicity. Some answers to your other questions:

• In the case of curves, $NS(X)\cong\mathbb{Z}$, with the isomorphism given by the degree map.

• More generally, there is a natural morphism $NS(X) \to H^2(X,\mathbb{Z})$ which is injective on the free part of the group. This can then be used to get an upper bound for the Picard number.

• Torsion in $NS(X)$ naturally lives inside $H^1(X,\mathbb{Z})$, so if this is torsion free then so is $NS(X)$. This is the case free for abelian varieties (over $\mathbb{C}$ say), since $H^1(X,\mathbb{Z})$ can identified with the dual of a lattice $\Lambda \subset \mathbb{C}^g$ such that $X\cong\mathbb{C}^g/\Lambda$.

• Finally, torsion divisors can never be ample because they are numerically trivial, and ampleness is preserved under numerical equivalence.

There were quite a few different questions, so forgive me if my answer is somewhat fragmented.

The Néron-Severi group $NS(X)$ (divisors modulo algebraic equivalence) is finitely generated over any field for any non-singular projective variety $X$, this is Severi's theorem of the base (at least for the case of characteristic zero). In what follows however I am assume that the variety is defined over $\mathbb{C}$ for simplicity. Some answers to your other questions:

• In the case of curves, $NS(X)\cong\mathbb{Z}$, with the isomorphism given by the degree map.

• More generally, there is a natural morphism $NS(X) \to H^2(X,\mathbb{Z})$ which is injective on the free part of the group. This can then be used to get an upper bound for the Picard number.

• Torsion in $NS(X)$ naturally lives inside $H_1(X,\mathbb{Z})$, so if this is torsion free then so is $NS(X)$. This is the case free for abelian varieties (over $\mathbb{C}$ say), since $H_1(X,\mathbb{Z})$ can identified with a lattice $\Lambda \subset \mathbb{C}^g$ such that $X\cong\mathbb{C}^g/\Lambda$.

• Finally, torsion divisors can never be ample because they are numerically trivial, and ampleness is preserved under numerical equivalence.

There were quite a few different questions, so forgive me if my answer is somewhat fragmented.

• The Néron-Severi group $NS(X)$ (divisors modulo algebraic equivalence) is finitely generated over any field for any non-singular projective variety $X$, this is Severi's theorem of the base.

• In the case of curves, it is always isomorphic to $\mathbb{Z}$, with the isomorphism given by the degree map.

• There is a natural morphism $NS(X) \to H^2(X,\mathbb{Z})$, which is injective on the free part of the group. This will then give you an upper bound for the Picard number.

• Torsion in $NS(X)$ naturally lives inside $H^1(X,\mathbb{Z})$, so if this is torsion free then so is $NS(X)$. This is the case free for abelian varieties (over $\mathbb{C}$ say), since $H^1(X,\mathbb{Z})$ can identified with the dual of a lattice $\Lambda \subset \mathbb{C}^g$ such that $X\cong\mathbb{C}^g/\Lambda$.

• Finally, torsion divisors can never be ample because they are numerically trivial, and ampleness is preserved under numerical equivalence.

There were quite a few different questions, so forgive me if my answer is somewhat fragmented.

The Néron-Severi group $NS(X)$ (divisors modulo algebraic equivalence) is finitely generated over any field for any non-singular projective variety $X$, this is Severi's theorem of the base (at least for the case of characteristic zero). In what follows however I am assume that the variety is defined over $\mathbb{C}$ for simplicity. Some answers to your other questions:

• In the case of curves, $NS(X)\cong\mathbb{Z}$, with the isomorphism given by the degree map.

• More generally, there is a natural morphism $NS(X) \to H^2(X,\mathbb{Z})$ which is injective on the free part of the group. This can then be used to get an upper bound for the Picard number.

• Torsion in $NS(X)$ naturally lives inside $H^1(X,\mathbb{Z})$, so if this is torsion free then so is $NS(X)$. This is the case free for abelian varieties (over $\mathbb{C}$ say), since $H^1(X,\mathbb{Z})$ can identified with the dual of a lattice $\Lambda \subset \mathbb{C}^g$ such that $X\cong\mathbb{C}^g/\Lambda$.

• Finally, torsion divisors can never be ample because they are numerically trivial, and ampleness is preserved under numerical equivalence.