Enriques Surface
Here are the two outputs revised with LaTeX style: inline formulas in
$…$ and display formulas in $$…$$.
Earlier Answer (on existence of torsion in Néron–Severi groups):
Yes. A classic example is an Enriques surface (over an algebraically closed field, char ≠ 2). For an Enriques surface \(X\), the canonical bundle is nontrivial but \[2K_X \sim 0,\] so the class of \(K_X\) is a nonzero 2-torsion element in \(\mathrm{NS}(X)\). In fact the geometric Picard/Néron–Severi group is \[\mathrm{NS}(X)\ \cong\ \mathbb Z^{10}\ \oplus\ \mathbb Z/2\mathbb Z,\] with the \(\mathbb Z/2\mathbb Z\) generated by \(K_X\).
There are also bielliptic (Bagnera–de Franchis) surfaces whose Néron–Severi groups contain torsion; depending on the type one can get \(\mathbb Z/2\mathbb Z\), \((\mathbb Z/2\mathbb Z)^2\), \(\mathbb Z/3\mathbb Z\), etc. (so \(\mathbb Z/2\mathbb Z\) certainly occurs).
For contrast, on abelian varieties the Néron–Severi group is torsion-free.
Detailed Enriques Example:
Below is a detailed explanation of the Enriques example with proper LaTeX style formatting.
1) What an Enriques surface is (and its torsion)
A (classical) Enriques surface \(S\) (over an algebraically closed field of characteristic \(\neq 2\)) is a smooth projective surface with
- \(q(S)=h^1(\mathcal O_S)=0\),
- \(p_g(S)=h^0(K_S)=0\),
- \(2K_S \sim 0\) but \(K_S \not\sim 0\).
Hence \(K_S\) is a nontrivial \(2\)-torsion line bundle. Because \(q=0\) we have \(\mathrm{Pic}^0(S)=0\), so \(\mathrm{NS}(S) \cong \mathrm{Pic}(S)\); the torsion in \(\mathrm{NS}(S)\) is therefore exactly the class \(\langle K_S\rangle \cong \mathbb Z/2\mathbb Z\). Modulo torsion, \[\mathrm{Num}(S) = \mathrm{NS}(S)/\mathrm{Tors} \ \cong\ E_8(-1)\ \oplus\ U,\] (an even unimodular lattice of signature \((1,9)\)).
2) Why \(K_S\) is \(2\)-torsion via the K3 cover
Every Enriques surface \(S\) admits an étale double cover \(\pi: X \to S\) with \(X\) a K3 surface; conversely, the quotient of a K3 surface by a fixed-point-free involution (an “Enriques involution”) is an Enriques surface. For that étale cover, \[\pi^*K_S \cong K_X \cong \mathcal O_X,\] so \(K_S\) pulls back to the trivial bundle — hence \(K_S\) is \(2\)-torsion. It is nontrivial because \(p_g(S) = h^0(K_S) = 0\) (if \(K_S\) were trivial, we would get \(p_g = 1\)).
3) An explicit model you can write down
One hands-on construction is: start with a K3 surface in \(\mathbb P^5\) and divide by a free involution.
- K3 with a free involution. Let \[X := \big{P_1(u,v,w) + Q_1(x,y,z) = 0,\ P_2(u,v,w) + Q_2(x,y,z) = 0,\ P_3(u,v,w)+Q_3(x,y,z)=0\big}\subset \mathbb P^5,\] where \(P_i, Q_i\) are general quadrics in the indicated variables. Define the involution \[ \iota : (u:v:w:x:y:z) \mapsto (-x:-y:-z : u:v:w). \] For a general choice of the quadrics \(P_i, Q_i\), this involution acts freely (i.e. no fixed points). Then the quotient \[ S := X / \langle \iota \rangle \] is an Enriques surface. In particular, \(\pi^*K_S = K_X = \mathcal O_X\), so \(K_S\) is the nontrivial 2-torsion class in \(\mathrm{NS}(S)\).
(There are other explicit constructions too — e.g. using K3 double planes or double covers of \(\mathbb P^1 \times \mathbb P^1\) arranged so that a natural involution is fixed-point-free — but the principle is the same: a free involution on a K3 gives an Enriques quotient with a 2-torsion canonical class.)
4) Reading the Néron–Severi lattice in this example
For any Enriques surface \(S\) (char \(\neq 2\)), \[\mathrm{NS}(S) \cong \big(E_8(-1) \oplus U\big) \oplus \underbrace{\mathbb Z/2\mathbb Z}_{\langle K_S \rangle},\] so the unique torsion is generated by the canonical class. Equivalently, the numerical Néron–Severi lattice \(\mathrm{Num}(S)\) is torsion-free and equals \(E_8(-1)\oplus U\).
Takeaway. The Enriques surface \(S = X / \langle \iota \rangle\), constructed from a K3 surface \(X\) with a fixed-point-free involution \(\iota\), is the standard explicit example of a surface whose Néron–Severi group has a \(\mathbb Z/2\mathbb Z\) torsion factor, namely the canonical class \(K_S\). The rest of \(\mathrm{NS}(S)\) is the even unimodular lattice \(E_8(-1)\oplus U\) of rank 10.
If you like, I can also give you an even more concrete coordinate example or check intersection pairings on \(S\).