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Every torsion class $D$ on a surface is the difference of two curves. Choose a very ample divisor $A$ so that $D+nA$ is also very ample. Then $(D+nA)-nA=D$ so $D$ is the difference of two curves. They may even be chosen smoothly by Bertini's Theorem, but I'm unsure whether they can be guaranteed to be connected.

Jim's example is good; I think one can construct more examples by using elliptic fibrations. Just start with $$X\times E\rightarrow X$$ and introduce logarithmic transformaions of equal multiplicity $m$ at two fibers $F_1$ and $F_2$. Then $F_1-F_2$ is $m$-torsion. Unfortunately, this construction doesn't preserve algebraicity.

To construct algebraic examples, maybe one can begin with a trivial bundle $$X\times E\rightarrow X.$$ Choose $X$ so that $\textrm{Aut}(X)\supset\mathbb{Z}/p\mathbb{Z}=G$. Maybe one can construct an elliptic fibration by declaring that the monodromy around a ramification point of $X\rightarrow X/G:=Y$ is a translation on $E$ by a $p$-torsion element. Then, we have an elliptic fibration $$S\rightarrow Y$$ with multiple fibers of multiplicity $p$ over the branch locus, whose base-change to $X$ is given by $X\times E$. Since $X\times E$ is an etale cover of $S$, can we can conclude that $S$ is algebraic?

Every divisor class $D$ on a surface is the difference of two smooth, connected curves. Choose a very ample divisor $A$ and an $n$ so that $D+nA$ is also very ample. Then $(D+nA)-nA=D$ so $D$ is the difference of two curves. They may be chosen smoothly by Bertini's Theorem.

ADDED LATER: They may also be chosen to be connected. The Lefschetz hyperplane theorem shows that hyperplane sections of surfaces are connected.

Jim's example is good; I think one can construct examples by using elliptic fibrations. Just start with $$X\times E\rightarrow X$$ and introduce logarithmic transformaions of equal multiplicity $m$ at two fibers $F_1$ and $F_2$. Then $F_1-F_2$ is $m$-torsion. Unfortunately, this construction doesn't preserve algebraicity.

To construct algebraic examples, maybe one can begin with a trivial bundle $$X\times E\rightarrow X.$$ Choose $X$ so that $\textrm{Aut}(X)\supset\mathbb{Z}/p\mathbb{Z}=G$. Maybe one can construct an elliptic fibration by declaring that the monodromy around a ramification point of $X\rightarrow X/G:=Y$ is a translation on $E$ by a $p$-torsion element. Then, we have an elliptic fibration $$S\rightarrow Y$$ with multiple fibers of multiplicity $p$ over the branch locus, whose base-change to $X$ is given by $X\times E$. Since $X\times E$ is an etale cover of $S$, can we can conclude that $S$ is algebraic?

Every torsion class $D$ on a surface is the difference of two curves. Choose a very ample divisor $A$ so that $D+nA$ is also very ample. Then $(D+nA)-nA=D$ so $D$ is the difference of two curves. They may even be chosen smoothly by Bertini's Theorem, but I'm unsure whether they can be guaranteed to be connected.

Jim's example is good; I think one can construct more examples by using elliptic fibrations. Just start with $$X\times E\rightarrow X$$ and introduce logarithmic transformaions of equal multiplicity $m$ at two fibers $F_1$ and $F_2$. Then $F_1-F_2$ is $m$-torsion. Unfortunately, this construction doesn't preserve algebraicity.

To construct algebraic examples, maybe one can begin with a trivial bundle $$X\times E\rightarrow X.$$ Choose $X$ so that $\textrm{Aut}(X)\supset\mathbb{Z}/p\mathbb{Z}=G$. Maybe one can construct an elliptic fibration by declaring that the monodromy around a ramification point of $X\rightarrow X/G:=Y$ is a translation on $E$ by a $p$-torsion element. Then, we have an elliptic fibration $$S\rightarrow Y$$ with multiple fibers of multiplicity $p$ over the branch locus, whose base-change to $X$ is given by $X\times E$. Since $X\times E$ is an etale cover of $S$, can we can conclude that $S$ is algebraic?

Jim's example is good; one can easily construct examples by using elliptic fibrations. Just start with $$X\times E\rightarrow X$$ and introduce logarithmic singularities of equal multiplicity $m$ at two fibers $F_1$ and $F_2$. Then $F_1-F_2$ is $m$-torsion. Unfortunately, this construction doesn't preserve algebraicity.

To construct algebraic examples, maybe one can begin with a trivial bundle $$X\times E\rightarrow X.$$ Choose $X$ so that $\textrm{Aut}(X)\supset\mathbb{Z}/p\mathbb{Z}=G$. Maybe one can construct an elliptic fibration by declaring that the monodromy around a ramification point of $X\rightarrow X/G:=Y$ is a translation on $E$ by a $p$-torsion element. Then, we have an elliptic fibration $$S\rightarrow Y$$ with multiple fibers of multiplicity $p$ over the branch locus, whose base-change to $X$ is given by $X\times E$. Since $X\times E$ is an etale cover of $S$, can we can conclude that $S$ is algebraic?

Jim's example is good; I think one can construct examples by using elliptic fibrations. Just start with $$X\times E\rightarrow X$$ and introduce logarithmic transformaions of equal multiplicity $m$ at two fibers $F_1$ and $F_2$. Then $F_1-F_2$ is $m$-torsion. Unfortunately, this construction doesn't preserve algebraicity.

To construct algebraic examples, maybe one can begin with a trivial bundle $$X\times E\rightarrow X.$$ Choose $X$ so that $\textrm{Aut}(X)\supset\mathbb{Z}/p\mathbb{Z}=G$. Maybe one can construct an elliptic fibration by declaring that the monodromy around a ramification point of $X\rightarrow X/G:=Y$ is a translation on $E$ by a $p$-torsion element. Then, we have an elliptic fibration $$S\rightarrow Y$$ with multiple fibers of multiplicity $p$ over the branch locus, whose base-change to $X$ is given by $X\times E$. Since $X\times E$ is an etale cover of $S$, can we can conclude that $S$ is algebraic?