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Let $C$ be a complex projective curve with atmost nodes as singularities and let $v:C'\rightarrow C$ be its normalization. Can a non-trivial line bundle $L$ on $C$ pullback to a trivial line bundle on $C'$? In particular, consider the pull back $Pic\, C\rightarrow Pic\, C'$. Is the kernel of this morphism nontrivial? If so, is there a description of line bundles that pull back to the trivial bundle?

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In general, the answer is yes. Let's assume that $C$ is geometrically irreducible curve over any field $k$.

Note that we have a short exact sequence of sheaves

$$1\to \mathcal{O}_C^\times\to v_\ast \mathcal{O}_{C'}^\times\to (v_\ast\mathcal{O}_{C'}^\times)/\mathcal{O}_C^\times\to 1$$

Taking the long exact sequence in cohomology gives

$$1\to k^\times\to k^\times\to ((v_\ast \mathcal{O}_{C'}^\times)/\mathcal{O}_C^\times)(C)\to \mathrm{Pic}(C)\to \mathrm{Pic}(C')\to 1$$

where we note that $(v_\ast \mathcal{O}_{C'}^\times)/\mathcal{O}_C^\times$ is finitely supported which is why its cohomology is zero. In particular, if $C$ has $n$ double points then it's not hard to see that

$$((v_\ast \mathcal{O}_{C'}^\times)/\mathcal{O}_C^\times)(C)=\bigoplus_{x_i\text{ node}}((v_\ast \mathcal{O}_{C'}^\times)/\mathcal{O}_C^\times)_{x_i}=(k^\times)^n$$

essentially because locally around a node $x_i$, with preimage points $p_1$ and $p_2$, we have that the curve has ring of functions $\{f(t)\in k[t]:f(p_1)=f(p_2)\}$ and locally around each of the two points $p_1$ and $p_2$ in $v^{-1}(x_i)$ we have that the functions just look $k[t]$ so the isomorphism $((v_\ast \mathcal{O}_{C'}^\times)/\mathcal{O}_C^\times)_{x_i}\to k^\times$ is something like $f\mapsto f(p_1)f(p_2)^{-1}$.

All in all we see that we have a short exact sequence

$$1\to (k^\times)^n\to\mathrm{Pic}(C)\to\mathrm{Pic}(C')\to 1$$

so that $\mathrm{Pic}(C)\to\mathrm{Pic}(C')$ is essentially never injective.

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  • $\begingroup$ Thank you! Is there any characterizations of line bundles in the kernel? $\endgroup$ – user52991 Apr 22 at 12:22

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