# Kernel of pullback map of line bundles

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?

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.

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