Let \(C\) be a singular curve with normalization \(\nu : \tilde{C} \to C\). Comparing units gives the exact sequence \[ 0 \to \OO_C^* \to \nu_* \OO_{\tilde{C}}^* \to \mcs \to 0 \] with \(\mcs\) a skyscraper at the singular points, and its cohomology sequence computes \(\Pic(C)\) from \(\Pic(\tilde{C})\) and the local unit groups.
For the cusp \(A = k[t^2,t^3] \subseteq k[t]\), the conductor square gives \(\Pic(\Spec A) \cong k^+ = \GG_a\). For the node \(k[x,y]/(y^2 - x^2(x+1))\), the same computation gives \(\GG_m\).
At a node \(p\) with \(\nu^{-1}(p)=\ts{p_1,p_2}\), the stalk \(\mcs_p\) is \((k^\times\times k^\times)/k^\times\cong k^\times\), the ratio of the values of a unit of \(\OO_{\tilde C}\) at \(p_1\) and \(p_2\); so the node contributes \(\GG_m\). At the cusp of \(\Spec k[t^2,t^3]\), with \(\nu^{-1}(p)=\ts{\tilde p}\) and \(\OO_{C,p}\) the functions whose derivative in \(t\) vanishes at \(\tilde p\), the stalk \(\mcs_p\) is \((k[t]/(t^2))^\times/k^\times\cong k^+\), via \(a+bt\mapsto b/a\); so the cusp contributes \(\GG_a\).
For the projective cubics, \(\Pic^0\) of the nodal cubic is \(\GG_m\) and \(\Pic^0\) of the cuspidal cubic is \(\GG_a\); in each case the smooth locus, with the chord-tangent group law, is isomorphic to \(\Pic^0\).
Dependencies
D-5PQ5WWeil and Cartier divisors, and when they agreeuses