Let \(C\) be a real algebraic curve of genus 2 with at least two real components \(B_1\) and \(B_2\). An embedding of \(C\) into the projective plane blown-up in a point allows an explicit description of the neutral real component \(\text{Pic}^0 (C)^0\) of the Jacobian of \(C\). The author uses an isomorphism \(\text{Pic}^0 (C)^0 \simeq B_1\times B_2\) which is a particular case of an isomorphism found by J. Huisman.
In particular the group law on \(B_1\times B_2\) is given by intersecting with conics when \(C\) is mapped as a quartic curve into \(\mathbb{P}^2\), and finally the author describes the 2- and 3-torsion points on \(B_1\times B_2\).