  This paper aims to study the decomposition group of a nonsingular plane cubic
under the light of the log Calabi-Yau geometry. Using this approach we prove
that an appropriate algorithm of the Sarkisov Program in dimension 2 applied to
an element of this group is automatically volume preserving. From this, we
deduce some properties of the (volume preserving) Sarkisov factorization of its
elements. We also negatively answer a question posed by Blanc, Pan and Vust
asking whether the canonical complex of a nonsingular plane cubic is split.
Within a similar context em dimension 3, we exhibit in detail an interesting
counterexample for a possible generalization of a theorem by Pan in which there
exists a Sarkisov factorization obtained algorithmically that is not volume
preserving.
