On the canonical volume of 3-folds of general type with $P_{12}\ge 2$

Let $V$ be a nonsingular projective 3-fold of general type. When the pluricanonical section index $\delta (V)>12$, Chen-Chen \cite{Chen3} has a complete list of the possibility for the weighted basket $\mathbb{B}(V)$. However the possibility of $\mathbb{B}(V)$ could be infinite in the situation $\delta (V)\le 12$, which is the main challenge to the classification. In this paper we mainly study the case with $\delta (V)=12$ and show that $Vol(V)\geq \frac{31}{48048}$, which improves the corresponding result of Chen–Chen \cite[Prop. 4.9(5)]{Chen3}.