We consider an extremal contraction from a non-singular projective 3-folds onto a smooth curve. This is so-called a del Pezzo fibration. The degree of a del Pezzo fibration is defined to be the anti-canonical degree of general fibers and Mori proved that the degree can not be seven. To study del Pezzo fibrations, it is convenient to construct embeddings over the base into more simple fibration. Mori and Fujita proved that every del Pezzo fibration is a relative weighted complete intersection of weighted projective bundle if the degree is not five or six. If the degree is five, Takeuchi asserted that it is relatively defined by Pfaffian.

In this talk, we will mainly discuss on del Pezzo fibrations of degree six. The main result of this talk establish embeddings of those into $\mathbb{P}^2 \times \mathbb{P}^2$-fibrations and $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$-fibrations as relative linear sections. We will construct such $\mathbb{P}^2 \times \mathbb{P}^2$-fibrations and $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$-fibrations as extremal contractions and classify their singular fibers. As the application of those embeddings, we will completely classify the singular fibers of sextic del Pezzo fibrations.