Theorem 1.11 [014B]

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Theorem 1.11

Let $f:X\rightarrow B$ be a well-behaved $T^{3}$ -fibration, $b_{0}\in\Delta_{d}$ , with $X_{b_{0}}$ a semi-stable fibre, $U,T_{1},\ldots,T_{n}$ as above. Then

(1) If $X_{b_{0}}$ is of type $(2,2)$ , then $n=2$ and in a suitable basis, $T_{1}=\pmatrix{1&0&1\cr 0&1&0\cr 0&0&1\cr}$ and $T_{2}=T_{1}^{-1}$ .

(2) If $X_{b_{0}}$ is of type $(1,2)$ or $(2,1)$ , then $n=3$ and in a suitable basis, $T_{1}=\pmatrix{1&0&1\cr 0&1&0\cr 0&0&1\cr}$ , $T_{2}=\pmatrix{1&0&0\cr 0&1&1\cr 0&0&1\cr}$ and $T_{3}=\pmatrix{1&0&-1\cr 0&1&-1\cr 0&0&1\cr}$ in the $(1,2)$ case, and the transpose of these matrices in the $(2,1)$ case.

(3) If $X_{b_{0}}$ is a fibre of type $(1,1)$ , then $n=4$ . Furthermore, in a suitable basis, possibly after relabelling the $p_{i}$ ’s, $T_{1}=\pmatrix{1&1&0\cr 0&1&0\cr 0&0&1\cr}$ , $T_{2}=\pmatrix{1&0&0\cr 0&1&1\cr 0&0&1\cr}$ , $T_{3}=\pmatrix{1&-1&a\cr 0&1&0\cr 0&0&1\cr}$ , and $T_{4}=\pmatrix{1&0&-a-1\cr 0&1&-1\cr 0&0&1\cr}$ .