Normal factorization in $SL(2,\mathbb{Z})$ and the confluence of singular fibers in elliptic fibrations

Carlos A. Cadavid and Juan D. Vélez

Abstract: In this article we obtain a result about the uniqueness of factorization in terms of conjugates of the matrix $U=\left[\begin{array}{cc} 1 & 1
0 & 1\end{array}\right]$, of some matrices representing the conjugacy classes of those elements of $SL(2,\mathbb{Z})$ arising as the monodromy around a singular fiber in an elliptic fibration (i.e. those matrices that appear in Kodaira's list). Namely we prove that if $M$ is a matrix in Kodaira's list, and $M=G_1\cdots G_r$ where each $G_i$ is a conjugate of $U$ in $SL(2,\mathbb{Z})$, then after applying a finite sequence of Hurwitz moves the product $G_1\cdots G_r$ can be transformed into another product of the form $H_1\cdots H_nG_{n+1}'\cdots G_r'$ where $H_1\cdots H_n$ is some fixed shortest factorization of $M$ in terms of conjugates of $U$, and $G_{n+1}'\cdots G_r'=Id_{2\times 2}$. We use this result to obtain necessary and sufficient conditions under which a relatively minimal elliptic fibration without multiple fibers $\phi:S\rightarrow D=\{z \in \mathbb{C}:\left|z\right|<1 \}$, admits a weak deformation into another such fibration having only one singular fiber.

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