genus-one fibered surfaces were classified by Kodaira [@Kod1; @Kod2]. Methods to determine the singular fibers of elliptic surfaces can be found in [@Ner; @Tate].

In F-theory compactifications on genus-one fibrations, the non-Abelian gauge groups that form on the 7-branes correspond to the singular fibers of genus-one fibrations [@MV2; @BIKMSV]. The correspondences of the singular fibers and the singularity types of the compactification spaces are shown in Table \[tablemonodromy in 2.1\] below. The corresponding monodromies and j-invariants of the singular fibers are also presented in the table.

  --------------------------------------------------------------------------------------
   Fiber type   J-invariant            Monodromy  Order of Monodromy   Singularity Type
  ------------ ------------- ------------------- -------------------- ------------------
    $I^*_0$       regular      $-\begin{pmatrix}          2                 $D_4$
                                        1 & 0 \\                      
                                        0 & 1 \\                      
                                  \end{pmatrix}$                      

     $I_m$       $\infty$       $\begin{pmatrix}       infinite           $A_{m-1}$
                                        1 & m \\                      
                                        0 & 1 \\                      
                                  \end{pmatrix}$                      

    $I^*_m$      $\infty$      $-\begin{pmatrix}       infinite           $D_{m+4}$
                                        1 & m \\                      
                                        0 & 1 \\                      
                                  \end{pmatrix}$                      

      $II$           0          $\begin{pmatrix}          6                 none.
                                        1 & 1 \\                      
                                       -1 & 0 \\                      
                                  \end{pmatrix}$                      

     $II^*$          0          $\begin{pmatrix}          6                 $E_8$
                                       0 & -1 \\                      
                                        1 & 1 \\                      
                                  \end{pmatrix}$                      

     $III$         1728         $\begin{pmatrix}          4                 $A_1$
                                        0 & 1 \\                      
                                       -1 & 0 \\                      
                                  \end{pmatrix}$                      

    $III^*$        1728         $\begin{pmatrix}          4                 $E_7$
                                       0 & -1 \\                      
                                        1 & 0 \\                      
                                  \end{pmatrix}$                      

      $IV$           