#%Format: latex
#% This file is marked for TeX processing
#[caption]"Computing $\pi_1$ for sextics with a type $\bE_7$ point"
#================================================================
#: The computation of the fundamental groups of plane sextics with a%
#: type $\bE_7$ singularity.
#: We use the type~$\tA$ regions $\reg_-(e\ra\zeta)$, where%
#: $\zeta=\x$, $\x^2$, $\x\y\x^2$, and%
#:   either $\x(\y\x^2)^2$, $(\y\x^2)^2$ or $(\x\y)^2\x^2$.%
#: Their widths (in the order listed) are the parameters.
#----------------------------------------------------------------
Reread("pi1sc.txt");   #: (see \file{pi1sc})
#: Relations at infinity
fixed := [G.3*rho^2/G.1, G.1*G.2/(G.2*G.1), G.1^2*G.2*G.3/(G.3*G.1^2*G.2)];
#: Regions and the central element to be used
list := [x, x^2, x*y*x^2, x*(y*x^2)^2, (y*x^2)^2];  kappa := G.1^2*G.2;
#: For irreducible curves%
#: (see \kwd{AA} and \kwd{DS} in \file{pi1sc})
size := function(arg) return Size(DS(AA(list, arg))); end;
#: For the splittings $D_5+D_1$ and $D_3+D_3$,%
#: consider the quotient $\pi_1/\slope$, $\slope:=\Ga_1^2\Ga_2$
size2 := function(arg) return Size(DS(AA(list, arg), [kappa])); end;
#: For the splitting $D_4+D_2$,%
#: consider the quotient $\pi_1/(\Ga_1\Ga_3)^s$,%
#: where $s=\width\reg_-(e\ra\x^2)$
#: Also change the last chain to $(\x\y)^2\x^2$
list3 := [x, x^2, x*y*x^2, (x*y)^2*x^2];
AAc := function(w) return Union(AA(list3, w), [(G.1*G.3)^w[2]]); end;
group3 := function(arg) return DS(AAc(arg)); end;
size3 := function(arg) return Size(CallFuncList(group3, arg)); end;
#: Finally, for $D_3+D_2+D_1$, use both central elements%
#: $\slope$ and $(\Ga_1\Ga_3)^s$
group4 := function(arg) return DS(AAc(arg), [kappa]); end;
size4 := function(arg) return Size(CallFuncList(group4, arg)); end;
#================================================================
#[endinput]
