Trac #34002: Sirocco and projective curves

Let `C`be a proyective plane curve with equation `F(x,y,z)=0` of degree
`d`. The method `fundamental_group` computes the fundamental group of
the complement. When the point `[0:1:0]` belongs to the curve it may
produce wrong results.

The computation works as follows. Consider the affine curve
`f(x,y):=F(x,y,1)=0` and compute its fundamental group via the
projection `(x,y)->x`. In this group we have an ordered system of
generators `t1,...,td` and to pass from the fundamental group of the
complement of the affine curve to the fundamental group of the
complement of the projective curve it is enough to add as relation the
ordered product of the generators.

In order to apply it directly we need `[0:1:0]` outside the curve which
is equivalent to have `f` to be of degree `d` in the variable `y` but up
to now the method only checks if the maximal coefficient of `y` is a
non-zero constant.

The following example shows the problem. It computes the fundamental
group of a quintic which is known to be finite and non-abelian by
Degtyarev.

{{{
sage: P.<x,y,z>=ProjectiveSpace(QQ,2)
sage: f=z^2*y^3-z*(33*x*z+2*x^2+8*z^2)*y^2+(21*z^2+21*x*z-x^2)*(z^2+11*x
*z-x^2)*y+(x-18*z)*(z^2+11*x*z-x^2)^2
sage: C=Curve(f)
sage: C.fundamental_group()
Finitely presented group < x0 | x0^3 >
}}}

If we make a change of coordinates to avoid this problem, everything is
OK
{{{
sage: f1=f(z=z+y)
sage: g1p=Curve(f1).fundamental_group()
Singularity detected. Abort Sirocco
sage: g1p.order()
320
}}}

URL: https://trac.sagemath.org/34002
Reported by: enriqueartal
Ticket author(s): Miguel Marco
Reviewer(s): Enrique Artal

Author: Release Manager

src/sage/schemes/curves/projective_curve.py

@@ -1691,6 +1691,15 @@ def fundamental_group(self):
         .. WARNING::
 
             This functionality requires the ``sirocco`` package to be installed.
+
+        TESTS::
+
+            sage: P.<x,y,z>=ProjectiveSpace(QQ,2)
+            sage: f=z^2*y^3-z*(33*x*z+2*x^2+8*z^2)*y^2+(21*z^2+21*x*z-x^2)*(z^2+11*x*z-x^2)*y+(x-18*z)*(z^2+11*x*z-x^2)^2
+            sage: C = P.curve(f)
+            sage: C.fundamental_group() # optional - sirocco
+            Finitely presented group < x1, x3 | (x3^-1*x1^-1*x3*x1^-1)^2*x3^-1, x3*(x1^-1*x3^-1)^2*x1^-1*(x3*x1)^2 >
+
         """
         from sage.schemes.curves.zariski_vankampen import fundamental_group
         F = self.base_ring()

src/sage/schemes/curves/zariski_vankampen.py

@@ -1018,7 +1018,12 @@ def fundamental_group(f, simplified=True, projective=False):
         sage: fundamental_group(f) # optional - sirocco
         Finitely presented group < x0 |  >
     """
-    bm = braid_monodromy(f)
+    g = f
+    if projective:
+        x,y = g.parent().gens()
+        while g.degree(y) < g.degree():
+            g = g.subs({x : x + y})
+    bm = braid_monodromy(g)
     n = bm[0].parent().strands()
     F = FreeGroup(n)