|
Existence of Singular Hypersurfaces - An
Example
We compute the spectrum of T3,3,3 by considering the representative x3+ y3+ z3.
The computation can be done over the rationals and has to be performed in a local ring.
Hence, we have to choose a local ordering on Q[x,y,z], e.g. the negative degree lexicographical ordering
ds:
LIB "gaussman.lib";
ring R=0,(x,y,z),ds;
poly f=x^3+y^3+z^3;
list s1=spectrum(f);
s1;
|
==>
|
[1]:
0, 1/3, 2/3, 1 // spectral numbers
[2]:
1, 3, 3, 1 // multiplicities
|
Now, we compute the spectrum of x7+ y7+ z7.
(Any hypersurface of degree 7 in P3 is a small deformation of this equation.)
poly g = x^7+y^7+z^7;
list s2 = spectrum(g);
s2;
|
==>
|
[1]:
-4/7, -3/7, -2/7, -1/7, 0, 1/7, 2/7, 3/7,
4/7, 5/7, 6/7, 1, 8/7, 9/7, 10/7, 11/7
[2]:
1, 3, 6, 10, 15, 21, 25, 27,
27, 25, 21, 15, 10, 6, 3, 1
|
Evaluating semi-continuity is very easy:
spsemicont(s2,list(s1));
This tells us that there are at most 18 singularities of type
T3,3,3.
Since x7+y7+z7
is semi-quasihomogeneous, we can apply the stronger
form of semi-continuity:
spsemicont(s2,list(s1),1);
So, a septic has at most 17 triple points of type T3,3,3.
Theoretical Background
|