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4 Monoid Hypersurfaces 65
7. A double line and another line, f 3 = x 2 x 2
3
8. A triple line f 3 = x 3
3
9. A smooth curve, f 3 = x + x + x +3ax 0 x 1 x 3 where a = −1
3
3
3
3
1 2 3
To each quartic monoid we can associate, in addition to the type, several integer
invariants, all given as intersection numbers. From [19] we know that, for the types
1–3, 5, and 9, these invariants will determine the singularity type of O up to right
equivalence. In the other cases the singularity series, as defined by Arnol’d in [1] and
[2], is determined by the type of f 3 . We shall use, without proof, the results on the
singularity type of O due to [19]; however, we shall use the notations of [1] and [2].
We complete the classification begun in [19] by supplying a complete list of
the possible singularities occurring on a quartic monoid. In addition, we extend the
results to the case of real monoids. Our results are summarized in the following
theorem.

Theorem 9. On a quartic monoid surface, singularities other than the monoid point
can occur as given in Table 4.1. Moreover, all possibilities are realizable on real
quartic monoids with a real monoid point, and with the other singularities being real
−
and of type A .

Proof. The invariants listed in the “Invariants and constraints” column are all non-
negative integers, and any set of integer values satisfying the equations represents
one possible set of invariants, as described above. Then, for each set of invariants,

(positive) intersection multiplicities, denoted m i , m and m , will determine the
i
i
singularities other than O. The column “Other singularities” give these and the equa-
tions they must satisfy. Here we use the notation A 0 for a line L a on Z(F) where O
is the only singular point.
The analyses of the nine cases share many similarities, and we have chosen not
to go into great detail when one aspect of a case differs little from the previous one.
We end the section with a discussion on the possible real forms of the tangent cone
and how this affects the classification of the real quartic monoids.
In all cases, we shall write

4 3 3 2 2 2
f 4 = a 1 x + a 2 x x 2 + a 3 x x 3 + a 4 x x + a 5 x x 2 x 3
1 1 1 1 2 1
+ a 6 x x + a 7 x 1 x + a 8 x 1 x x 3 + a 9 x 1 x 2 x + a 10 x 1 x 3
3
2
2
2 2
1 3 2 2 3 3
+ a 11 x + a 12 x x 3 + a 13 x x + a 14 x 2 x + a 15 x 4
4
3
2 2
3
2 2 2 3 3 3
and we shall investigate how the coefficients a 1 ,...,a 15 are related to the geometry
of the monoid.
Case 1. The tangent cone is a nodal irreducible curve, and we can assume
f 3 (x 1 ,x 2 ,x 3 )= x 1 x 2 x 3 + x + x .
3
3
2 3
The nodal curve is singular at (1 :0:0).If f 4 (1, 0, 0) =0, then O is a T 3,3,4
singularity [19]. We recall that (1 :0:0) cannot be a singular point on Z(f 4 ) as