Simple Cohen–Macaulay Codimension 2 Singularities

Abstract

In this article, we provide a complete list of simple isolated Cohen–Macaulay codimension 2 singularities together with a list of adjacencies which is complete in the case of fat point and space curve singularities.

Key Words:

• Adjacencies
• Classification of singularities
• Isolated singularities
• Simple singularities

2000 Mathematics Subject Classification:

• Primary 32S05
• Secondary 14B05, 14B07, 32S30

ACKNOWLEDGMENTS

We should like to thank Gerhard Pfister and the whole algebraic geometry group at the University of Kaiserslautern as well as the developers of the computer algebra system Singular [14] for many fruitful discussions. We would also like to thank Jan Stevens for pointing out an omission in an earlier version of this article and for several helpful remarks, and the referee of the article for many helpful remarks and in particular for the simplification of a proof in the case of fat points.

Notes

¹We use the symbol ∼ _C to indicate contact-equivalence.

²If we are considering a singularity X, 0 in the notation of its presentation matrix M, we often also denote by T ¹(M).

³Any matrix whose 1-jet only involves p of the m variables is adjacent to this matrix, and hence this is the set of weights to consider for determining the least number of variables appearing in the 1-jet of a simple singularity of given size n × (n + 1) and number of variables m.

⁴At this point it is important to observe that the roles of y and w may harmlessly be interchanged.

⁵Note that the roles of x and y can harmlessly be exchanged, which explains why we can assume in the second case that the mixed term is ux.

⁶The adjacency lists for the singularities of types G ₅ and G ₇ did not contain any mistakes.

⁷The classification of these singularities can be found in [6].

Communicated by R. Piene.