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Monoid Hypersurfaces

P˚ al Hermunn Johansen, Magnus Løberg, and Ragni Piene

Centre of Mathematics for Applications and Department of Mathematics
University of Oslo
P. O. Box 1053 Blindern
NO-0316 Oslo, Norway
{hermunn,mags,ragnip}@math.uio.no

Summary. A monoid hypersurface is an irreducible hypersurface of degree d which has a
singular point of multiplicity d − 1. Any monoid hypersurface admits a rational parameteriza-
tion , hence is of potential interest in computer aided geometric design . We study properties
of monoids in general and of monoid surfaces in particular. The main results include a de-
scription of the possible real forms of the singularities on a monoid surface other than the
(d − 1)-uple point. These results are applied to the classification of singularities on quartic
monoid surfaces , complementing earlier work on the subject.

4.1 Introduction

A monoid hypersurface is an (affine or projective) irreducible algebraic hypersurface
which has a singularity of multiplicity one less than the degree of the hypersurface.
The presence of such a singular point forces the hypersurface to be rational: there
is a rational parameterization given by (the inverse of) the linear projection of the
hypersurface from the singular point.
The existence of an explicit rational parameterization makes such hypersurfaces
potentially interesting objects in computer aided design. Moreover, since the “space”
of monoids of a given degree is much smaller than the space of all hypersurfaces
of that degree, one can hope to use monoids efficiently in (approximate or exact)
implicitization problems. These were the reasons for considering monoids in the
paper [17]. In [12] monoid curves are used to approximate other curves that are close
to a monoid curve, and in [13] the same is done for monoid surfaces. In both articles
the error of such approximations are analyzed – for each approximation, a bound on
the distance from the monoid to the original curve or surface can be computed.
In this article we shall study properties of monoid hypersurfaces and the classi-
fication of monoid surfaces with respect to their singularities. Section 4.2 explores
properties of monoid hypersurfaces in arbritrary dimension and over an arbitrary base
field. Section 4.3 contains results on monoid surfaces, both over arbritrary fields and
over . The last section deals with the classification of monoid surfaces of degree
four. Real and complex quartic monoid surfaces were first studied by Rohn [15], who