"On algorithmic desingularization of hypersurfaces"
Gabor Bodnar and Josef Schicho.
Research Institute for Symbolic Computation Linz
RISC-Linz, Joh. Kepler. Univ., A-4040 Linz
Linz, Austria, A-4040
Austria
E-mail:{Bodnar,Schicho}@risc.uni-linz.ac.at
The need to desingularize algebraic varieties has arised from the
desire to extend the rich results from the nonsingular theory
to the general case. For curves, the problem is easy, and the
common answer is normalization (see \cite{Trager:76}).
But for higher dimensional varieties, the problem is much more difficult.
For instance, algorithmic desingularization of surfaces
was posed as an open problem in \cite{Eisenbud:93}.
Theoretically, the problem was solved in the famous paper
\cite{Hironaka:64}, which proved existence of desingularizations
for varieties in characteristic zero. Other proofs have been
suggested by
\cite{Jung:08,Walker:35,Zariski:39,Abhyankar:69,Hironaka:84,Lipman:78}
in the surface case and by
\cite{Villamayor:89,Bierstone_Milman:91} for
arbitrary dimensions.
Unfortunately, almost all approaches to the desingularization
problem are not algorithmic. But a desingularization algorithm
is essential for effective application of all the results mentioned
above. For instance, \cite{Schicho:97} uses surface desingularization
for the parameterization of rational surfaces.
In the same paper, desingularization is used to compute ``adjoints'',
a powerful tool in computational algebraic geometry
(see also \cite{Blass_Lipman:79}). Other applications would be possible in the
theory of differential equations (see \cite{Cano_Cerveau:92}).
>From an algorithmic point of view, the breakthrough was the
paper~\cite{Villamayor:89}. It describes a desingularization
of hypersurfaces by blowing up similar to Hironaka's desingularization
(still characteristic zero).
But unlike Hironaka, Villamayor gives a way to determine the
blowing up centers explicitly. Hence the paper contains a
desingularization algorithm, at least implicitly (see
also \cite{Villamayor:91}).
Up to now, Villamayor's algorithm was not implemented. The reasons
are two-fold. First, the common way of representing blowing ups embeds
the blowup variety in some higher dimensional space (see \cite{Hartshorne:77})
.
This is not convenient for performing the differential operations
that are necessary in the algorithm. Second, the algorithm itself
is rather complicated, even without taking the problem of representing
blowing up into account.
In this paper we present an implementation of Villamayor's algorithm,
and a solution to the technical problem of representing blowing ups.
The idea is to work simultanously with two coordinate frames.
The affine coordinates generate the coordinate ring,
and they are used for all kinds of algebraic computations,
based on the method of Gr\"obner bases
(see \cite{Buchberger:65,Buchberger:85}).
The differentiable frame is a set of regular functions
that form a regular system of parameters for each point, modulo translation.
It is used for differentiating and for defining all objects in the
algorithm that are supposed to be nonsingular (i.e. the blowing up
centers).