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This chapter describes several packages of functionality developed for working
with reduced (hyper)surfaces in three-dimensional projective space P3.
At the core is a package to compute a formal desingularization of such a
hypersurface X, expressed via a collection of algebraic power series giving the
formal completion of the components of some desingularization lying
over the components of the singular subscheme of the hypersurface. This allows
the computation of important birational invariants of any desingularization
of X like the arithmetic and geometric genera and higher geometric plurigenera.
The algorithm is based on the method of Jung and was designed and implemented
by Tobias Beck. It is fully described in [Bec07].
An important application of the desingularization data is the computation of
m-adjoint maps as rational maps on X. A function is provided
for this. Theoretical and algorithmic details may be found in [BS08].
There are functions to determine whether X is of Kodaira dimension -∞,
ie birationally ruled. For the important special case of rational
surfaces, there is a suite of functions to determine whether a parameterization
exists over the base field and to explicitly construct one in the affirmative case.
This is based on the work of Josef Schicho described in [Sch98] and
[Sch00].
X is mapped to a standard model by applying an appropriate m-adjoint map.
These are then parameterized by special case code. The main special cases are Del
Pezzo surfaces (including some singular cases) and line and conic bundles. The
functions for these are made available and can be called separately by the user.
Apart from the previously existing Del Pezzo code (for degrees 6, 8 and 9), these
functions were implemented by Tobias Beck again and Josef Schicho.
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