Beitr\"age zur Algebra und Geometrie
Contributions to Algebra and Geometry
Volume 37 (1996), No. 1, 149-159.
The Divisor of the Resultant
G\"unter Scheja and Uwe Storch
Abstract.
Let $A$ be an integrally closed noetherian domain, $A[T_{0}, \ldots , T_{n}]$
a graded polynomial algebra with indeterminates having arbitrary positive
(integral) weights and $F_{0}, \ldots , F_{n}\in A[T]:=A[T_{0}, \ldots , T_{n}]$ homogeneous polynomials of
positive degrees, which form a regular sequence. The properties of
the subvariety ${\rm Proj}\, C$, $C:=A[T]/(F_{0},\ldots , F_{n})$, of the
weighted projective space ${\rm Proj}\, A[T]$ and its canonical image in
${\rm Spec}\, A$ are best described in terms of the {\it resultant ideal} ${\frac R}$
which is a principal ideal and can be defined either by a suitable Fitting
ideal or a cancelling rule for determinants.
By means of duality available for regular sequences the following is proved:
The divisor of ${\frac R}$ equals the divisor defined by the algebra of global sections of
${\rm Proj}\, C$ (which is an $A$--torsion--module). Some applications on
the degree of elimination and on the fibres of the elimination mapping are
given.