Über den \(n\)-dimensionalen Residuensatz. (On the \(n\)-dimensional residuum theorem). (German) Zbl 0789.14003

The author gives a survey on geometric applications of the residue theorem in the \(n\)-dimensional affine or projective space. Let \(f:=(f_ 1,\dots,f_ n)\) be a sequence of polynomials in the polynomial ring \(K[X_ 1,\dots,X_ n]\) over an algebraically closed field \(K\) such that the homogeneous components form a regular sequence \(Gf=(Gf_ 1,\dots,Gf_ n)\). Then \((f_ 1,\dots,f_ n)\) is also a regular sequence, and the algebraic set \({\mathcal V} (f_ 1, \dots, f_ n) \subset \mathbb{A}^ n\) consists of finitely many closed points \(P_ i\). For any differential form \(\omega=hd X_ 1 \dots dX_ n\) and each point \(P \in {\mathcal V} (f_ 1,\dots,f_ n)\) there is the Grothendieck residue \(\text{Res}_ P [{\omega \over f}]\), for which the author cites several sources in the literature, especially the elementary construction of Scheja and Storch. (In the classical case \(K=\mathbb{C}\) they coincide with the analytically defined residues.) One defines \(G \omega:=Gh \cdot dX_ 1 \dots dX_ n\) and \(\det \omega:=\deg h+ \sum^ n_{i= 1} \deg X_ i\). Then the residue theorem states that for \(\deg \omega \leq \sum^ n_{i=1} \deg X_ i\) one has \[ \sum_{P \in {\mathcal V} (f)} \text{Res}_ P \left[ {\omega \over f}\right]=\text{Res}_ O \left[ {G \omega \over Gf} \right]. \] The right hand side of this equation vanishes for \(\deg \omega< \sum^ n_{i=1} \deg X_ i\). If one takes the standard graduation \(\deg X_ i=1\) for all \(i\) then the left hand side of the residue theorem depends only on the points at infinity of the hypersurfaces \(h=0\) und \(f_ i=0\), \(i=1,\dots,n\).
By choosing suitable differential forms \(\omega\) which allow a geometric interpretation one can get many classical results in intersection theory of hypersurfaces in \(\mathbb{A}^ n\) and generalizations of them as corollaries of the residue theorem. The author gives several impressing examples. Among others he proves a theorem of Newton, a formula of Reiss (1837), a formula of Jacobi (1835), a theorem of Humbert (1885) and even the well known theorems of Pappus and Pascal. He also gives generalizations of some of these theorems.


14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14A05 Relevant commutative algebra
13N05 Modules of differentials