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Associate forms, joins, multiplicities and an intrinsic elimination theory. (English) Zbl 0762.13006
Topics in algebra. Part 2: Commutative rings and algebraic groups, Pap. 31st Semester Class. Algebraic Struct., Warsaw/Poland 1988, Banach Cent. Publ. 26, Part 2, 71-108 (1990).
[For the entire collection see Zbl 0716.00007.]
The first time a mathematician hears about “multiplicity” \(m_ r\in\mathbb{Z}^ +\), it refers to an \(m_ r\)-ple root of a polynomial \(f(x)\) or binary form \(\varphi(x_ 0,x_ 1)\): \(f(x)=a_ 0\prod_{r\in\mathbb{P}_ 1(\mathbb{C})}(x-r)^{m_ r}\), \(\varphi(x_ 0,x_ 1)=\prod_{r\in\mathbb{P}_ 1(\mathbb{C})}\left|{x_ 0\atop r_ 0} {x_ 1\atop r_ 1}\right|^{m_ r}\), \(m_ r\in\mathbb{Z}\), \(m_ r\geq 0\). It is natural to ask whether or not this exponent is also the natural intersection multiplicity of an irreducible component \(I\) in the proper intersection \(V\cap W\) \((V,W\) irreducible affine variety in \(\mathbb{P}_ n(\mathbb{C})=\mathbb{P}(E))\). An affirmative answer is found in the book by B. L. van der Waerden: “Einführung in die algebraische Geometrie” (2nd edition 1973; Zbl 0264.14001) only for two irreducible plane curves. This idea of the exponent intersection multiplicity is developed in this paper in the general case by showing that the form \(F_{V\cdot W}=\prod F_ I^{m_ I}\) (I proper irreducible component of \(V\cap W)\) can be computed by restriction of the \(F_ J\) (associated to the join \(J(V\times W)\subset\mathbb{P}(E\oplus E))\) to the diagonal subspace \(\Delta\subset\mathbb{P}(E\oplus E)\). The method extends naturally to \(h(\geq 2)\) affine varieties \(V^{(j)}\subset\mathbb{P}_ n\), \(j=1,\dots,h\) provided \(c=\sum c_ j\leq n\) \((c_ j=\text{codim} V^{(j)}\) in \(\mathbb{P}_ n)\). The geometric interpretation of \(F_ V\) in terms of the complex \({\mathfrak C}(V^ c)=\{\mathbb{P}_{c-1}\subset\mathbb{P}_ n|\mathbb{P}_{c-1}\cap V^ c\neq\emptyset\}\) leads naturally to an equivalence of the exponent multiplicity with van der Waerden’s theory.
Since \(c=\text{codim} J\) in \(\mathbb{P}(E\oplus\cdots\oplus E)\) \((h\) copies of \(E)\) a natural discussion arises also in the case \(c>n\). Then the old elimination theory can be replaced by intrinsic constructions. Natural applications are made to Bézout’s theorem as well to possible future relations with the “length multiplicity”.

MSC:
13H15 Multiplicity theory and related topics
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
13B25 Polynomials over commutative rings
14A05 Relevant commutative algebra
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